The paragraph after Equation (51) looks out of place, maybe it shou...
Self-testing of quantum systems: a review
Ivan Šupić
1
and Joseph Bowles
2
1
Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
2
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona),
Spain
September 21, 2020
Self-testing is a method to infer the underlying physics of a quantum experiment
in a black box scenario. As such it represents the strongest form of certification for
quantum systems. In recent years a considerable self-testing literature has developed,
leading to progress in related device-independent quantum information protocols and
deepening our understanding of quantum correlations. In this work we give a thorough
and self-contained introduction and review of self-testing and its application to other
areas of quantum information.
1 Introduction
In contrast to classical theories, states in quan-
tum physics can be entangled and sets of mea-
surements can be incompatible. As shown by
Bell in 1964 [Bel64], these features imply striking
observable phenomena. In particular, the out-
comes of incompatible measurements made on
the local subsystems of an entangled quantum
state can exhibit correlations that are provably
stronger than any resulting from a classical the-
ory, a phenomenon known as Bell nonlocality.
The field of Bell nonlocality has since grown con-
siderably (see [BCP
+
14] for a recent review ar-
ticle), and the existence of Bell nonlocal corre-
lations in nature is now a well established fact
[HBD
+
15, GVW
+
15, SMSC
+
15].
As more was understood about Bell nonlocal-
ity, a number of works [SW87, PR92, BMR92,
Tsi93] eventually pointed out that there exist Bell
nonlocal correlations that—as well as requiring
entanglement and incompatibility—can only be
produced by making particular sets of incompati-
ble measurements on particular entangled states.
These works have since given birth to the field
of self-testing, which broadly speaking aims to
understand the structure of the set of quantum
correlations and identify those correlations that
admit a unique physical realisation.
An important milestone in the development of
self-testing was the 2004 work of Mayers and
Yao [MY04]. This work set the terminology
and formalism that was to be adopted by later
works, and includes the first usage of the term
‘self-testing’ in this context. A similar idea was
already present in [MY98] in a cryptographic
context, using the term ‘self-checking’ instead
of ‘self-testing’. These early works also intro-
duced the paradigm of device-independence, to
which self-testing is intimately related. In par-
ticular, a self-testing protocol can be seen as a
device-independent—or black box—certification
of a quantum system, assuming that the sys-
tem can be prepared many times in an inde-
pendent, identically distributed manner. Self-
testing is consequently relevant to many device-
independent quantum information protocols and
has led to related progress in this area. More re-
cently, self-testing has become synonymous with
any protocol for certifying any type of quantum
system under a small set of assumptions.
In this work we give a up-to-date review of the
field of self-testing. We hope that it will be of
use to people both unfamiliar with the field, as
well as serving as a reference for those within
it. The review is organised as follows. In sec-
tion 2 we give a gentle introduction to device-
independence and its connection to self-testing.
We then formally introduce self-testing, giving
the mathematical definitions in section 3 and a
simple example in section 4 that illustrates many
important concepts. Sections 5 to 8 are a thor-
ough literature review of state and measurement
self-testing, explaining the tools and techniques
that are commonly used along the way. In sec-
tion 9 we review extensions of self-testing to other
Accepted in Quantum 2020-23-08, click title to verify 1
arXiv:1904.10042v4 [quant-ph] 18 Sep 2020
scenarios, and in section 10 the application of
self-testing to other fields in quantum informa-
tion theory. In section 11 we cover experimental
realisations of self-testing protocols. Finally, in
section 12 we discuss some possible future direc-
tions for the field and a number of open problems.
We point the reader to the related review ar-
ticles [MdW16] and [Sca12] where discussions
about self-testing can also be found. [MdW16]
deals with the classical and quantum certifi-
cation of both classical and quantum proper-
ties of an object, with self-testing being identi-
fied as classical certification of quantum prop-
erties. [Sca12] provides a pedagogical review
of the device-independent approach to quan-
tum physics. Self-testing is discussed as one of
device-independent protocols. We also recom-
mend [McK10, Kan17, Kan16] as valuable texts
for first time readers.
Contents
1 Introduction 1
2 Self-testing as a device-independent
protocol 3
3 Definitions 5
3.1 Notation . . . . . . . . . . . . . . . 5
3.2 The self-testing scenario . . . . . . 6
3.3 Self-testing of states . . . . . . . . 7
3.4 Self-testing of measurements . . . . 8
3.5 Self-testing via a Bell inequality
and the geometry of the set of
quantum correlations . . . . . . . . 8
3.6 Robust self-testing . . . . . . . . . 9
3.7 Generalisations and alternative
definitions . . . . . . . . . . . . . . 10
4 A first example 11
4.1 Geometrical proof of anticommu-
tativity . . . . . . . . . . . . . . . 13
4.2 Algebraic proof of anticommuta-
tivity . . . . . . . . . . . . . . . . . 13
4.3 Swap gate . . . . . . . . . . . . . . 13
4.4 Self-testing of measurements . . . . 16
5 Self-testing of bipartite states 16
5.1 Self-testing of two-qubit states . . 16
5.2 Self-testing of qudit states . . . . . 18
5.3 Self-testing n maximally entangled
pairs of qubits . . . . . . . . . . . . 19
6 Self-testing of multipartite states 21
6.1 Self-testing of graph states from
stabilizer operators . . . . . . . . . 21
6.2 Tailoring Bell inequalities . . . . . 22
6.3 Reductions to bipartite methods . 23
6.4 Parallel self-testing of multipartite
states . . . . . . . . . . . . . . . . 23
6.5 Self-testing using only marginal in-
formation . . . . . . . . . . . . . . 23
7 Robust self-testing of states 23
7.1 Robust self-testing methods . . . . 24
7.2 Robust certification of large entan-
glement . . . . . . . . . . . . . . . 28
8 Self-testing of measurements 29
8.1 Measurement self-testing results . . 29
8.2 Methods in measurement self-testing 30
8.3 Robust measurement self-testing . 32
9 Extensions of self-testing to other
scenarios 33
9.1 Self-testing of quantum gates and
circuits . . . . . . . . . . . . . . . . 33
9.2 Semi-device-independent scenarios 34
10 Applications of self-testing 37
10.1 Device-independent randomness
generation . . . . . . . . . . . . . . 38
10.2 Device-independent quantum
cryptography . . . . . . . . . . . . 39
10.3 Entanglement detection . . . . . . 40
10.4 Delegated quantum computing . . 41
10.5 Structure of the set of quantum
correlations . . . . . . . . . . . . . 41
11 Experiments 42
12 Concluding remarks and open ques-
tions 43
Acknowledgements 45
A Appendix 45
A.1 Self-testing complex measurements 45
A.2 Regularisation trick . . . . . . . . . 45
A.3 Swap isometries . . . . . . . . . . . 46
A.4 Localising matrices in the Swap
method . . . . . . . . . . . . . . . 46
B State and measurement assumptions 47
B.1 State . . . . . . . . . . . . . . . . . 47
B.2 Measurements . . . . . . . . . . . . 47
Accepted in Quantum 2020-23-08, click title to verify 2
References 49
2 Self-testing as a device-independent
protocol
The treatment of complex systems as black boxes
is a powerful tool in many scientific domains, pro-
viding a minimalist level of abstraction that al-
lows one to focus on what a device or system does
without the need to model precisely how this is
achieved. In quantum information theory, this
approach is known as the device-independent (DI)
approach.
In order to explain the idea of the device-
independent approach we imagine the following
scenario. Consider two laboratories, run by two
experimenters called Carmela and Deng. In their
laboratories (let’s imagine they are quantum op-
tics laboratories) both Carmela and Deng have
access to some equipment (e.g. lasers, beamsplit-
ters, waveplates, photon detectors,...) which they
can use to perform different experiments. A given
experiment consists of a choice of settings (e.g.
laser intensity, angle of the waveplates, type of
beamsplitter,...) that after a run of the exper-
iment provides a result (e.g. photon detection
location, time of detection,...). Furthermore, a
source is positioned between the laboratories and
emits physical systems (e.g. photons) that are
sent to Carmela’s and Deng’s laboratories; see
figure 1, left.
Suppose that Carmela and Deng would like to
learn if the source is emitting entangled particles
(where the entanglement is with respect to the
two laboratories). One way to achieve this is to
use their equipment to perform tomography of
the state of the source, i.e. Carmela and Deng
perform a number of experiments each with dif-
ferent settings, collect statistics of the results, and
use quantum state tomography to reconstruct the
density matrix of the state, which can then be
checked to determine if it is entangled (for in-
stance using an entanglement witness). This is
indeed what is done in many experiments around
the world.
Imagine now however that two computer
scientists—called Alice and Bob—are visiting
each of the labs. Despite knowing the mathe-
matical definition of entanglement, they will have
problems convincing themselves that the source is
producing entanglement. Firstly, they do not un-
derstand the experimental setup, so they do not
know what the different settings do. Moreover,
even if they were told what the settings do, they
do not have a good understanding of quantum
optics. As a result, they will not be able to re-
construct the state of the source in order to check
if it is entangled, as was the case for Carmela and
Deng.
Alice, however, proposes the following: even
though they do not understand what the set-
tings do, they can still change them and observe
something. That is, they can simply model their
laboratories as black boxes. Each laboratory is
treated as a device (a black box) that takes an
input (the settings) and returns and output (the
result), but the physical mechanism behind how
this occurs is unknown (see figure 1, centre). Sim-
ilarly, they do not assume anything about the
source; all they know is that it is distributing
some physical systems that may or may not be
entangled. Alice denotes each of her possible set-
tings as x = 0, 1, . . . and Bob denotes each of his
possible settings as y = 0, 1, . . . . Similarly Al-
ice and Bob denote the possible results of their
experiments by a = 0, 1, . . . and b = 0, 1, . . . .
After trying the different settings sufficiently
many times and collecting statistics, Alice and
Bob can estimate the probabilities (also called the
correlations)
p(a, b|x, y), (1)
that is, the probabilities to see the results a and
b given that the settings x and y are used. It
is important here to stress that although Al-
ice and Bob can estimate these probabilities,
they are ignorant about the underlying physics;
from their perspective the experiments could have
been made on atoms, electrons, neutrinos or any
other physical system. This scenario is called the
device-independent scenario. Remarkably, even
with such little knowledge, Alice and Bob can still
conclude that the source emits entangled states.
The trick to achieving this is to use Bell nonlo-
cality, a counter-intuitive phenomenon discovered
by John Bell in 1964 [Bel64] (see also box 3.1).
At the heart of Bell nonlocality are objects called
Bell inequalities. A Bell inequality consists of a
function I of the probabilities {p(a, b|x, y)} such
that, for a source producing separable (i.e non-
entangled) states one has
I({p(a, b|x, y)}) ≤ β. (2)
Accepted in Quantum 2020-23-08, click title to verify 3
Figure 1: (left) A source produces systems and distributes them between two laboratories that can perform different
local experiments by varying the settings of their equipment. (centre) In the device-independent approach, each local
laboratory is treated as a black box that takes as input a label that corresponds to a particular choice of settings for the
experiment, and outputs a label that denotes the corresponding result. After repeating the experiments sufficiently
many times, the probabilities p(a, b|x, y) can be estimated. (right) When self-testing a state and measurements,
one aims to infer the form of the state of the source and measurement operators describing the laboratories from
knowledge of the probabilities alone, i.e. in a black box scenario.
Importantly, the bound β holds for any source
producing any kind of physical systems, provided
that these systems are not entangled. This is
a consequence of the fact that the definition of
separability is independent of the physical sys-
tem in question. Interestingly, Bell inequali-
ties can be violated by entangled sources. That
is, for some entangled sources one can achieve
I(p(a, b|x, y)) > β. This situation can be under-
stood geometrically in the vector space of prob-
abilities, where Bell inequalities are defined by
linear hyperplanes; see Fig. 2.
Alice and Bob can therefore do the following.
They compare their probabilities against as many
Bell inequalities as they know. If they see that
one is violated, then it must be that the source
is entangled! Moreover, they are able to con-
clude this despite knowing nothing about how
the experiment was actually performed; all the
information that was needed was the probabili-
ties {p(a, b|x, y)}. Such a procedure is called a
device-independent certification of entanglement.
Suppose now that we change the task: instead
of only detecting entanglement, Alice and Bob
want to know the particular entangled state of
the source. Due to the device-independent sce-
nario, they cannot know the type of physical sys-
tem that the source is producing. However, they
may hope to write down the state vector |ψi of the
source, without specifying which types of physical
degrees of freedom it describes. This often turns
out to be possible (up to some local transforma-
tions, see the definitions in the following section),
as long as one observes the maximum possible
violation achievable in quantum theory of a cor-
responding Bell inequality. Such a procedure is
called a device-independent self-test or simply a
self-test of the state. Often, this maximal viola-
tion also allows one to self-test the measurements,
i.e. to determine the form of the measurement op-
erators that describe how the outcomes a and b
are produced in the local laboratories.
Alice and Bob will nevertheless face a problem:
they will never see the maximum violation of a
Bell inequality due to experimental noise and fi-
nite statistics. At best, they will be able to lower
bound the violation up to some statistical con-
fidence. To become a practically relevant proto-
col, self-testing therefore has to be combined with
two other tools: (i) noise-resistant self-testing
methods (called robust self-testing; see section
3.6) which give distance upper bounds to the de-
sired state and/or measurements as a function
of experimental noise; and (ii) tools of statisti-
cal analysis (such as Chernoff bounds) that al-
low for statistically valid estimations of probabil-
ities and corresponding confidence levels. Self-
testing can thus be seen as a theoretical tool
that, when augmented by statistical techniques,
can be transformed into a protocol for device-
independent state and measurement certification.
A number of remarks are in order before pro-
ceeding to formal definitions of self-testing in the
next section. First, one may wonder whether self-
testing is possible using only a single device. That
is, given a single device from which one observes
the conditional probabilities p(a|x), is it possi-
ble to determine the quantum state |ψi inside?
Notice that one possibility is that there is a pre-
programmed classical computer inside the device
that simply simulates the statistics p(a|x). In the
device-independent scenario one cannot rule out
this possibility. Thus, one cannot hope to cer-
tify any non-classical properties of |ψi with only
Accepted in Quantum 2020-23-08, click title to verify 4
Bell inequality
L
I·p ≤ β
q =⇒|ψ, M
a|x
, N
b|y
Figure 2: Geometric representation of self-testing. The
figure is a 2-dimensional representation of the space of
probability vectors ~p = (p(00|00), p(01|00), . . . ). The
grey area L is the set of probability vectors that are ob-
tainable with separable states (commonly called the ‘lo-
cal set’). A Bell inequality (dotted line) consists of a pair
(
~
I, β) such that the half space
~
I·~p ≤ β contains L. En-
tangled quantum states are capable of producing proba-
bility vectors that lie outside of L (purple area) and thus
violate Bell inequalities. Often, extremal points (e.g.
~q) of the quantum set, which maximally violate some
Bell inequality, admit—up to local transformations—a
unique realisation in terms of a particular entangled state
and measurements. Self-testing involves identifying such
probability distributions and proving their unique reali-
sation.
a single device. It is for this reason that it is cru-
cial to move to a multipartite scenario in order to
certify non-classical states. Indeed, it is precisely
the phenomenon of Bell nonlocality that forbids
an explanation using local pre-programmed clas-
sical devices.
Second, it is worth mentioning some practical
advantages of the device-independent scenario.
Suppose Alice and Bob are sold two devices that
are claimed to contain a particular entangled
state and perform certain measurements on it.
Using self-testing, they will be able to conclude
that the devices are indeed working correctly
without having to understand precisely how they
operate. This is clearly desirable from the per-
spective of Alice and Bob, especially for crypto-
graphic applications where one would prefer not
to trust the devices. From a more experimen-
tal perspective, the device-independent scenario
naturally treats experimental errors at the level
of the observed statistics, meaning for example,
that a false positive detection of entanglement
will never occur. Being the strongest form of
device-independent certification, self-testing has
proven to be useful for many device-independent
protocols; see section 10 for more information.
Third, we note that the notion of device-
independence referred to by the large majority
of self-testing works assumes independent, iden-
tically distributed (i.i.d. ) rounds of the ex-
periment, i.e. the state and the measurements
are assumed to be the same in each round and
do not depend on past measurement results or
choices. Such an assumption allows us to talk
about the probability distribution p(a, b|x, y) that
is valid in every experimental round. This how-
ever is not the strongest possible notion of device-
independence where one further drops the i.i.d.
assumption (typically used in the field of quan-
tum cryptography). Self-testing under this more
stringent notion of device-independence is little
explored (although arguably relevant); see the
concluding remarks in section 12 for further dis-
cussion.
Finally, we note that aside from its motivation
for device-independent certification, self-testing
can also be viewed as part of the general study
of quantum correlations. In particular, quan-
tum correlations are defined via the Born rule,
whereby a state and measurement operators are
mapped to a probability distribution. Generally,
this map does not have an inverse, since many
combinations of state and measurements can lead
to the same probabilities. Self-testing identifies
those points of the set of quantum probability
distributions for which such an inverse exists, up
to the local transformations defined in the next
section. That is, it associates some (extremal)
probability distributions with a unique realisa-
tion using a particular state and measurements.
For this reason self-testing has been used to de-
rive important results about the set of quantum
correlations.
3 Definitions
3.1 Notation
We first introduce some notation. L(H) denotes
the set of linear operators acting on Hilbert space
H. Uppercase Roman letters denote a local party,
most often either Alice, A, or Bob, B. Roman let-
ters in either subscript or superscript denote the
Hilbert space in which a state lives or on which
Accepted in Quantum 2020-23-08, click title to verify 5
an operator acts e.g. |ψi
A
∈ H
A
. Consecutive
labels denote tensor products of Hilbert spaces,
e.g. |ψi
AB
∈ H
A
⊗ H
B
. Labels containing the
same letter are implicitly assumed to refer to dif-
ferent local Hilbert spaces of a single subsystem,
e.g. A and A
0
refer to two Hilbert spaces of Alice.
As a result, self-testing is often a useful tool
in a variety of device-independent protocols (see
section 10 for explicit examples). In the follow-
ing section, we formalise these ideas and define
precisely what it means to self-test a state and
measurements.
3.2 The self-testing scenario
The device-independent scenario described in the
previous section is commonly called a Bell test,
and the probabilities p(a, b|x, y) are called cor-
relations. From quantum theory, we know that
there exist measurement operators M
a|x
∈ L(H
A
)
acting on Alice’s local Hilbert space and satisfy-
ing
M
a|x
< 0 ∀x, a,
X
a
M
a|x
=
A
∀x (3)
that describe how the outcomes a are obtained
given settings x. Similarly there exist measure-
ment operators N
b|y
∈ L(H
B
) for Bob acting on
his local Hilbert space. From here on we work
with a Naimark dilation of each of the measure-
ments. The measurement operators are therefore
projective:
M
a|x
M
a
0
|x
= δ
a,a
0
M
a|x
∀x, a, a
0
,
N
b|y
N
b
0
|y
= δ
b,b
0
N
b|y
∀y, b, b
0
. (4)
This can be justified if one takes the position
that projective measurements and unitary evolu-
tion are the only fundamental operations in quan-
tum theory. From this perspective, any measure-
ment necessarily involves a projective measure-
ment over a dilated space, the degrees of freedom
of which belong to Alice and Bob’s local Hilbert
spaces. For further discussion on this assumption,
see Appendix B.
Now, from the Born rule, there must exist some
quantum state ρ
AB
∈ L(H
A
⊗H
B
) < 0, tr ρ
AB
=
1 such that
p(a, b|x, y) = tr
h
ρ
AB
M
a|x
⊗ N
b|y
i
. (5)
In self-testing, one aims to infer the form of the
state and the measurements in (5) from knowl-
edge of the correlations p(a, b|x, y) alone; i.e. in
the device-independent scenario.
In order to write (5), one nevertheless needs to
make some basic physical assumptions that con-
stitute the definition of the device-independent
scenario in most self-testing works. These are
1. The experiment admits a quantum descrip-
tion; i.e. there exists a quantum state and
measurement operators that lead to the ob-
served outcomes via the Born rule.
2. The laboratories of Alice and Bob are located
at separate locations in space and there is
no communication between the laboratories;
e.g. Alice cannot send the choice of setting x
to Bob or vice-versa.
3. The settings x and y are chosen freely and
independently of all other systems in the ex-
periment. For example, the physical system
used to generate x does not have any corre-
lations (quantum or classical) with the par-
ticles or the source or the laboratory of Bob.
4. Each round of the experiment is independent
from all other rounds and physically equiv-
alent to all others. That is, there exists a
single density matrix and measurement op-
erators that are valid in every round. The
statistics of all the rounds are thus indepen-
dent and identically distributed (i.i.d. ), so
that we may consider p(a, b|x, y) only.
Following the majority of self-testing works,
we will often choose to work with a purification
|ψi
ABP
of ρ
AB
, where the purification space H
P
is external to both Alice’s and Bob’s laboratories.
This will be quite convenient mathematically,
since working with state vectors over density ma-
trices will significantly shorten the length of some
equations. We stress that we do not assume that
the state shared between Alice and Bob is pure;
indeed it is given by tr
P
[|ψihψ|
ABP
] = ρ
AB
. Fur-
thermore, our definitions of self-testing will be
such that the purification space is untouched by
the self-testing protocol. Hence, working with
|ψi
ABP
and tracing out the purification space is
equivalent to working with ρ
AB
. For this rea-
son, the definitions in the following section can
be equivalently phrased using ρ
AB
by simply dis-
carding the purification space. Using such a pu-
rification, the probabilities (5) can be written
p(a, b|x, y) = hψ|M
a|x
⊗ N
b|y
⊗
P
|ψi
ABP
, (6)
Accepted in Quantum 2020-23-08, click title to verify 6
where
P
is the identity operator on the purifica-
tion space.
Now, let’s imagine we have in mind a par-
ticular pure state
1
|ψ
0
i
A
0
B
0
and projective mea-
surements {M
0
a|x
}, {N
0
b|y
} that we would like
to self-test. We call this state and measure-
ments the reference state and reference mea-
surements. The state ρ
AB
and measurements
{M
a|x
}, {N
b|y
} that correspond to the actual ex-
periment are called the physical state and phys-
ical measurements. Similarly, the realisation
{|ψ
0
i
A
0
B
0
, {M
0
a|x
}, {N
0
b|y
}} is called the reference
experiment and {ρ
AB
, {M
a|x
}, {N
b|y
}} the physi-
cal experiment.
Note that it will not be possible to infer exactly
the reference state and measurements from the
correlations alone. This is for the following two
reasons:
i Due to the unitary invariance of the trace, one
can reproduce the statistics of any state |ψ
0
i
and measurements {M
0
a|x
}, {N
0
b|y
} by instead
using the rotated state U ⊗ V |ψ
0
i and mea-
surements {UM
0
a|x
U
†
}, {V N
0
b|y
V
†
}, where U
and V are unitary transformations. Hence,
one can never conclude that the state is |ψ
0
i
since it may in fact be U ⊗ V |ψ
0
i.
ii One cannot rule out additional degrees of free-
dom on which the measurement operators act
trivially. That is, a state |ψ
0
i ⊗ |ξi and mea-
surements {M
0
a|x
⊗ }, {N
0
b|y
⊗ } (where the
identity operators act on the local spaces of
|ξi) gives the same correlations as |ψ
0
i and
{M
0
a|x
}, {N
0
b|y
}.
To be able to define what it means to infer a
particular state in the device-independent sce-
nario, we thus need to define an equivalence be-
tween states that takes into account the above
unknowns (that is, local unitary transformations
and additional unused degrees of freedom). To
do this, we make use of the concept of a local
isometry.
An isometry Φ : H
A
1
→ H
A
2
is a linear trans-
formation on quantum states that preserves the
inner product, and can thus be seen as a unitary
operator that can increase the dimension of the
space. A general isometry on a state |ψi
A
1
can be
achieved by embedding |ψi
A
1
in a larger Hilbert
1
self-testing of mixed states will not be possible; see
section 3.5
Figure 3: A local isometry applied to a quantum state.
Local ancillas are added to the state and unitary trans-
formations are applied locally.
space H
A
2
and performing a unitary transforma-
tion; i.e. Φ[|ψi
A
1
] = U |ψi
A
2
. A local isometry
Φ
A
1
⊗ Φ
B
1
: H
A
1
⊗ H
B
1
→ H
A
2
⊗ H
B
2
(7)
is an isometry that can be realised with local
quantum operations; i.e. it is a tensor product
of isometries acting locally. One way to imple-
ment a local isometry—often used in self-testing
works—is to embed the initial state using local
ancilla states |00i
A
0
B
0
, and then perform a local
unitary transformation (see figure 3):
Φ
A
⊗ Φ
B
[|ψi
AB
] = U
A
0
A
⊗ V
B
0
B
[|00i
A
0
B
0
⊗ |ψi
AB
] .
In the following sections we will use Φ[|ψi] to de-
note the action of an isometry on the pure state
|ψi and use Φ[ρ] to denote the corresponding
transformation on a density matrix. As we will
see, the notion of a local isometry will be central
to our definitions of self-testing.
3.3 Self-testing of states
We are now ready to define what it means to
self-test a quantum state.
Definition 1. (self-testing of pure states)
The correlations p(a, b|x, y) self-test the state
|ψ
0
i
A
0
B
0
if for any state ρ
AB
compatible with
p(a, b|x, y) (for some choice of local measure-
ments) and for any purification |ψi
ABP
of ρ
AB
there exists a local isometry
Φ
A
⊗ Φ
B
: H
A
⊗ H
B
→ H
A
0
¯
A
⊗ H
B
0
¯
B
such that
Φ
A
⊗ Φ
B
⊗
P
[|ψi
ABP
] =
ψ
0
A
0
B
0
⊗ |ξi
¯
A
¯
BP
(8)
for some state |ξi
¯
A
¯
BP
.
Accepted in Quantum 2020-23-08, click title to verify 7
The above can be understood via the idea of ex-
traction. If we trace out the purification space in
(8) then we have
Φ
A
⊗ Φ
B
[ρ
AB
] = |ψ
0
ihψ
0
|
A
0
B
0
⊗ σ
¯
A
¯
B
, (9)
where σ
¯
A
¯
B
= tr
P
|ξihξ|. Thus, if one self-tests
the state |ψ
0
i
A
0
B
0
, then there necessarily exists a
local channel (given by the isometry) that allows
one to extract |ψ
0
i
A
0
B
0
from ρ
AB
into the ancilla
space. The state |ξi
¯
A
¯
BP
in (8), which contains ev-
erything else from |ψi
AB
after the reference state
has been extracted, is called a junk state. We note
that it is not necessary that one actually perform
the isometry in the laboratory; all that is needed
is a mathematical proof that such as procedure
exists in principle. In section 4 we will see how
this is possible with an explicit example.
Definition 1 is the definition that is most com-
monly used in self-testing works, although the
identity channel on the purification space is often
left implicit. Note that it is important that the
isometry does not act on the purification space
since, for example, the purification of a mixed
separable state would result in an entangled state,
which would give the devices access to entangle-
ment for free. Finally, the above definitions can
be straightforwardly generalised to define self-
testing of multipartite states; one simply uses an
isometry that is local with respect to the subsys-
tems of the multipartite state to be self-tested.
3.4 Self-testing of measurements
The correlations that are used to self-test the ref-
erence state often allow one to identify the mea-
surements that were performed. Since our phys-
ical measurements are projective, this definition
will apply to self-testing of projective measure-
ments only. The idea is to prove that under the
action of the isometry, the physical measurements
map to the reference measurements acting on the
reference state. More specifically,
Definition 2. (self-testing of states and mea-
surements)
The correlations p(a, b|x, y) self-test the state
and measurements |ψ
0
i
A
0
B
0
, {M
0
a|x
}, {N
0
b|y
} if for
any state and measurements ρ
AB
, {M
a|x
}, {N
b|y
}
compatible with p(a, b|x, y) and for any purifica-
tion |ψi
ABP
of ρ
AB
there exists a local isometry
Φ
A
⊗ Φ
B
such that
Φ
A
⊗ Φ
B
⊗
P
h
M
a|x
⊗ N
b|y
⊗
P
|ψi
ABP
i
=
M
0
a|x
⊗ N
0
b|y
ψ
0
A
0
B
0
⊗ |ξi
¯
A
¯
BP
for all a, x, b, y, for some state |ξi
¯
A
¯
BP
.
If we wish to make a statement solely involving
the measurements, we will only be able to say
something about the part of the measurement
that acts on the support of the physical state.
Some recent works (see e.g. [Kan17]) adopt this
approach and use a definition of measurement
self-testing in the following spirit (for example,
for Alice’s measurements).
Definition 3. (self-testing of measurements)
The correlations p(a, b|x, y) self-test the measure-
ments {M
0
a|x
} for Alice if for any measurements
{M
a|x
} and state ρ
AB
compatible with p(a, b|x, y)
there exists a unitary operator U such that
U
A
[
˜
M
a|x
]U
†
A
= M
0
a|x
⊗
for all a, x, where
˜
M
a|x
= Π M
a|x
Π and Π is the
projector onto the support of ρ
A
= tr
B
ρ
AB
.
3.5 Self-testing via a Bell inequality and the
geometry of the set of quantum correlations
It is known that only those correlations that are
extremal points of the quantum set of correla-
tions and are achievable with finite dimensional
quantum systems can be used to self-test both a
state and measurements [GKW
+
18]. Such points
can often be witnessed by the maximum violation
of some Bell inequality over the set of quantum
correlations. As a result, one often does not need
the full set of probabilities p(a, b|x, y) in order
to prove self-testing statements; the maximum
quantum violation of a Bell inequality may al-
ready imply the existence of the desired isome-
try. One can thus consider self-testing relative to
a Bell inequality by replacing the observation of
the correlations by the value of a Bell inequality
I[p(a, b|x, y)] in the previous definitions. Many of
the well known Bell inequalities, such as CHSH
and CGLMP have been used to this effect (see
section 5 for such results).
In this light one might ask if the maximal vio-
lation of every nontrivial Bell inequality, i.e. one
Accepted in Quantum 2020-23-08, click title to verify 8
which can be violated in quantum theory, is also
a self-test of some entangled state. Or even more
generally, do all extremal points of the set of
quantum correlations self-test some state? These
questions are examined in [GKW
+
18] where it
was shown that the relation between self-testing,
maximisers of non-trivial Bell inequalities and the
boundary of the quantum set is not as simple as
one might hope for.
We also note here that bipartite mixed state
correlations can always be reproduced by a pure
state of the same dimension [SVW16]. This im-
plies that self-testing of bipartite mixed states fol-
lowing the same spirit as definition 1 above is im-
possible. Since the isometry preserves the purity
of the input, applying the isometry to the pure
state that gives the same correlations cannot re-
sult in the desired mixed state in tensor product
with a junk state.
3.6 Robust self-testing
One encounters two problems when trying to
prove self-testing statements as defined above
from experimental data: (i) the experiment will
inevitably contain some level of noise that will
dampen the correlations, and (ii) the precise val-
ues of p(a, b|x, y) will be uncertain due to a fi-
nite sample size. In practice, this will mean that
proving perfect self-testing of states and measure-
ments is impossible. Point (ii) can be addressed
using tools of statistical inference [LRZ
+
18].
Point (i) can be tackled by proving approximate
self-testing statements, and is known as robust
self-testing. In short, the aim of robust self-
testing is to prove that if the correlations are suf-
ficiently close to the ideal correlations, then the
state and measurements must be close (in some
well-defined sense) to the desired ones.
We will focus on two notions of ‘closeness’ that
are frequently used in the literature. Our first
definition is as follows. Imagine we have identified
an isometry that allows us to prove a self-testing
statement as in definition 1. If the correlations
are close to the ideal then one would expect that
the two vectors appearing on either side of (8)
be approximately equal up to some vector norm.
This leads to the following.
Definition 4. (robust self-testing of states, vec-
tor norm)
The correlations p(a, b|x, y) self-test the state
|ψ
0
i
A
0
B
0
with distance δ in the vector norm ||·|| if
for any state ρ
AB
compatible with p(a, b|x, y) and
for any purification |ψi
ABP
of ρ
AB
there exists a
local isometry Φ = Φ
A
⊗ Φ
B
such that
||Φ ⊗
P
[|ψi
ABP
] −
ψ
0
A
0
B
0
⊗ |ξi
¯
A
¯
BP
|| ≤ δ
for some state |ξi
¯
A
¯
BP
.
This definition was used as the first definition of
robust self-testing.
Our second definition follows the intuition that
if in the ideal case one can extract the reference
state |ψ
0
i, then in the noisy case one should be
able to extract something close to |ψ
0
i. Here, it
is usually easiest to adopt the fidelity, defined as
F (|ψi, ρ) = hψ|ρ |ψi as the notion of closeness.
First, define ρ
EXT
as the extracted state of the
ancillas after the application of the isometry, that
is,
ρ
EXT
A
0
B
0
= tr
¯
A
¯
B
Φ[ρ
AB
]. (10)
We then have the following definition.
Definition 5. (robust self-testing of states, fi-
delity)
The correlations p(a, b|x, y) self-test the state
|ψ
0
i
A
0
B
0
with fidelity f if for any state ρ
AB
com-
patible with p(a, b|x, y) there exists a local isom-
etry Φ = Φ
A
⊗ Φ
B
such that
F
ψ
0
A
0
B
0
, ρ
EXT
A
0
B
0
≥ f. (11)
Ideally, one would like replace the fidelity in the
above definition by the trace distance T (ρ, σ) =
1
2
kρ − σk
1
=
1
2
tr[
q
(ρ − σ)
†
(ρ − σ)], since the
trace distance is both a metric (unlike the fidelity)
and relates directly to the probability of distin-
guishing the two states. The fact that the fidelity
is much more commonly used is because its lin-
earity in the state ρ makes bounds generally much
easier to compute. One can nevertheless prove an
upper bound to T (ρ
EXT
, |ψ
0
ihψ
0
|) from a bound
on the fidelity using the relation T ≤
√
1 − F
[NC18]. We point the reader to [BLM
+
09] where
a useful discussion about appropriate figures of
merit for robust self-testing can be found.
Note that a local isometry can always prepare
any pure product state of the ancillas for free
by simply ignoring the physical state and apply-
ing the necessary unitaries on the ancilla space.
Accepted in Quantum 2020-23-08, click title to verify 9
Hence, the best bound achievable via this strat-
egy defines a trivial bound that can always be
achieved. As an example, consider the task of
self-testing the state |ψi = cos θ |00i + sin θ |11i
for some θ ∈ (0, π/4). This state has fidelity
cos
2
(θ) and euclidean distance
√
2 − 2 cos θ to the
state |00i. Thus a self-tested fidelity or distance is
interesting only if it surpasses the corresponding
bound. Taking definition 5, this trivial fidelity is
equal to the square of the largest Schmidt coeffi-
cient of the state.
3.6.1 Extractability relative to a Bell inequality
As with ideal self-testing statements, it is most
common to consider robust self-testing relative
to a Bell inequality I. For example, taking def-
inition 5 as the figure of merit, one aims to find
a function f(β) that gives a lower bound on the
fidelity as a function of the Bell inequality vio-
lation I(p(a, b|x, y)) = β. This can be linked to
the notion of extractability of the physical state
with respect to the reference state for the Bell in-
equality I. Note that any CPTP map can be re-
alised by performing an isometry and discarding
some degrees of freedom [Sti55]. Thus the map
tr
¯
A
¯
B
Φ[ρ
AB
] is equivalent to a general local CPTP
map Λ
A
⊗Λ
B
: H
A
⊗H
B
→ H
A
0
⊗H
B
0
applied to
ρ
AB
. Given a physical state ρ, the extractability
Ξ is the maximum fidelity of Λ
A
⊗Λ
B
[ρ] and |ψ
0
i
over all CPTP maps:
Ξ(ρ →
ψ
0
) = max
Λ
A
,Λ
B
F (Λ
A
⊗ Λ
B
[ρ],
ψ
0
). (12)
To get optimal robust self-testing statements for
a given inequality I one therefore needs to min-
imise the extractability over all states compatible
with I = β for all values of β. This leads to the
extractability-violation trade-off function
Q
ψ,B
I
= inf
ρ|tr[B
I
ρ]=β
Ξ(ρ →
ψ
0
). (13)
Finding the extractability-violation trade-off
function for a given Bell inequality is a difficult
task since it involves a minimisation of the fi-
delity over all compatible states and a maximi-
sation over all possible CPTP maps. Moreover,
the optimal CPTP map (or equivalently, isome-
try) generally depends on the observed violation.
More commonly, one fixes a single isometry for all
violations and minimises the fidelity only, leading
to a sub-optimal curve.
Finally, one can use similar ideas to the above
to define the robust self-testing of measurements.
We do not give any definition here, but point the
reader to section 8.3 for work in this direction.
3.7 Generalisations and alternative definitions
3.7.1 The issue of complex conjugation
When self-testing quantum states in the bipar-
tite scenario, it is sufficient to consider real refer-
ence states only, i.e. states such that |ψ
0
i = |ψ
0
i
∗
,
where ∗ denotes complex conjugation with re-
spect to a fixed basis. This follows since all pure
states are local unitary equivalent to a real state
via the Schmidt decomposition [Pre98]. A similar
argument for measurements however is not possi-
ble. As a result, definition 2 suffers from a serious
drawback; it can only be used to self-test sets of
measurements that are invariant under the com-
plex conjugation of all measurement operators.
To see this note that since p(ab|xy) = (p(ab|xy))
∗
then (assuming a real state |ψ
0
i)
p(ab|xy) =tr
h
|ψ
0
ihψ
0
| M
0
a|x
⊗ N
0
b|y
i
=tr
h
|ψ
0
ihψ
0
| (M
0
a|x
)
∗
⊗ (N
0
b|y
)
∗
i
. (14)
Thus any correlations obtained using
{|ψ
0
i, M
0
a|x
, N
0
b|y
} can also be obtained using
{|ψ
0
i, (M
0
a|x
)
∗
, (N
0
b|y
)
∗
}. These two realisa-
tions are generally not equivalent under local
unitary operations. In this case, one cannot
self-test the set {|ψ
0
i, M
0
a|x
, N
0
b|y
} using defini-
tion 2 since there is always another realisation
{|ψ
0
i, (M
0
a|x
)
∗
, (N
0
b|y
)
∗
} that is not related to the
first via a local isometry but results in the same
correlations.
A straightforward solution to this problem first
proposed in [MM11] is to generalise the definition
of measurement self-testing so that one self-tests
the measurements {M
0
a|x
, N
0
b|y
} if one can show
that on the support of |ψ
0
i, the physical measure-
ments act as some unknown convex combination
of {M
0
a|x
, N
0
b|y
} and {(M
0
a|x
)
∗
, (N
0
b|y
)
∗
}. This is in
line with the general spirit of self-testing in which
one aims to certify the measurements up to all
the intrinsic limitations of the device-independent
scenario. See appendix A.1 for a possible defi-
nition along these lines and section 8.2.1 for an
example of such a self-test.
In principle, there may be more state and
measurement transformations other than com-
Accepted in Quantum 2020-23-08, click title to verify 10
plex conjugation that do not affect the observed
probabilities. Determining this set is still an open
problem. While in the case of qubit bipartite
systems one can aim at self-testing states and
measurements up to local isometries and complex
conjugations, it is unclear if more transformations
may be present when considering higher dimen-
sional systems or multipartite scenarios.
3.7.2 Self-testing via simulation
Another recent approach presented in [Kan16] is
to adopt the philosophy that self-testing a ref-
erence state or measurements should imply that
the physical state or measurements be capable
to simulate the reference state or measurements.
For states, this translates to finding a local quan-
tum channel Λ
A
⊗Λ
B
that maps the physical state
to the reference state, thus allowing the simula-
tion of any measurement on the reference state by
first applying the channel followed by the desired
measurement. Note that this definition is equiva-
lent to definition 1 since via Stinespring’s dilation
theorem [Cho75, Sti55] any local channel can be
realised by first applying a local isometry then
tracing out any irrelevant degrees of freedom.
For measurements, one considers unital chan-
nels, i.e. quantum channels that preserve the
identity (and thus map sets of measurements to
sets of measurements). The idea is then (say,
for Alice) that if one can find a unital channel
such that Λ[M
a|x
] = M
0
a|x
, then one can simulate
the reference measurement M
0
a|x
on any state by
first applying the dual quantum channel Λ
†
on
the state followed by the physical measurement
M
a|x
. Since
tr
h
Λ
†
[%]M
a|x
i
= tr
h
%Λ[M
a|x
]
i
= tr
h
%M
0
a|x
i
(15)
one recovers the same statistics as making the
reference measurement on any state %. This ap-
proach was used in [RKB18] to self-test the Bell
state measurement (see section 8.2.4) .
3.7.3 Measurement self-testing based on commu-
tation
Yet another approach to measurement self-testing
focuses on certifying that the physical measure-
ments satisfy some desired commutation relation
on the support of the physical state. This can be
advantageous, as commutation relations are often
the only relevant features that one is interested
in, and since they are invariant under isometry
maps the approach can lead to simpler proofs of
self-testing. Furthermore, in the case of perfect
statistics, certifying a particular commutation re-
lation may be enough to prove full self-testing
statements of the form of definition 2. This ap-
proach has been used to prove measurement self-
testing statements for anti-commuting qubit ob-
servables [Kan17] and sets of mutually unbiased
bases in dimension 3 [KŠT
+
19]. It is very close
in spirit to one of the earliest self-testing state-
ments given in [PR92]. For further discussion on
this technique, see section 8.2.2.
4 A first example
The maximally entangled state of two-qubits
φ
+
=
1
√
2
(|00i + |11i) (18)
is the quantum state which is most emblematic
of the significance entanglement has in quantum
theory and is used in a wide class of information
processing protocols [BBC
+
93, BW92, Eke91]. In
this section we show how to prove formal self-
testing statements for this state and locally anti-
commuting observables. Many of the techniques
used to self-test more complex states and mea-
surements can be understood as a generalisation
of those presented here. We work in a simple
scenario in which Alice and Bob each have two
inputs (x, y = 0, 1) and two outputs (a, b = ±1).
We chose the convention of having ±1 valued out-
comes since it will be convenient to work with the
observables
A
x
= M
+|x
− M
−|x
; B
y
= N
+|y
− N
−|y
, (19)
where M
a|x
, N
b|y
are the physical measurements
operators in (5). Note that since the physical
measurement operators are projective (see section
3.2), the operators A
x
, B
y
are by construction
Hermitian and unitary. We thus have
A
†
x
= A
x
; A
2
x
= ; B
†
y
= B
y
; B
2
y
= .
(20)
Following definition 1 we work with a purification
|ψi
ABP
of the physical state where the measure-
ments act trivially on H
P
. In the following we
Accepted in Quantum 2020-23-08, click title to verify 11
Box 3.1: Bell Nonlocality and the CHSH inequality
Bell nonlocality is a counter-intuitive property of quantum correlations discovered by John Bell
in 1964 [Bel64]. The correlations p(a, b|x, y) are called local if they can be reproduced by shared
classical information. To formalise this, we represent the shared information by a classical
random variable Λ ∼ π(λ). Averaging over this information, the possible correlations that Alice
and Bob can achieve is given by (see (a) below for the corresponding classical causal network)
p(a, b|x, y) =
Z
Λ
π(λ)p
A
(a|x, λ)p
B
(b|y, λ)dλ. (16)
Notice that local measurements on any separable state ρ =
R
Λ
dλπ(λ)σ
A
λ
⊗σ
B
λ
lead to correlations
of the above form, and so (16) is precisely those correlations that can be achieved using separable
states. If we collect all of the probabilities into a single vector p = (p(00|00), p(01|00), ···)
then the set of local correlations forms a convex polytope, the facets of which are called Bell
inequalities (see (b), below). Remarkably, if Alice and Bob share an entangled quantum system,
they may produce correlations that are nonlocal, i.e. which violate a Bell inequality and therefore
cannot be written in form (16). An important Bell inequality, called the CHSH Bell inequality
[CHSH69], already exists in the simplest scenario in which Alice and Bob have two inputs
(x, y = 0, 1) and two outcomes (a, b = ±1). It is given by
β
CHSH
= hA
0
B
0
i + hA
1
B
0
i + hA
0
B
1
i − hA
1
B
1
i ≤ 2, (17)
where hA
x
B
y
i =
P
a,b
a · b p(ab|xy) denotes the correlator for the inputs x, y. Measurements
on the maximally entangled state
φ
+
= [|00i + |11i]/
√
2 can violate this inequality up to
β
CHSH
= 2
√
2. For a comprehensive review on Bell nonlocality, see [BCP
+
14].
do not explicitly write the identity on the pu-
rification space, e.g. A
x
should be understood as
A
x
⊗
P
.
The central object we will use for self-testing is
the CHSH Bell inequality [CHSH69] (see box 3.1
for a summary of Bell nonlocality)
β
CHSH
= hA
0
B
0
i+hA
1
B
0
i+hA
0
B
1
i−hA
1
B
1
i ≤ 2.
Performing certain local measurements on the
state
φ
+
leads to a violation of β
CHSH
= 2
√
2.
More specifically, both Alice and Bob use anti-
commuting measurement observables to achieve
this violation. Alice measures A
0
= σ
x
and A
1
=
σ
z
, while Bob measures B
0
= (σ
z
+ σ
x
)/
√
2 and
B
1
= (σ
x
− σ
z
)/
√
2. The reverse statement, that
the violation 2
√
2 can only be achieved by mea-
surements applied on
φ
+
, represented the first
self-testing statement. Early proofs of this state-
ment can be found in [SW87], [PR92], [BMR92]
and [Tsi93], a decade and a half before the term
self-testing was coined.
In the following we show how to self-test
φ
+
from correlations achieving the maximal violation
of the CHSH inequality. The central step in the
proof will be to show that Alice and Bob’s local
observables anticommute on the support of their
shared state, i.e. {A
0
, A
1
}|ψi = {B
0
, B
1
}|ψi = 0.
We present two methods to achieve this; (i) a
geometrical argument for anticommutativity of
Bob’s observables (section 4.1), and (ii) an al-
gebraic argument (section 4.2). Once this is
achieved, the anticommuting observables can be
Accepted in Quantum 2020-23-08, click title to verify 12
used to build the required local isometry that is
needed to prove a formal self-testing statement
(section 4.3).
4.1 Geometrical proof of anticommutativity
In this section we give a simple geometric proof
that the correlations maximally violating the
CHSH inequality can be achieved only by mea-
suring locally anticommuting observables. This
will require knowing all correlations, not only the
observation that β
CHSH
= 2
√
2. The ideal corre-
lations achieving this violation are
hψ|A
0
B
0
|ψi =
1
√
2
, hψ|A
0
B
1
|ψi =
1
√
2
,
hψ|A
1
B
0
|ψi =
1
√
2
, hψ|A
1
B
1
|ψi = −
1
√
2
.
(21)
Let us define vectors
a
0
≡
1
√
2
(A
0
+ A
1
) |ψi, a
1
≡
1
√
2
(A
0
− A
1
) |ψi,
b
0
≡ B
0
|ψi, b
1
≡ B
1
|ψi.
Equations (21) imply the following inner product
values:
a
0
· b
0
†
= 1, a
1
· b
1
†
= 1. (22)
The Cauchy-Bunyakovski-Schwarz inequality a ·
b
†
≤ |a||b| implies
|a
i
||b
i
| ≥ 1 for i = 0, 1,
where |a
i
| =
p
a
i
· a
i
†
. Since operators A
i
and B
j
are unitary, vectors b
0
and b
1
have unit norm,
which implies
|a
i
| ≥ 1, for i = 0, 1. (23)
The norms of the vectors a
0
and a
1
satisfy
|a
0
|
2
+ |a
1
|
2
= 2 (24)
by construction, which together with (23) implies
|a
0
| = |a
1
| = 1. Since eqs. (22) represent the sat-
uration of the Cauchy-Bunyakovski-Schwarz in-
equality, vectors a
i
and b
i
for i = 0, 1 must be
parallel, i.e. b
i
= a
i
. This implies
{B
0
, B
1
}|ψi = (B
0
B
1
+ B
1
B
0
) |ψi (25)
=
(A
0
− A
1
)B
0
+ (A
0
+ A
1
)B
1
√
2
|ψi
=
(A
0
− A
1
)(A
0
+ A
1
) + (A
0
+ A
1
)(A
0
− A
1
)
2
|ψi
=0,
and thus B
0
and B
1
anti-commute on the sup-
port of |ψi. Note that since the correlations are
symmetric, the same result holds for Alice’s ob-
servables.
4.2 Algebraic proof of anticommutativity
In principle, it is not easy to find correlations
which self-test some state and measurements.
Natural candidates, however, are correlations
that maximally violate a particular Bell inequal-
ity. Moreover, the structure of the Bell inequal-
ity can be useful for proving self-testing state-
ments, especially in cases when simple geometric
considerations are not possible. Here we show
how one can deduce an anticommutation rela-
tion for Bob’s observables from the observation
β
CHSH
= 2
√
2. As a starting point we take the
SOS decomposition of the shifted CHSH Bell op-
erator (see box 4.1 for a summary of SOS decom-
positions):
2
√
2 − B
CHSH
=
1
√
2
"
A
0
+ A
1
√
2
− B
0
2
+
A
0
− A
1
√
2
− B
1
2
#
,
(30)
which follows from the properties (20). For
any state |ψi leading to β
CHSH
= 2
√
2, i.e.
hψ|B
CHSH
|ψi = 2
√
2 we thus have
A
0
± A
1
√
2
|ψi = B
0/1
|ψi, (31)
as explained in equation (28) in box 4.1. With
these relations we can prove that B
0
and B
1
an-
ticommute in the same way we did in (25).
4.3 Swap gate
We now prove a formal self-testing statement for
the state
φ
+
in the form of definition 1. This
will require proving the existence of an isometry
Φ mapping the physical state |ψi to our reference
state |ψ
0
i =
φ
+
. In the majority of self-testing
proofs the isometry is explicitly constructed and
in most cases it takes the form of the partial Swap
gate given in figure 4.
The main idea behind this particular isometry
is as follows. In the case that the physical state is
a two-qubit state and the operators are Z
A
= σ
A
z
,
X
A
= σ
A
x
, Z
B
= σ
B
z
and X
B
= σ
B
x
, the action of
the circuit is to swap the physical state with the
state |00i of the registers A and B. Of course,
given the device-independent scenario we cannot
Accepted in Quantum 2020-23-08, click title to verify 13
Box 4.1: SOS decompositions
To every Bell inequality I =
P
a,b,x,y
w
xy
ab
p(a, b|x, y) corresponds a Bell operator
B =
X
a,b,x,y
w
xy
ab
M
a|x
⊗ N
b|y
(26)
such that the violation is obtained as β = tr [Bρ]. If the maximal violation achievable by using
quantum resources (i.e. the quantum bound) is β
Q
the shifted Bell operator is defined as β
Q
−B.
Every shifted Bell operator is by construction positive semidefinite since hψ|B|ψi ≤ β
Q
for all
|ψi. Imagine the shifted Bell operator admits a decomposition
β
Q
− B =
X
λ
P
†
λ
P
λ
, (27)
where each P
λ
is a polynomial in the operators M
a|x
and N
b|y
. The decomposition (27) is called
a sum of squares (SOS) decomposition of the shifted Bell operator. If the polynomials are of
degree at most n in either M
a|x
or N
a|x
we say the SOS decomposition is of n-th degree.
SOS decompositions for Bell inequalities are typically hard to find. One can use numerical
methods to find SOS decompositions of various degrees via the NPA hierarchy [NPA07, NPA08]
(in particular, see [PNA10] for a link to SOS decompositions).
SOS decompositions allow one to extract potentially useful information about the physical state
|ψi and measurements used to achieve the maximal violation of the corresponding Bell inequality.
From (27) we have
hψ|β
Q
− B|ψi = 0 ⇒
X
λ
hψ|P
†
λ
P
λ
|ψi = 0 ⇒
X
λ
kP
λ
|ψik
2
= 0
⇒ P
λ
|ψi = 0 ∀λ (28)
Since P
λ
is a function of the operators used to obtain the maximal violation, the relations of the
form {P
λ
|ψi = 0}
λ
often represent nontrivial statements about the strategy used to maximally
violate the Bell inequality under consideration. Additionally, if a non-maximal violation β
Q
−
is observed the approximate relations analogous to (28) can be obtained:
hψ|β
Q
− B|ψi = ⇒ kP
λ
|ψik ≤
√
∀λ (29)
These relations are often significant for proving robust self-testing statements.
assume that the physical state is a two-qubit state
or any particular form of the operators. However,
from sections 4.1 and 4.2, we know that like σ
z
,
σ
x
, the operators A
0
, A
1
and B
0
, B
1
anti-commute
on the support of the state. The idea is then to
use these operators to create new operators Z
A
,
X
A
, Z
B
, X
B
which act in an analogous way to
σ
z
, σ
x
on |ψi. Since we expect our physical state
to be
φ
+
(up to a local isometry), the hope is
that by using these operators in the place of σ
z
and σ
x
one will still be able to extract
φ
+
into
the ancilla space. Indeed, this is the case. More
precisely, we choose
Z
A
=
1
√
2
(A
0
+ A
1
), X
A
=
1
√
2
(A
0
− A
1
),
Z
B
= B
0
, X
B
= B
1
. (32)
Note that we have
{Z
A
, X
A
} = 0 (33)
by construction and
{Z
B
, X
B
}|ψi = 0 (34)
from (25). Furthermore from (31) we have
Z
A
|ψi = Z
B
|ψi, X
A
|ψi = X
B
|ψi. (35)
Accepted in Quantum 2020-23-08, click title to verify 14
|0i
A
0
|0i
B
0
|ψi
AB
H
H
Z
A
Z
B
H
H
X
A
X
B
|ξi
φ
+
Figure 4: The partial swap gate isometry used to self-test the maximally entangled state of two qubits. H is the
Hadamard gate. After the application of the circuit, the maximally entangled state is extracted from |ψi to the
ancilla qubits.
In order for Φ to be a valid isometry, Z
A
, X
A
,
Z
B
and X
B
must be unitary. This is automatically
the case for Z
B
and X
B
(see (20)), however it is
not necessarily the case for Z
A
and X
A
. To deal
with this problem, we need to regularise these
operators so that they are unitary. Formally, to
regularise a Hermitian operator Z, one changes
all zero eigenvalues of Z to 1, resulting in a new
Hermitian operator Z
∗
. The regularised opera-
tor is then obtained by normalising the eigen-
values of Z
∗
, i.e.
ˆ
Z = |Z
∗
|
−1
Z
∗
. Note that
ˆ
Z
is unitary by construction. One can often show
that the regularised operators act in same way as
the original operators on the physical state, i.e.
ˆ
Z |ψi = Z |ψi. This is indeed the case for our
example (see appendix A.2). From hereon we
therefore take Z
A
and X
A
to be the regularised
unitary operators and continue to use the substi-
tutions (32) without problem.
After a straightforward calculation one finds
that the output of the isometry is
Φ[|ψi] =
1
4
|00i ⊗ ( + Z
A
)( + Z
B
) |ψi
+ |01i ⊗( + Z
A
)X
B
( − Z
B
) |ψi
+ |10i ⊗X
A
( − Z
A
)( + Z
B
) |ψi
+ |11i ⊗X
A
( − Z
A
)X
B
( − Z
B
) |ψi
,
or in more compact form
Φ[|ψi] =
X
i,j∈{0,1}
|iji
A
0
B
0
⊗
ˆ
f
ij
|ψi
ABP
, (36)
where
ˆ
f
ij
=
1
4
X
i
A
( + (−1)
i
Z
A
)X
j
B
( + (−1)
j
Z
B
).
Using (35) expressions of the form ( ± Z
A
)( ∓
Z
B
) |ψi are automatically equal to zero, setting
ˆ
f
01
|ψi =
ˆ
f
10
|ψi = 0. The expression
ˆ
f
11
|ψi can
be simplified in the following manner:
ˆ
f
11
|ψi =
1
4
X
A
( − Z
A
)X
B
( − Z
B
) |ψi
=
1
4
( + Z
A
)X
A
( + Z
B
)X
B
|ψi
=
1
4
( + Z
A
)( + Z
B
) |ψi
=
ˆ
f
00
|ψi
The second line is obtained by using anticommu-
tativity relations (33) and (34), while (35) and
the unitarity of X
B
was used to obtain the third
line. Finally we see that the output of the Swap
isometry is
Φ[|ψi
ABP
] =
φ
+
A
0
B
0
⊗ |ξi
ABP
, (37)
where |ξi =
√
2
ˆ
f
00
|ψi. Note that |ξi is nec-
essarily normalised since the circuit of figure 4
is unitary. We have thus self-tested the state
φ
+
in the sense of definition 1. Although we
have worked with a purification of the physical
state, the isometry does not act on the purifica-
tion space, as needed from definition 1. This is
because Φ is constructed from the measurement
operators themselves, which by assumption act
only on the local Hilbert spaces of Alice and Bob
and therefore not on the purification space of |ψi.
4.3.1 Partial vs full Swap gates
The partial Swap gate was used in self-testing
protocols for the first time in [MYS12] and in
a large number of self-testing proofs since then.
The full Swap gate differs from the partial one in
that it contains another controlled gate before the
first Hadamard is applied to the ancillary qubit.
This controlled gate can be omitted if the ancilla
Accepted in Quantum 2020-23-08, click title to verify 15
is initiated in the state |0i. In order to get bet-
ter robust self-testing protocols it might be use-
ful that Alice and Bob each have a local pair of
maximally entangled ancillas. In this case the full
Swap gate has to be used (see [McK16a]). The
generalisation of the Swap gate useful for self-
testing states of local dimension larger than two
is introduced in [YN13]. For more details on dif-
ferent types of Swap gates used for self-testing see
Appendix A.3.
4.4 Self-testing of measurements
The measurements Alice and Bob use to maxi-
mally violate the CHSH inequality can also be
self-tested via the Swap isometry. Here we ex-
plicitly show how to self-test Bob’s measurement
observable B
0
. For that purpose we check how
the partial Swap gate transforms the state B
0
|ψi,
which can also be written as Z
B
|ψi:
Φ[B
0
|ψi] =
X
i,j∈{0,1}
|iji
A
0
B
0
ˆ
f
ij
B
0
|ψi
=
X
i,j∈{0,1}
|iji
A
0
B
0
ˆg
ij
|ψi,
where
ˆg
ij
=
1
4
X
i
A
( + (−1)
i
Z
A
) ⊗ X
j
B
( + (−1)
j
Z
B
)Z
B
=
1
4
X
i
A
( + (−1)
i
Z
A
) ⊗ X
j
B
((−1)
j
+ Z
B
)
= (−1)
j
ˆ
f
ij
.
This relation implies ˆg
01
= ˆg
10
= 0 and ˆg
11
=
−ˆg
00
. Thus the output of the Swap isometry will
be
Φ(B
0
|ψi) =
⊗ σ
z
φ
+
A
0
B
0
⊗ |ξi
ABP
, (38)
i.e. the measurement observable acts on the sup-
port of |ψi as σ
z
. A similar method can be used to
self-test all other measurement observables used
for the maximal CHSH violation. Note that (38)
implies a self-test of the measurement operators.
Since from (19) one has N
b|0
= ( + bB
0
)/2 it
follows by linearity of Φ that
Φ(N
b|0
|ψi) =
⊗
+ bσ
z
2
φ
+
A
0
B
0
⊗ |ξi.
(39)
Combining this and the previous section, we can
prove a full state and measurement self-testing
statement as follows. This concludes the intro-
ductory sections of the review.
Self-testing statement for the CHSH inequality
Let {|ψi
ABP
, A
0
, A
1
, B
0
, B
1
} be the state and the
±1 valued observables maximally violating the
CHSH inequality. Then there exists a local isom-
etry Φ such that
Φ(|ψi) =
φ
+
⊗ |ξi,
Φ(A
0
|ψi) = (
σ
x
+ σ
z
√
2
⊗
φ
+
) ⊗ |ξi
Φ(A
1
|ψi) = (
−σ
x
+ σ
z
√
2
⊗
φ
+
) ⊗ |ξi,
Φ(B
0
|ψi) =
⊗ σ
z
φ
+
⊗ |ξi,
Φ(B
1
|ψi) =
⊗ σ
x
φ
+
⊗ |ξi.
for some state |ξi.
5 Self-testing of bipartite states
In this section we give an overview of the exist-
ing results in the self-testing of bipartite quantum
states. All of the results are for the self-testing
of pure states, since mixed states cannot be self-
tested (see section 3.5). In 5.1 we present the
known results from self-testing qubit states, fo-
cusing first on the large literature dedicated to
the maximally entangled pair of qubits. In 5.2
we move to self-testing of bipartite states of a
higher local dimension. Finally, in 5.3 we review
the results and methods to self-test many copies
of the maximally entangled pair of qubits.
5.1 Self-testing of two-qubit states
5.1.1 The maximally entangled pair of qubits
The fact that the maximal violation of the CHSH
inequality can be obtained only by using the max-
imally entangled pair of qubits or a mixture of
maximally entangled qubit states corresponding
to different degrees of freedom was reported al-
ready in [SW87, PR92, BMR92, Tsi93]. An alter-
native method to self-test the maximally entan-
gled pair of qubits is presented in [MY04], today
mostly known as the Mayers-Yao self-test. While
[SW87, PR92, BMR92, Tsi93] can be considered
as the avant-garde self-testing papers, [MY04]
stands out as the founding work which defined
self-testing as a protocol ‘on its own’ and pointed
out its importance. It is worth mentioning that
Mayers and Yao made a similar statement already
Accepted in Quantum 2020-23-08, click title to verify 16
in [MY98], where they called the reference corre-
lations ‘self-checking’. In the Mayers-Yao proto-
col, Alice and Bob both measure three observ-
ables, σ
z
, σ
x
and (σ
z
+ σ
x
)/
√
2. The proof is
geometric in spirit and the isometry the authors
use does the same job as the Swap gate, but the
authors do not make the connection to the idea of
applying a swap unitary. The self-test was made
robust in [MMMO06]. A simplified proof of the
Mayers-Yao self-test, in which Alice makes the
same measurements, while Bob measures only σ
z
and σ
x
appeared in the supplementary material
of [McK14].
The concept of robustness and relevant figures
of merit when self-testing the maximally entan-
gled pair of qubits were introduced in [BLM
+
09],
alongside with some explicitly calculated ro-
bustness bounds. The first completely device-
independent robust self-test of the maximally en-
tangled pair of qubits, both CHSH and Mayers-
Yao based, appeared in [MYS12]. Further in-
equivalent proofs for self-testing the maximally
entangled pair of qubits were reported in [MS13],
where the authors gave a condition for a given
binary XOR game to be a robust self-test, and
in [ŠASA16], where the chained Bell inequali-
ties were used to self-test the maximally entan-
gled state and an arbitrary number of real mea-
surements. An improvement of the robustness
bounds were provided numerically in [YVB
+
14]
and [BNS
+
15], and analytically in [Kan17], which
is currently the best self-test of the maximally en-
tangled pair of qubits in terms of robustness. An
important contribution to the self-testing of the
maximally entangled pair of qubits is [WWS16],
which characterises all the correlations that self-
test the state using two dichotomic measurements
per party. The robustness of these self-tests was
estimated in [LWH
+
19].
All the results presented so far used only real
measurements. The self-testing of maximally
entangled pairs of qubits using σ
y
observables
was introduced in [MM11] based on the chained
Mayers-Yao conditions, also in [APVW16] and
[ABB
+
17] based on the elegant Bell inequality
[Gis09], and in [Kan17] based on the extended
version of CHSH introduced in [Slo11] (more on
the issue of self-testing complex measurements
will follow in section 8.2.1).
5.1.2 Self-testing of partially entangled states
All pure entangled states of two qubits admit a
Schmidt decomposition
|ψ
θ
i = cos(θ) |00i + sin(θ) |11i θ ∈ (0, π/4].
(40)
Such states are known as partially entangled pairs
of qubits, and they maximally violate the tilted
CHSH inequalities [AMP12]:
αhA
0
i+hA
0
B
0
i+hA
0
B
1
i+hA
1
B
0
i−hA
1
B
1
i ≤ 2+α.
The maximal quantum violation
√
8 + 2α
2
is
achieved by the corresponding partially entan-
gled state (40) for tan 2θ =
p
2α
−2
− 1/2. To
achieve the maximal violation Alice measures
A
0
= σ
z
and A
1
= σ
x
while Bob measures B
0
=
cos µ σ
z
+ sin µ σ
x
and B
1
= cos µ σ
z
− sin µ σ
x
,
with tan µ = sin 2θ. The proof that the maxi-
mal violation of the tilted CHSH inequality self-
tests the corresponding partially entangled pair of
qubits appeared in [YN13]. It relied on an SOS
decomposition of the shifted Bell operator, but
the proof appeared to have an error which made
the self-testing proof invalid. The work [BP15]
introduced a systematic way to find SOS decom-
positions for arbitrary shifted Bell operators. A
whole family of SOS decompositions correspond-
ing to the tilted CHSH Bell operator is introduced
which was used to show that every tilted CHSH
inequality self-tests the corresponding partially
entangled pair of qubits. Improved robustness
bounds for self-testing partially entangled pairs
of qubits through violation of the tilted CHSH
inequalities were presented in [CKS19]. Two dif-
ferent Bell inequalities, inequivalent to the tilted
CHSH inequality and useful for self-testing the
partially entangled pairs of qubits appeared in
[BAŠ
+
20] and [WBSS18].
The nonlocal character of partially entangled
pairs of qubits can be assessed through the Hardy
test [Har92, Har93]. In [RZS12] it is proven that
Hardy test can be used as a robust self-test for
the following states
|ψ
ϕ
i = α(|01i + |10i) + e
iϕ
p
1 − 2α
2
|11i,
where a =
q
(3 −
√
5)/2 and ϕ is a free parame-
ter.
A recent contribution [WKB
+
19] presents a
self-test for any partially entangled pair of qubits
and all three Pauli measurements (up to com-
plex conjugation) on Alice’s side. Bob needs to
Accepted in Quantum 2020-23-08, click title to verify 17
apply six measurements. The self-test is proven
from the value of three Bell inequalities; two max-
imally violated tilted CHSH inequalities and one
non-maximally violated CHSH inequality.
5.2 Self-testing of qudit states
The self-testing of bipartite entangled states of
higher local dimension (qudits) is more compli-
cated task than the self-testing of qubit states.
The good understanding of the qubit case has in-
spired the use of methods that we call ‘subspace
methods’ in which different two-qubit subspaces
of the state are self-tested until enough informa-
tion is gained to self-test the full state. In subsec-
tion 5.2.1 we review this approach, before focus-
ing on more genuinely d-dimensional methods in
subsections 5.2.2 and 5.2.3. Some states of local
dimension 2
n
can be seen as a tensor product of n
qubit states. In such cases the so-called parallel
self-testing is often used, described in section 5.3.
5.2.1 Subspace methods
Self-testing of maximally entangled states of any
dimension is discussed for the first time in [YN13].
The isometry for self-testing introduced there is a
high-dimensional generalisation of the Swap gate.
The authors provided a set of correlations which
self-test the maximally entangled state of two qu-
dits
Φ
+
d
E
=
1
√
d
d−1
X
i=0
|iii. (41)
One party performs three measurements and
the other four. The idea is to self-test sepa-
rately maximally entangled subnormalised sub-
states |ψ
0,1
i = |00i + |11i, |ψ
2,3
i = |22i + |33i,
···, |ψ
d−2,d−1
i = |d − 2, d − 2i + |d − 1, d − 1i.
This can be done if all of the substates max-
imally violate the CHSH inequality (although
in [YN13] the authors used a different corre-
lation to test the substates). For this, both
parties apply the measurements which are di-
rect sums of the ideal CHSH measurements.
This step is not enough, since the mixed state
1/d
P
d−2
i=0
|ψ
i,i+1
ihψ
i,i+1
| could also pass the test.
Another necessary step is self-testing of the sub-
states |ψ
d−1,0
i, |ψ
1,2
i and so on. It is clear that
the mixed state given above cannot provide cor-
relations necessary for this step, where the two
parties use again measurements which are the
direct sum of the ideal CHSH measurements in
a shifted basis, i.e. they now self-test the states
|ψ
1,2
i, |ψ
3,4
i, ··· , |ψ
d−1,0
i. The direct sum of σ
z
measurements are the same in both bases, thus
one party applies three measurements in total
while the other applies four.
An arbitrary pure bipartite state admits the
Schmidt decomposition
|ψi =
d−1
X
i=0
λ
i
|iii. (42)
The generalisation of the above explained method
to self-testing states of the form (42) is given
in [CGS17]. In the first step the sub-states
|ψ
i,i+1
i = λ
i
|iii + λ
i+1
|i + 1, i + 1i for i =
0, ··· , d − 2 are self-tested via the maximal vi-
olation of the tilted CHSH inequalities. The sec-
ond step self-tests the shifted states |ψ
i,i+1
i =
λ
i
|iii + λ
i+1
|i + 1, i + 1i for i = 1, ··· , d − 1.
This result completed the problem of self-testing
all bipartite pure states. The Bell inequalities
corresponding to this type of the self-test for max-
imally entangled states are described in [Col18].
5.2.2 Self-testing from qudit correlations
The method for self-testing all pure bipartite
entangled states presented in the previous sec-
tion relied on self-testing two-qubit sub-states.
The measurements used in the self-test were also
block-diagonal, where all blocks were either 2 ×2
or 1 × 1. It is surprisingly difficult to prove self-
testing statements about high-dimensional states
without resorting to such methods. In this sec-
tion we outline a few protocols for self-testing qu-
dit states that use genuinely qudit measurements.
The first such results were proven in [BNS
+
15],
[YVB
+
14] and [SAT
+
17] where two-qutrit states
were self-tested by using the numerical Swap
method (for details see section 7.1.4). In
[BNS
+
15] and [YVB
+
14] the maximal violation
of the CGLMP inequality [CGL
+
02] was used
to self-test the partially entangled state of two
qutrits:
|ψi =
1
p
2 + γ
2
(|00i + γ |11i + |22i) (43)
where γ = (
√
11 −
√
3)/2, and in [SAT
+
17]
the SATWAP Bell inequality is introduced and
used to self-test the maximally entangled pair of
qutrits.
Accepted in Quantum 2020-23-08, click title to verify 18
An important contribution in this direction is
the analytic self-test presented in [KŠT
+
19]. The
maximally entangled pair of qutrits is self-tested
through the maximal violation of a generalised
CHSH inequality. These inequalities, introduced
in [KŠT
+
19] can be seen as a special class of
Buhrman-Massar inequalities [BM05], represent
good candidates for self-testing maximally entan-
gled states in any prime dimension d. Alice and
Bob, both have d inputs, and the measurements
necessary for the maximal violation are mutually
unbiased bases. For higher dimensions, the SOS-
decomposition of the shifted Bell operator is pro-
vided, but the self-testing statement is still lack-
ing. In fact, for d = 5 and d = 7, it is proven
that the maximal violation can be achieved by
using inequivalent quantum realisations, however
all of them involve the maximally entangled state
in dimensions 5 and 7, respectively.
Another contribution to self-testing maximally
entangled states of qudits in the context of
nonlocal games is given in [Man14]. There,
the author considers a specific type of nonlocal
games, the so-called pseudo-telepathy weak pro-
jection games. A nonlocal game is called pseudo-
telepathy game if it can be won with probability
equal to one by using quantum finite dimensional
strategy, but cannot be won by using classical
strategies [BBT05]. Weak projection games be-
long to a sub-class of pseudo-telepathy games and
[Man14] shows that every such game can be used
to self-test maximally entangled states in finite
dimensions.
5.2.3 Group theoretic tools
Self-testing properties of non-local games were
elaborately explored in [Slo11] and [CS17b]. The
common method for both works is the ‘algebraisa-
tion’ of the winning strategies in nonlocal games.
The idea of relating representations of a Clifford
algebra to the optimal strategies to win the CHSH
game was used already in [SW87] and [Tsi87]. In
[Slo11], to each XOR game G is associated a C
∗
algebra A, such that optimal strategies to win
G correspond to representations of A. Further-
more, there is a relation between near-optimal
strategies and approximate representations. Us-
ing these techniques a self-testing statement for
high-dimensional maximally entangled states via
a generalisation of the CHSH game is implicitly
given in [Slo11].
In [CS17b], the authors study self-testing prop-
erties of a class of pseudo-telepathy games, known
as linear-constraint system games, of which the
magic square and magic pentagram games are
two popular examples [Per90, Mer90b]. In these
games, the players are asked for assignments to a
subset of variables in a system of linear equations,
and they win the game if they return consistent
and valid assignments. The authors extend the
representation theoretic framework of [CM14],
[CLS17] and [Slo20] and obtain a generic self-
testing result for linear-constraint system games
of a certain kind. They apply this result to ob-
tain a self-testing protocol for a tensor product of
n maximally entangled pairs of qubits. The self-
testing condition is the perfect score in either the
magic square game or the magic pentagram game.
It is proven in [CM14] that perfect strategy for ev-
ery linear-constraint system game which is also a
pseudo-telepathy game must involve a maximally
entangled state. On the other side in [CLS17] it is
shown that a solution group can be associated to
every linear-constraint system game. Moreover,
the operators used in the winning strategy must
satisfy certain algebraic relations determined by
the solution group. In [CS17b] the authors use
these results and by exploiting algebraic proper-
ties of the solution group corresponding to the
magic square and magic pentagram games prove
the self-testing statement for a tensor product of
maximally entangled pairs of qubits. The self-test
is also proven to be robust.
5.3 Self-testing n maximally entangled pairs of
qubits
In this section we outline methods and results for
self-testing n copies of the maximally entangled
state of two qubits (which itself is a maximally
entangled state of dimension 2
n
). Here, there are
two main approaches; sequential self-testing and
parallel self-testing.
5.3.1 Sequential self-testing
The first result relating to the self-testing of n
maximally entangled pairs of qubits (here also
called EPR pairs) appeared in [RUV13]. In this
scheme, in each round of the experiment the
devices receive inputs (x
i
, y
i
) for i = 1, ··· , n,
labelling the measurement bases for i-th maxi-
mally entangled pair. The inputs are given to
the devices sequentially: first the inputs (x
1
, y
1
)
Accepted in Quantum 2020-23-08, click title to verify 19
are given and the outputs (a
1
, b
1
) are returned;
then the inputs (x
2
, y
2
) are given and the out-
puts (a
2
, b
2
) are returned. This process is contin-
ued until the n-th pair of outcomes is collected
and it is characterised by the following transcript
a
1
, b
1
given x
1
, y
1
,
a
2
, b
2
given a
1
, x
1
, x
2
, b
1
, y
1
, y
2
,
··· ,
a
n
, b
n
given a
1
, x
1
, ··· , a
n−1
, x
n−1
, x
n
, b
1
, y
1
,
··· , b
n−1
, y
n−1
, y
n
There is no assumption that in each round the
source emits the same state and the measure-
ment strategies in the rounds may depend on
the inputs and outputs in all previous rounds.
In [RUV13] the authors prove that if the parties
win CHSH game in ω
∗
n rounds, where ω
∗
is
the optimal probability to win the CHSH game,
there is isometry mapping the state the parties
shared at the beginning of the procedure to
the tensor product of n EPR pairs. The result
is stated in its robust form: if the parties win
CHSH game in (1 − )ω
∗
n rounds, then at
the beginning of any randomly chosen block of
m = n
Ω(1)
rounds the state of the parties can
be mapped to a state f ()-close to the tensor
product of m EPR pairs. The drawback of
the work is a very low robustness, i.e. as in-
creases the number of rounds necessary to extract
a state f()-close to m EPR pairs grows very fast.
5.3.2 Parallel self-testing
A more popular approach for self-testing n EPR
pairs is parallel self-testing (see figure 5). Here,
the inputs are not given sequentially but all at
the same time, i.e. the devices receive input
vectors x = (x
1
, ··· , x
n
), y = (y
1
, ··· , y
n
) and
return outputs vectors a = (a
1
, ··· , a
n
), b =
(b
1
, ··· , b
n
). In principle this makes it more dif-
ficult to prove self-testing statements than in the
sequential scenario, since one assumes less struc-
ture on how the outcomes are generated. To self-
test a single maximally entangled pair we saw
in section 4 that it is enough to identify a pair
of anticommuting observables. In the case of n
pairs, one has not only to find n pairs of anti-
commuting obervables, but also to show that ob-
servables from different pairs mutually commute.
An important feature of parallel self-tests is their
robustness. A parallel self-test is robust if any
Figure 5: Scenario for parallel self-testing. Alice and
Bob both receive a list of n inputs and provide n out-
puts, which correspond to measurements made on n in-
dependent copies of a physical state. The aim is to prove
that the statistics self-test n independent copies of the
reference state
strategy producing correlations that are -close to
the ideal ones must use a state which is f(, n)-
close to |ψ
0
i
⊗n
, where f(, n) is a monotonically
increasing function in . How quickly the func-
tion f (, n) increases with and n determines how
good robustness is.
The first parallel self-test was proven for 2 EPR
pairs in [WBMS16]. The work gives two different
self-tests: one based on the optimal success in a
double use of the CHSH game and the other on
the optimal success in the Magic Square game
[Mer90b, Per90]. The result was subsequently
generalised for arbitrary n: via parallel repetition
of the CHSH game in [Col17] and [McK17] and
via parallel repetition of the Magic Square Game
in [Col17] and [CN16]. The usefulness of the
Magic Pentagram game [Mer90b] for self-testing
a tensor product of three EPR pairs was proven in
[KM18]. Self-testing of n EPR pairs via parallel
repetition of the Mayers-Yao self-test is given in
[McK16b]. We also note that the self-test [CS17b]
discussed in the previous section belongs to this
class of parallel self-tests.
In recent years, several works appeared which
also managed to self-test n EPR pairs shared by
two parties but not by parallel repetition of any
single self-test of a single pair. The first of such
results was presented in [OV16], based on the
XOR games introduced in [Slo11]. Later, self-
testing of n EPR pairs with measurements per-
formed on few of them in each round was the sub-
ject of [CRSV18]. This self-test is nondestructive:
not all entanglement is consumed in the self-test,
but can be used for eventual later protocols.
A combination of self-testing based on nonlo-
cal games with the quantum version of the linear-
ity test from [BLR93], named Pauli braiding test
[NV17] led to the first self-test of n EPR pairs in
which robustness does not get worse if the num-
Accepted in Quantum 2020-23-08, click title to verify 20
ber of EPR pairs tested increases. Another paral-
lel self-test keeping this desirable property is pre-
sented in [NV18]. The test can be seen as a quan-
tum version of the classical plane-vs-point test for
multivariate low-degree polynomials [RS97].
Finally, while all self-tests presented in this sec-
tion certify n EPR pairs and tensor products of
the observables σ
x
and σ
z
, it is possible to extend
them to involve certification of σ
y
also. This was
first done in [CGJV17] and later in [BŠCA18b],
up to the uncertainty of the complex conjugation
of the full n-qubit measurement operators as ex-
plained in section 3.7.
5.3.3 Overlapping qubits
A standard parallel self-test of n EPR pairs
proves the existence of n pairs of anticommuting
observables, where any two observables belong-
ing to different anticommuting pairs necessarily
commute. As explained in [CRSV17] each an-
ticommuting pair of observables defines a qubit.
Hence the dimension of the underlying Hilbert
space in this case must be at least 2
n
. The main
contribution of [CRSV17] is the estimation of the
dimension of the underlying Hilbert space if ob-
servables from different anticommuting pairs do
not commute exactly, which might happen when
the self-testing conditions are approximately sat-
isfied. This leads to the concept of ‘overlapping
qubits’, which, depending on the amount of the
overlap can be ’packed’ in the Hilbert space whose
dimension grows polynomially with n.
6 Self-testing of multipartite states
All bipartite pure states admit a Schmidt de-
composition, which simplifies the characterisa-
tion of bipartite entanglement and the self-testing
of bipartite pure states. Multipartite states do
not admit such a simple characterisation, al-
though some generalisations of the Schmidt de-
composition exist in the entanglement literature
[AAC
+
00, Kra10]. While all bipartite pure en-
tangled states can be self-tested, when it comes
to self-testing of multipartite states, only some
partial results exist. Furthermore, from a ‘loop-
hole free’ perspective, multipartite self-testing is
considerably more demanding than bipartite self-
testing, since it requires space-like separation be-
tween multiple measurement devices.
In this chapter we identify four main methods
for self-testing multipartite entangled states: self-
testing of graph states based on the structure of
their stabilizer operators (section 6.1); tailoring
Bell inequalities to self-test specific states (section
6.2); reductions to bipartite methods (section
6.3); parallel self-testing of multipartite states
(section 6.4); and self-testing from marginal in-
formation only (section 6.5).
6.1 Self-testing of graph states from stabilizer
operators
The first multipartite states to be self-tested were
graph states [McK14]. Formally, given a graph G
defined by a set of vertices V = {1, ··· , N} and a
set of edges E (pairs of connected vertices of V ),
the graph state corresponding to G is given by
|Gi =
Y
(i,j)∈E
CZ
i,j
|+i
⊗N
, (44)
where CZ
i,j
is the controlled-σ
z
two-qubit unitary
CZ = diag(1, 1, 1, −1) acting on qubits i and j.
Equivalently, |Gi can be defined as the unique
state that is stabilized by (i.e. is a +1 eigenstate
of) a set of N local stabiliser operators σ
i
x
⊗
j∈n(i)
σ
j
z
, where n(i) is the neighbourhood of vertex i;
the set of vertices connected to i on G.
A self-testing protocol for any graph state cor-
responding to a connected graph is provided in
[McK14]. Note that graph states corresponding
to graphs that are not connected must be sep-
arable with respect to at least one bipartition.
The reference measurements needed for the self-
testing are given by the stabilizer operators them-
selves aided by a few measurements generalising
those from Mayers-Yao self-test. More specif-
ically, for an arbitrary graph state, one party
has to measure three observables: σ
z
, σ
x
and
(σ
x
+ σ
z
)/
√
2, while all the other parties mea-
sure only σ
x
and σ
z
. The self-test is robust to
small imperfections and the isometry is the mul-
tipartite generalisation of the Swap gate.
The approach from [BAŠ
+
20] can be also
placed in the following subsection, but since it
is intrinsically related to stabilizers we discuss it
in this group. Starting from any graph state, the
authors introduce a method to construct a Bell
inequality that is maximally violated by the cor-
responding state. Moreover, the derived Bell in-
equality can be used to self-test the state. Each
Accepted in Quantum 2020-23-08, click title to verify 21
Robustness Inputs size (in bits) Outputs size (in bits)
[BŠCA18b] poly(n, ) O(n) n
[CRSV18] poly(n, ) O(log n) 1
[Col17] poly(n, ) O(n) n
[CS17b] poly(n, ) O(n) n
[CGJV17] poly() O(n log n) 2
[CN16] poly(n, ) O(n) n
[McK16b] poly(n, ) O(n) n
[McK17] poly(n, ) O(n) n
[NV17] poly() O(n) 2
[NV18] poly() O(poly(log n)) poly(log log n)
[OV16] poly(n, ) O(log n) 1
Table 1: Comparative properties of different self-tests of n EPR pairs. The most important aspect of
a self-test of n EPR pairs when it comes to practical usefulness is its robustness to noise (or rigidity).
The other relevant property is its complexity, in terms of the size of the inputs. The size of the outputs
is also a relevant factor, especially in possible applications for randomness expansion. For now, the
self-testing protocol presented in [NV18] has the best properties in terms of the total number of inputs
(polynomial) and robustness bounds (independent on n). The papers use different distance measures,
but all the bounds given here are in terms of the Euclidean distance using definition 4. The work
[CGJV17] self-tests σ
y
measurements on each EPR pairs, and the number of inputs increases in order
to deal with the issue of complex conjugation (see 3.7.1.) If one omits self-testing of σ
y
from the
protocol the number of inputs is O(n).
party measures an anti-commuting pair of observ-
ables from the real plane of the Bloch sphere. Be-
yond graph states, the method can be used to self-
test the so-called partially entangled GHZ states
cos θ |0i
⊗n
+ sin θ |1i
⊗n
for any n ≥ 2.
6.2 Tailoring Bell inequalities
In [PVN14] the authors introduce a method to
build permutationally invariant Bell inequalities
with two measurement settings per party useful
for self-testing multipartite states. The method
is tailored for a specific state |ψ
0
i and the mea-
surements leading to the maximal violation are
chosen from the real plane of the Bloch sphere.
A linear program can be used to find a Bell oper-
ator, whose eigenstate is |ψ
0
i and maximises the
ratio of the quantum and classical bound. The
derived Bell inequality is just a suitable candidate
for self-testing, which further must be checked by
utilising the numerical Swap method technique
(see section 7.1.4). Since the self-testing proof
relies on the Swap method, it becomes too costly
when the number of parties becomes larger than
four. Examples of the successful implementation
of this method involve the tripartite W state,
the tripartite and four-partite GHZ state and the
four-qubit linear cluster state.
Another method for developing Bell inequali-
ties, tailored for self-testing of multipartite qubit
states is described in [SBWS18]. As in [PVN14],
all parties can perform two different measure-
ments and the constructed Bell inequalities are
suitable candidates for self-testing applications.
The starting point for choosing a Bell operator is
not the permutational invariance, but the struc-
ture of the stabilizers of the state. The method
can be applied to multipartite states that are not
graph states, in which case these stabilizer op-
erators will not all be tensor products of Pauli
operators. Of all the Bell operators mimicking
the structure of the stabilizers, constructed from
the arbitrary two real measurements per party,
the optimal candidate is the one whose maximum
eigenvalue is the local maximum with respect to
the small perturbation of the local measurement
directions. The robust self-test is then checked by
using semidefinite programming to find the lower
bound to the fidelity of the state providing the
maximal violation and |ψ
0
i (see section 7.1.3). As
example the authors apply the method to self-test
a family of four qubit states CU
φ
φ
+
⊗
φ
+
,
where CU
φ
= |0ih0| ⊗ + |1ih1| ⊗ exp[−iφσ
x
].
Since the self-testing is proven by employing nu-
merical methods, it becomes infeasible when the
number of parties increase.
Accepted in Quantum 2020-23-08, click title to verify 22
6.3 Reductions to bipartite methods
Self-testing protocols for multipartite states can
be constructed by reusing self-testing protocols
for bipartite states. The idea is as follows: when
n−2 parties perform appropriate projective mea-
surements, they might collapse the state of the
remaining two parties into some pure bipartite
entangled state, which in principle can be self-
tested by using methods from section 5. By re-
peating the process of projecting and self-testing
for different pairs of parties one might expect to
gather enough information to self-test the whole
multipartite state. An important restriction is
that the measurement used by any party in the
projecting part must be some of the measurement
the same party uses in the self-testing part of the
protocol.
The idea was first used in [WCY
+
14] to self-
test W -state |W i = (|001i + |010i + |100i)/
√
3.
Whenever one of the parties performs the mea-
surement in the computational basis and obtains
outcome +1 the state of the remaining two par-
ties becomes maximally entangled ∼ (|01i+|10i).
This state can be self-tested by maximally vio-
lating the CHSH inequality, for example. The
authors of [WCY
+
14] show that by repeating
the above process twice for different parties mea-
suring in the computational basis, the whole
state can be self-tested using the Swap isom-
etry. They also show that a similar method,
based on self-testing of partially entangled two-
qubit states, can be used to self-test states of the
form |W
γ
i = (|001i + |010i + γ |100i)/
p
2 + γ
2
.
The method was generalised in [ŠCAA18] to
prove self-testing of all permutationally invari-
ant qubit Dicke states, all qubit graph states,
and all multipartite states of any local dimen-
sion admitting the Schmidt decomposition |ψ
λ
i =
P
d−1
i=0
λ
i
|i, i, ··· , ii, representing the first self-test
of a high-dimensional multipartite state. Self-
testing of W -states for any number of parties was
also proven in [Wu17], and self-testing of all Dicke
state was proven in [Fad17].
The self-testing of graph states whose under-
lying graph is a triangular lattice is shown in
[HH18]. The whole graph is shared by three
parties and if one party measures its qubits in
the σ
z
basis it prepares maximally entangled
pairs of qubits for the remaining two parties,
which are in [HH18] self-tested through the
Mayers-Yao criterion.
6.4 Parallel self-testing of multipartite states
In section 5.3 we saw many ways to self-test n
EPR pairs by using parallel repetition of CHSH or
Magic Square game. Up to date, the only parallel
self-test of some multipartite state is shown in
[BKM19]. The authors use diagramatic proofs
based on categorical quantum mechanics [CK17],
to prove that parallel repetition of the GHZ game
robustly self-tests n copies of the GHZ state.
6.5 Self-testing using only marginal informa-
tion
Almost all the protocols for self-testing multipar-
tite states presented so far require measuring full-
body correlators, that is, they depend on corre-
lations between all parties. This quickly becomes
a practical problem since measuring such correla-
tions is typically experimentally very challenging.
The possibility of self-testing by measuring only
few-body correlators is the subject of [LCH
+
18].
The authors use the numerical Swap method
(see section 7.1.4) to self-test the tripartite W -
state, a class of W -like states (|001i + |010i +
γ |100i)/
p
2 + γ
2
and the states maximally vio-
lating Bell inequalities defined in [TSV
+
14] by
using only two-body correlators. The four-partite
W -state is also self-tested using three-body cor-
relators.
7 Robust self-testing of states
It is impossible to meet exactly the conditions
for ideal self-testing. On one hand, experimental
noise and imperfections undermine hope to repro-
duce exactly the reference correlations (i.e. those
obtained by performing reference measurements
on the reference quantum state). On the other
hand, even if all noise contributions are elimi-
nated, one must work with a finite sample size
and so the precise probabilities cannot be known,
but only estimated up to some statistical confi-
dence level. In order to make self-testing proto-
cols practically meaningful, it is therefore crucial
to make them robust to deviations from the ideal
case. For possible definitions of robust self-testing
see Defs. 4 and 5.
The first self-testing protocol to be made ro-
bust was the Mayers-Yao self-test of maximally
Accepted in Quantum 2020-23-08, click title to verify 23
Ideal self-testing Robust self-testing
Ideal probabilities
Relations between Alice’s and Bob’s operators
Local isometry mapping exactly to the reference state
Approximately ideal probabilities
Approximate relations between Alice’s and Bob’s operators
Local isometry mapping approximately to the reference state
Figure 6: The program for robust self-testing. The aim is to show that approximately satisfied self-testing conditions
imply the existance of the local isometry which approximately maps the physical state to the reference one.
entangled pair of qubits [MMMO06]. Robust
self-testing through the CHSH inequality was ex-
plored in [BLM
+
09] and simple robust self-testing
protocols based on both the Mayers-Yao and the
CHSH criterion were presented in [MYS12]. The
techniques presented therein remained the main
tool for making self-testing protocols robust in
the majority of later contributions. As more is
known about ideal self-testing the focus of the
research is shifting towards finding better tech-
niques for assessing robustness. Arguably, it re-
mains a principal challenge in the field.
In this section we review the main contribu-
tions to robust self-testing and the techniques
predominantly used in the literature. In section
7.1 we identify and explain five approaches:
• An approach based on the vector norm in-
equalities (Section 7.1.1);
• Methods relying on the use of Jordan’s
lemma (Section 7.1.2);
• An approach based on the operator inequal-
ities (Section 7.1.3);
• The numerical Swap method (Section 7.1.4);
• An algebraic method (Section 7.1.5).
Finally, in Section 7.2 we discuss recent progress
on noise-tolerant self-testing of a tensor product
of many EPR pairs.
7.1 Robust self-testing methods
7.1.1 Norm inequalities method
The bulk of self-testing protocols start from the
observed probabilities p(a, b|x, y) = hψ|M
a|x
⊗
N
b|y
|ψi or the maximal violation of some Bell in-
equality and deduce equations of the type (see
e.g. (34) and (35) from section 4)
f({M
a|x
}, {N
b|y
}) |ψi = 0, (45)
where f is some polynomial function in the
measurement operators. Such relations can be
drawn from either geometrical arguments (like in
[MY04]), algebraic identities (like in [McK14]) or
SOS decompositions (like in [BP15]). The rela-
tions (45) are a necessary step in proving that
the appropriate isometry (the Swap gate in most
cases) maps the physical state to the reference
one,
Φ(|ψi) =
ψ
0
⊗ |ξi. (46)
Although the self-testing proof requires that the
correlations be ideal, one could hope to follow
the same proof in which the exact relations are
exchanged with approximate ones, leading to a
noise-dependent bound on the self-tested fidelity.
More precisely, when the observed probabilities
are within ε distance from the ideal ones (or the
violation of the Bell inequality is ε-far from the
maximal value), analogously to (45), the aim is
to find approximate relations:
kf({M
a|x
}, {N
b|y
}) |ψik ≤ g
f
(ε), (47)
where k·k is a vector vector norm (usually taken
to be Euclidean) and g
f
() are some increasing
functions for which g
f
(0) = 0. One can then
often guarantee that the appropriate isometry Φ
satisfies
kΦ(|ψi) −
ψ
0
⊗ |ξik ≤ g
Φ
(ε), (48)
where g
Φ
is obtained by propagating the uncer-
tainties (47) through the isometry circuit.
Accepted in Quantum 2020-23-08, click title to verify 24
Asymptotic behaviour of g
Φ
[MYS12] O(
1
4
)
[McK14] O(
1
4
)
[BP15] O(
1
2
)
[ŠASA16] O(
1
2
)
[WCY
+
14] O(
1
4
)
Table 2: Comparative properties of different robust self-
tests based on vector norm inequalities.
Relations of the type (47) are usually obtained
via various vector norm inequalities, such as
Cauchy-Bunyakovski-Schwarz, triangle or Hölder
inequalities. These techniques were first used in
[MYS12] and later in [McK14, WCY
+
14]. If a
self-testing proof relies on the maximal violation
of a Bell inequality, the relations (45) can be con-
veniently obtained from the SOS decomposition
of the shifted Bell operator (see box 4.1 and equa-
tion (29) therein). The usefulness of SOS decom-
positions for robust self-testing was first noted in
[YN13] and later used in e.g. [BP15, ŠASA16].
Techniques based on vector norm inequalities
are useful in making self-testing protocols ro-
bust, but the robustness bounds are typically not
very good due to large constants appearing in
them. The asymptotic behaviour of the function
g
Φ
for different self-testing protocols based on this
method is given in Table 2.
7.1.2 Utilising Jordan’s lemma
One of the main difficulties in the device-
independent description of quantum experiments
is related to the inability to fix the dimension of
the underlying Hilbert space, which prevents the
parameterisation of the measurements and states
used in the experiment. This difficulty stays
the prime hurdle towards calculating robust self-
testing bounds. A very useful theoretical asset
enabling a solution in scenarios where each party
has two dichotomic measurements is the Jordan
lemma [PAB
+
09] (see lemma 2 therein). It allows
to effectively reduce an arbitrary-dimensional ex-
periment to the one in which the local subsystems
are qubit systems.
For the purposes of robust self-testing the Jor-
dan lemma was first time used in [BLM
+
09] to
obtain robust self-testing of the maximally en-
tangled pair of qubits through violation of the
CHSH inequality. Later, it was used in [SBWS18]
for the robust self-testing of a arbitrary multipar-
tite states using the Bell inequalities introduced
therein and described in section 6.2 of this review.
For simplicity, here we give a short description of
the method to the bipartite scenario, while keep-
ing in mind that, as described in [SBWS18], it can
straightforwardly be applied to the multipartite
case.
The Jordan lemma states that given two Her-
mitian matrices of finite or countably infinite di-
mension and with eigenvalues ±1, there exists a
unitary transformation that simultaneously block
diagonalises them, where each block is of size at
most 2 × 2. Consider a self-testing protocol in
which Alice and Bob each have a pair of ±1 val-
ued observales A
x
, x = 0, 1 for Alice and B
y
,
y = 0, 1 for Bob. It follows there is a choice
of local basis in which these observables take the
block structure described above. One can further
assume that each of the blocks is of size 2 × 2,
since a one-dimensional block is equivalent to a
two-dimensional block where the state has sup-
port only on one of these dimensions. One can
then apply additional unitary rotations to each
of the blocks so that they take real values only.
Given this structure, one can paramaterise the
observables as follows
A
x
=
M
i
A
i
=
M
i
cos α
i
σ
x
+ (−1)
x
sin α
i
σ
z
,
B
y
=
M
j
B
j
=
M
j
cos β
j
σ
x
+ (−1)
y
sin β
j
σ
z
(49)
This parameterisation covers all possibilities:
α
i
= 0 implies that the observables commute in
that block, whereas α
i
= π/4 implies anticom-
mutation in that block. Consequently, the Bell
operator can be written as B = ⊕
ij
B(A
i
, B
j
).
Following such parametrisation the Bell violation
can be written as
β =
X
ij
p
ij
tr[B(A
i
, B
j
)ρ
ij
] (50)
where p
ij
ρ
ij
are projections of the physical state
ρ onto the blocks of Alice’s and Bob’s observ-
ables. Each block can then be treated separately
to achieve an expression of the form
F (Λ
i
A
⊗ Λ
j
B
(ρ
ij
), |ψ
0
ihψ
0
|) ≥ f(β). (51)
In [SBWS18] it is proven that if f is a convex
function of β there exist maps Λ
A
and Λ
B
such
Accepted in Quantum 2020-23-08, click title to verify 25
that the fidelity between Λ
A
⊗Λ
B
(ρ) and |ψ
0
ihψ
0
|
given the violation β is lower bounded by f(β).
In [BLM
+
09] a similar convexity argument is used
to obtain the final bound.
The remaining challenge is to obtain relations
of the form (51). In [BLM
+
09] the problem is
solved analytically and the isometry used is just
the one that rotates the blocks of the observ-
ables to obtain the form given in (49). The work
[SBWS18] provides a general recipe: (51) can be
solved by using a nonlinear optimisation with one
variable per party.
7.1.3 Operator inequalities method
An analytic approach to robust self-testing, in-
troduced in [Kan16] currently gives the best ro-
bustness bounds for the self-testing of two-qubit
states. It is suited for self-testing protocols based
on a Bell inequality violation. The method uses
the notion of extraction (see section 3.6.1) and
works by proving an operator inequality of the
form
K ≥ sB
I
+ µ , (52)
for all Bell operators B
I
for the Bell inequality I
in question, where K = Λ
†
A
⊗ Λ
†
B
(|ψ
0
ihψ
0
|) and
Λ
†
is the dual channel of Λ with respect to the
Hilbert-Schmidt inner product. This allows one
to make linear robust self-testing statements, that
is, to prove the existence of real parameters s and
µ such that the extractability-violation trade-off
defined in (13) satisfies
Q
ψ,B
I
(β) ≥ sβ + µ. (53)
One thus has
F (Λ
A
⊗ Λ
B
(ρ),
ψ
0
) ≥ sβ + µ (54)
for all states ρ achieving violation greater that β.
In principle it is a difficult task to prove the
operator inequality (52) for all Bell operators re-
gardless of the dimension. In [Kan16] Jordan’s
lemma is exploited to derive the current best ro-
bustness bounds for self-testing the maximially
entangled state of two qubits. The method uses
the CHSH inequality. The local channel Λ
A
⊗Λ
B
appearing in (54) is as follows. First, local uni-
tary transformations are applied to Alice and
Bob’s subsystems so that via the Jordan lemma,
their local observables take a block diagonal form
as in (49). Then, for each block, one applies the
α-dependent dephasing channel
Λ
α
[ρ] =
1 + g(α)
2
ρ +
1 − g(x)
2
Γ(α)ρΓ(α).
Here g(α) = (1 +
√
2)(sin α + cos α − 1) and
Γ(α) =
(
σ
x
α ∈ [0, π/4]
σ
z
α ∈ (π/4, π/2].
(55)
Bob’s channel Λ
B
is defined analogously. This
choice is shown to imply the lower bound (54) to
the fidelity with s = (2 +
√
2)/8 and µ = −(1 +
2
√
2)/4 (see figure 7 for a plot).
In [Kan16] inequality (52) is also proven for
Mermin inequalities in order to self-test the tri-
partite GHZ state. Moreover, the fidelity lower
bound for the Mermin inequality is proven to be
optimal in the sense that for any violation there
always exists a state achieving that violation with
the self-tested fidelity to the reference state. The
method has also been used for robust self-testing
of partially entangled pairs of qubits [CKS19] and
to assess the performance of different self-tests of
a maximally entangled pair of qubits [LWH
+
19].
7.1.4 Numerical Swap method
The analytic techniques presented in the previous
two subsections are only useful for either small
amounts of noise (norm inequalities method), or
(for now) solvable in simple cases, mostly when
each party applies two binary measurements (op-
erator inequalities method). For self-testing pro-
tocols which cannot be made robust with an-
alytic methods, one can resort to a numerical
method called the Swap method, introduced in
[BNS
+
15, YVB
+
14]. While its applicability is
still limited to simpler protocols due to computa-
tional resource requirements, it is responsible for
the majority of practically relevant robust self-
testing bounds.
The Swap method uses the Swap gate isometry
(see section 4.3) and makes use of the fidelity of
the extracted state as a figure of merit, as in defi-
nition 5. To get a lower bound on the fidelity, one
needs to minimise the fidelity between the state
of the output registers of the Swap gate and the
reference state, given that the input state to the
Swap gate provides the violation β. For exam-
ple, for two-qubit states we have seen in section
Accepted in Quantum 2020-23-08, click title to verify 26
Box 7.1: The NPA Hierarchy
In its most general form, the NPA hierarchy [NPA07, NPA08, PNA10] is a method to tackle
optimisation problems involving polynomials of non-commuting variables, and as a result is
suited to certain optimisation problems in quantum theory. Define optimisation problems of the
form
min
|ψi,M
a|x
,N
b|y
hψ|P({M
a|x
, N
b|y
}) |ψi subject to hψ|F
i
({M
a|x
, N
b|y
}) |ψi ≥ 0 ∀i , (56)
where P and F
i
are polynomials in the measurement operators {M
a|x
, N
b|y
}
a,x,b,y
and where
the dimension of the state and measurement can be potentially infinite. The NPA hierarchy
is a sequence of convex optimisation problems that provide increasingly better lower bounds
to the optimal solution of the above by relaxing the problem to a minimisation over a larger
set. Each of these relaxations can be solved via a corresponding semi-definite program [BV04].
Many problems in quantum information can be cast in the above form, particularly in the
device-independent setting where the state and measurements are unknown.
The NPA hierarchy works as follows. Consider a generic state and measurement operators
{|ψi, {M
a|x
}, {N
b|y
}}. Then, define sets S
k
(each corresponding to a level of the hierarchy)
comprised of the identity operator and all (non-commuting) products of operators M
a|x
, N
b|y
up
to degree k; e.g. S
1
= { } ∪
a,x
{M
a|x
} ∪
b,y
{N
b|y
}, S
k+1
= S
k
∪
i,j
{S
(i)
k
S
(j)
1
}, where S
(i)
k
is the
i
th
element of S
k
. Define the moment matrix of order k, Γ
k
, by Γ
k
i,j
= hψ|S
(i)†
k
S
(j)
k
|ψi. For any
state and measurements {|ψi, {M
a|x
}, {N
b|y
}}, the matrix Γ
k
is Hermitian positive semidefinite
and satisfies some linear constraints given by the orthogonality conditions of the measurement
operators. One can thus tackle optimisation problems of the form (56) by minimising the
corresponding elements of the matrix Γ
k
, under linear constraints on Γ
k
and Γ
k
0. Such a
problem is an instance of a semidefinite program which, via duality theorems, can be run to
obtain certified lower bounds to the optimal solution.
4, equation (36) that this fidelity is given by
F (ρ
swap
,
ψ
0
) =
X
i,j,k,l∈{0,1}
c
kl
ij
hψ|
ˆ
f
†
kl
ˆ
f
ij
|ψi,
where ρ
swap
= tr
AB
[Φ[ρ]], and c
kl
ij
=
hkl|ψ
0
ihψ
0
|iji. Note that from the definition of
f
ij
, the above is equal to hψ|P({A
x
a
, B
y
b
}) |ψi,
where P is a polynomial in the measurement op-
erators. As a result, lower bounds to the mini-
mum fidelity (subject to a Bell inequality viola-
tion) can be found numerically via a correspond-
ing semi-definite program defined by the NPA hi-
erarchy; see box 7.1 for more details.
The first applications of the method are given
in [YVB
+
14, BNS
+
15], which involve the follow-
ing self-testing results: the self-test of the singlet
state and Bob’s measurements from the CHSH
inequality, the self-test of the singlet state from
the Mayers-Yao criterion, the self-test of partially
entangled pair of qubits from the tilted CHSH in-
equality, the self-test of a pure two qutrit state
maximally violating CGLMP [CGL
+
02] inequal-
ity and the self-test of entangling measurements.
Subsequently the method has been used to devise
robust self-tests of the maximally entangled pair
of qutrits [SAT
+
17], the whole family of pure en-
tangled qutrit states [WPD
+
18], the three qubit
W-state [WCY
+
14, PVN14], three- and four-
qubit GHZ states and the four-qubit linear cluster
state [PVN14], a family of tripartite pure states,
including the W-state from only marginal infor-
mation [LCH
+
18], and a tensor product of two
singlet states [WBMS16]. It has also been used to
compare the performance of different types of self-
tests of the singlet state, presented in [WWS16].
7.1.5 Algebraic method
In section 5.2.2 we discussed self-testing through
the ‘algebraisation’ of the winning strategies in
nonlocal games. Let us briefly recall that the
crux of the method is associating an algebraic
Accepted in Quantum 2020-23-08, click title to verify 27
2 2.2 2.4 2.6 2.8
0.5
0.6
0.7
0.8
0.9
1
CHSH violation
Fidelity lower bound
[Kan16]
[BLM
+
09]
[BNS
+
15]
Figure 7: Lower bounds on the self-tested fidelity with
the maximally entangled pair of qubits as a function of
the observed violation of the CHSH inequality for three
methods. A trivial lower bound on the fidelity is 0.5,
achievable with the separable state |00i. Finding the
optimal curve remains as an open question. The im-
possibility to obtain a fidelity higher than 0.5 for ev-
ery CHSH violation > 2 is proven in [CKS19]. The
proof is constructive: there exists a state ρ providing
the CHSH violation of ≈ 2.0014, nevertheless there is
no local channel Λ such that fidelity between Λ(ρ) and
Φ
+
is higher than 0.5.
invariant, called the solution group, to each lin-
ear constraint system (LCS) nonlocal game. The
rules of the nonlocal game allow one to define an
abstract group whose representations correspond
to the winning quantum strategy of the game.
The correspondence between the group represen-
tations and winning strategies then allows for the
use of techniques from group theory to prove self-
testing statements.
In [CS17b] this reasoning is taken one step fur-
ther: a quantum strategy winning the generalised
magic square game with high probability allows
to extract an approximate representation of the
solution group, or equivalently, a mapping be-
tween the group elements and unitary operators
which is approximately a homomorphism. The
closeness between the approximate and the ex-
act representation is then used to make a robust
self-testing statement.
The most important ingredient for construct-
ing robust self-tests in this way is the sta-
bility theorem for approximate representations
from [GH17]. It states that for any approxi-
mate n-dimensional representation f of a finite
group G = {g
i
}
i
there exists an exact unitary
m-dimensional representation h such that the
Hilbert-Schmidt distance between f (g
i
) and h(g
i
)
is small and m is close to n. The distance between
these two representations is related to the score
the physical strategy gained in the LCS game un-
der consideration. The full robustness statement
is obtained through the use of the van Kampen
diagrams [Kam33].
7.2 Robust certification of large entanglement
In this section we discuss few contributions deal-
ing with robustly certifying large amounts of en-
tanglement without explicitly stating any self-
testing result. There are two main reasons why
such results merit attention in a review like this.
The first is that for many purposes they could be
used instead of robust self-testing protocols and
moreover the self-testing statement is implicitly
present for noiseless correlations. The second rea-
son is the possible influence they could have on
future approaches to robust self-testing.
A difference between robustness and noise tol-
erance when it comes to self-testing a tensor prod-
uct of n entangled pairs |ψ
0
i
⊗n
is emphasised in
works [AFB19] and [AFY18]. The known self-
testing protocols are robust in the sense that any
strategy producing correlations that are -close to
the ideal ones must use a state which is f(, n)-
close to |ψ
0
i
⊗n
, where f(, n) scales as an
b
c
for
appropriate constants a,b and c. This is, how-
ever, not the same as noise-tolerance since noisy
source producing the state ρ
⊗n
, where ρ is -close
to |ψ
0
i is not f(, n)-close to |ψ
0
i
⊗n
. The fidelity
of such state with |ψ
0
i
⊗n
drops exponentially with
n, so there is very little hope to make any non-
trivial self-testing statement about such highly
entangled state. Instead of self-testing
φ
+
⊗n
[AFY18] designs a one-shot test which is able to
certify states whose entanglement of formation
[BDSW96] is Ω(n). This certification method is
noise-tolerant in the sense that the states ρ
⊗n
are able to pass the test with high probability.
A method to bound the one-shot distillable en-
tanglement [BD10] of the states produced by an
uncharacterised source is presented in [AFB19].
The protocol is operationally useful since not all
entanglement is consumed for certification. Both
these results are implicit self-tests since the max-
imal score in the introduced games implies that
the state produced by a source must be
φ
+
⊗n
.
Accepted in Quantum 2020-23-08, click title to verify 28
8 Self-testing of measurements
In many cases the correlations which self-test
a quantum state also self-test the applied mea-
surements. As a result, many of the state self-
testing results presented in the previous sections
are accompanied by a corresponding statement
for the measurements. In this section we give
an overview of such results. In section 8.1 we
review the known self-testing results for various
sets of measurements, and in section 8.2 we dis-
cuss the different methods that have been used
to achieve these results. We end the section with
an overview of robustness techniques in measure-
ment self-testing in section 8.3.
8.1 Measurement self-testing results
8.1.1 Qubit measurements
The simplest set of incompatible qubit measure-
ments is given by a pair of Pauli observables σ
x
and σ
z
. Self-testing of these measurements (to-
gether with their rotated versions (σ
x
± σ
z
)/
√
2
for the other party) can be achieved through the
maximum violation of the CHSH Bell inequal-
ity (see Section 4) or related self-tests. Such
self-testing statements can be found in [MYS12,
Kan17, BNS
+
15, WCY
+
14]. Self-testing of the
set of local observables {σ
x
, σ
z
, (σ
x
±σ
z
)/
√
2} can
be achieved through the so called ‘Mayers-Yao’
self test and its generalisations [MY04, McK14,
MYS12]. A method to self-test large sets of
qubit observables that are equally spaced on
the equator of the Bloch sphere was given in
[ŠASA16] based on the maximum violation of the
chained Bell inequalities [Pea70, BC90]. A pro-
tocol for self-testing an arbitrary measurement
from the real plane of the Bloch sphere is given
in [McK16a]. Self-testing of pairs of observables
of the form cos µ σ
x
± sin µ σ
z
is given in [BP15]
and [Kan17] through the maximal violation of the
tilted or weighted CHSH inequalities [AMP12,
LLP10]. The first self-testing of the set of three
local Pauli observables {σ
x
, σ
y
, σ
z
} first appeared
in [MM11], using the ‘phase kick-back’ (see sec-
tion 8.2.1) method and a generalised definition of
self-testing to deal with the issue of complex con-
jugation. Other examples of such self-tests can
be found in [ABB
+
17, Kan17, WKB
+
19].
8.1.2 Qudit measurements
Self-testing results for measurements of dimen-
sion larger than two are much less common.
The only self-test of mutually unbiased bases in
a prime dimension higher that 2 was given in
[KŠT
+
19] for dimension d = 3. Self-testing of the
Bell state measurement was first achieved analyt-
ically in [RKB18, BSS18] (see section 8.2.4 for an
outline of the method). Self-tests of sets of mea-
surements in high dimension can be achieved us-
ing the same techniques as in parallel self-testing
of states (section 5.3). In this way, n-fold ten-
sor products of the measurements {σ
x
, σ
z
} and
{σ
x
, σ
y
, σ
z
} in dimension 2
n
have been achieved
[BŠCA18b, CGJV17, WBMS16, McK17, Col17,
NV17, CN16, BKM19, KM18, CS17b, NV18].
8.1.3 Non-projective measurements
Although definition 2 of measurement self-testing
assumes that the physical measurements are pro-
jective, one can nevertheless aim to prove that
on the support of the reduced state of the self-
tested state they act as some desired POVM.
More specifically, suppose we have self-tested the
reference state |ψ
0
i. Since the trace is invari-
ant under isometry maps, the correlations can be
written
p(a, b|x, y) = tr
|ψ
0
ihψ
0
|
A
0
B
0
⊗ σ
¯
A
¯
B
M
a|x
⊗ N
b|y
,
where the local measurements are projective and
may act on both the primed and bared spaces.
Taking the trace over the barred spaces we have
p(a, b|x, y) = tr
h
|ψ
0
ihψ
0
|
˜
M
a,b|x,y
i
, (57)
where
˜
M
a,b|x,y
= tr
¯
A
¯
B
A
0
B
0
⊗ σ
¯
A
¯
B
M
a|x
⊗ N
b|y
. (58)
To ‘self-test’ non-projective measurements, one
aims to show that
˜
M
a,b|x,y
= M
0
a|x
⊗ N
0
b|y
, where
now the reference measurements can be non-
projective. Essentially, one is self-testing a Stine-
spring dilation [Sti55] of the non-projective mea-
surement.
In this manner, a self-test of the ‘tetrahe-
dral’ qubit POVM first appeared in [APVW16],
with rigorous proofs appearing later in [ABB
+
17]
and [ABDC18], and an experimental demon-
stration presented in [SMN
+
20]. These results
were proven using the method of ‘post-hoc’ self-
testing, that we describe in 8.2.3. To self-test
Accepted in Quantum 2020-23-08, click title to verify 29
measurements which are neither projective nor
rank-one POVMs [WBSS18] use the approach de-
veloped by the same authors for the self-testing of
quantum channels, described here in section 9.1.
8.2 Methods in measurement self-testing
In this section we outline some of the methods
that have been used to prove measurements self-
testing statements.
8.2.1 Phase kickback method for self-testing com-
plex measurements
As mentioned in section 3.7, definition 2 is not
suitable for self-testing complex-valued measure-
ment operators. Take for example the prob-
lem of self-testing
φ
+
, the maximally entan-
gled state of dimension 2, and {σ
z
, σ
x
, σ
y
}, the
three Pauli observables for say Alice. In sec-
tion 4, we have seen how one can self-test the
state
φ
+
and {σ
x
, σ
z
}. Here, the issue of com-
plex conjugation is not a problem since there
exists a local basis in which the measurements
and state are both real. However, there is no
local basis in which the observables {σ
z
, σ
x
, σ
y
}
are all real. Thus, we have two distinct pos-
sibilities for Alice’s measurements, {σ
z
, σ
x
, σ
y
}
and {σ
∗
z
, σ
∗
x
, σ
∗
y
}={σ
z
, σ
x
, −σ
y
}, both of which are
compatible with the observed correlations.
A natural question to ask is, given this uncer-
tainty, what is the strongest possible self-testing
statement that one could hope to prove? This
question was first tackled by [MM11], see also
[CGS17, BŠCA18b]. The basic idea is as fol-
lows. Consider a self-testing scenario in which
Alice has (at least) three measurements given by
the observables A
0
, A
1
, A
2
. Take a known self-
testing protocol for the state
φ
+
and observ-
ables {σ
x
, σ
z
} for Alice. Use this self-testing pro-
tocol three times for the pairs {A
0
, A
1
}, {A
0
, A
2
},
{A
1
, A
2
}, introducing new measurements for Bob
and Alice if necessary. Since this proves that each
pair A
i
, A
j
anti-commute, one proves that the ob-
servables {A
0
, A
1
, A
2
} pairwise anti-commute and
should essentially be {σ
z
, σ
x
, σ
y
} or {σ
z
, σ
x
, −σ
y
}.
More precisely, one introduces a pair or local an-
cillas |00i
A
00
A
0
for Alice and another pair |00i
B
00
B
0
for Bob and proves the existence of an isometry
Φ such that
Φ[|ψi] =
φ
+
A
0
B
0
⊗ |ξi
Φ[A
0
|ψi] = (σ
z
⊗
φ
+
A
0
B
0
) ⊗ |ξi
Φ[A
1
|ψi] = (σ
x
⊗
φ
+
A
0
B
0
) ⊗ |ξi
Φ[A
2
|ψi] = (σ
y
⊗
φ
+
A
0
B
0
) ⊗ σ
A
0
z
|ξi (59)
where the state |ξi has the form
|ξi = |ξ
0
i
AB
⊗ |00i
A
00
B
00
+ |ξ
1
i
AB
⊗ |11i
A
00
B
00
. (60)
In (59) the additional σ
z
measurement on the
junk state acts as an effective ‘controlled con-
jugation’ for the measurement of σ
y
on
φ
+
,
where the probability to perform the conjuga-
tion is given by hξ
1
|ξ
1
i, which remains unknown.
Note that if we consider only the space H
A
⊗H
B
then the action of A
2
is to perform some unknown
convex combination of σ
y
and −σ
y
, as expected.
Similar statements can be proven for Bob, where
the register B
0
acts as a control for a possible con-
jugation of his measurements. Note that given
the form of (60), Alice and Bob will conjugate
their measurement operators in a correlated fash-
ion, as required from (14). The isometry, intro-
duced in [MM11] and later used in [JMS20] to
prove the above self-testing statement is an ex-
tension to the Swap isometry (see figure 4) intro-
duced in section 4. The full isometry consists of
the regular Swap isometry, followed by two extra
‘phase kickback’ controlled unitaries; see figure 8.
8.2.2 Self-testing measurements based on com-
mutation
The majority of self-testing protocols first prove
self-testing of the state, and then move to self-test
the form of measurements on the support of the
self-tested state. An alternative approach for bi-
nary observables was used already in [PR92] and
revived in [Kan17]. In this approach the violation
of a Bell inequality is directly related to the com-
mutation properties of measurement observables
without the need to prove a statement of the form
of definition 2. In the two input, two output sce-
nario the figure of merit (for Alice’s observables)
is directly relatable to the maximal violation of
the CHSH inequality and is given by
t
01
=
1
2
tr(|[A
0
, A
1
]|ρ
A
), (61)
where ρ
A
= tr
B
ρ
AB
is Alice’s reduced state of the
physical state. The maximal CHSH violation im-
plies t
01
= 1, which further can be used to infer
Accepted in Quantum 2020-23-08, click title to verify 30
|0i
A
0
|0i
A
00
|0i
B
00
|0i
B
0
|ψi
AB
H
H
Z
A
Z
B
H
H
X
A
X
B
H
H
iY
A
X
A
iY
B
X
B
H
H
φ
+
A
0
B
0
Figure 8: Swap gate with a phase kick back at the end of the circuit . The Y operators are constructed from the
additional measurement operators in a similar fashion to (32)
anticommutativity of A
0
and A
1
on ρ
A
. Beyond
CHSH, it is also proven that the maximal vio-
lation of the Mermin-Ardehali-Belinskii-Klyshko
inequalities [Mer90a, Ard92, BK93] implies the
parties use anticommuting observables to achieve
the maximal violation.
The method is also applied to self-test a set
of three mutually anti-commuting observables.
Note that there exists a basis in which any two an-
ticommuting observables A
0
and A
1
can be writ-
ten
A
0
= σ
z
⊗ , A
1
= σ
x
⊗ . (62)
If an inequality involving three observables
A
0
, A
1
, A
2
can be used to certify t
01
= t
02
= t
12
=
1 then besides relation (62) the following relation
can be extracted
A
2
= σ
y
⊗ A
Y
, (63)
where A
Y
is a Hermitian ±1-eigenvalue operator.
Hence, when measuring A
2
the measurement re-
sult for A
Y
on the ‘junk’ Hilbert spaces takes care
of the complex conjugation issue in the same way
as (59).
Once the form (62) is extracted for measure-
ment observables of all parties it is possible to
also make statements about the underlying state.
Furthermore, in [Kan17] this method is useful to
certify observables which are not anticommuting
but maximally violate weighted CHSH inequal-
ities introduced in [LLP10]. Going beyond bi-
nary measurements, the extension of this method
is applied also in [KŠT
+
19] to self-test mutually
unbiased bases in the dimension d = 3.
8.2.3 Post-hoc self-testing of measurements
Once a state and sufficiently many measurements
have been self-tested, further measurements can
often be self-tested ‘for free’ through a method
that we call post-hoc self-testing. As an exam-
ple, the maximal violation of the CHSH inequal-
ity assures that up to an isometry the shared
state is
φ
+
, Alice’s measurement observables
are σ
z
and σ
x
, and Bob’s measurement observ-
ables (σ
z
± σ
x
)/
√
2. The self-testing protocol
can be extended to certify any other real qubit
observable applied by either Alice or Bob. As-
sume Bob uses another measurement observable
B
2
which we want to self-test as cos θσ
x
+ sin θσ
z
on his half of the maximally entangled state.
If Bob performs this measurement, we will ob-
serve the correlations hψ|A
0
B
2
|ψi = cos θ and
hψ|A
1
B
2
|ψi = sin θ. Now, the maximal viola-
tion of the CHSH inequality implies that A
0
|ψi
and A
1
|ψi are orthogonal. If we take these
two states to be the first two states in an or-
thonormal basis of the full space, it must be that
B
2
|ψi = cos θA
0
|ψi + sin θA
1
|ψi. Applying the
isometry to this, we have
Φ[B
2
|ψi] =Φ [cos θA
0
|ψi + sin θA
1
|ψi]
=(cos θσ
z
⊗ + sin θσ
x
⊗
φ
+
) ⊗ |ξi
=( ⊗ cos θσ
z
+ ⊗ sin θσ
x
φ
+
) ⊗ |ξi,
as required, where we have used the property M⊗
φ
+
= ⊗ M
T
φ
+
.
This technique can be understood from the per-
spective of measurement tomography. Given a
set of linearly independent pure states that are
tomographically complete on some subspace, one
can infer the form of any measurement in this
Accepted in Quantum 2020-23-08, click title to verify 31
subspace from the statistics of measurement out-
comes on the set of states. Given the CHSH
self-test, we know that conditioned on Alice’s in-
put and output, the reduced states of Bob up
to a unitary transformation are π
B
a|x
⊗
B
, where
π
a|x
= |0ih0|, |1ih1|, |+ih+|, |−ih−| depending on
the value of x, a. These four states are informa-
tionally complete for real qubit measurements,
and can thus be used to infer further such mea-
surements for Bob when interpreted as states in
a measurement tomography protocol. The tech-
nique can be applied in a similar way for higher-
dimensional self-testing protocols and to the post-
hoc self-testing of complex measurements. The
first time such an approach was used was for
self-testing real measurements applied on a graph
state in [McK16a], expanding the protocol for
self-testing graph states from [McK14]. This
technique has also been particularly useful to self-
test non-projective measurements (see 8.1.3).
8.2.4 Self-testing of entangling measurements
An entangling measurement is one whose mea-
surement operators are non-separable with re-
spect to some bipartition of the Hilbert
space. Two recent works [RKB18, BSS18]
have presented analytic methods to robustly
self-test the Bell state measurement (BSM),
the entangling measurement whose eigenvectors
are the four maximally entangled Bell states
{|φ
+
ihφ
+
|, |φ
−
ihφ
−
|, |ψ
+
ihψ
+
|, |ψ
−
ihψ
−
|}. Here
the reference scenario is an entanglement swap-
ping protocol: Bob possesses two particles, one
maximally entangled with Alice’s particle and the
other with Carmela’s. Bob performs the BSM
on his two particles, and depending on his out-
come projects the particles of Alice and Carmela
onto one of the Bell states. These four Bell states
can be self-tested by maximally violating the four
different CHSH inequalities (mutually related by
relabelling) conditioned on the outcome of Bob.
The idea for self-testing is simple: if Alice and
Carmela maximally violate all CHSH inequali-
ties then it must be that Bob’s measurement is
the BSM. Importantly, to maximally violate each
of the four different Bell inequalities, Alice and
Carmela use the same measurements.
In [RKB18, BSS18], the entanglement swap-
ping scenario is used to self-test the BSM using
the notion of measurement self-testing via simu-
lation described in section 3.7.2. Here, one nec-
essarily needs to identify some well defined local
Hilbert spaces for Bob in order to define entangle-
ment. This is achieved by assuming that there are
two independent sources between Alice and Bob,
and between Bob and Carmela. Entanglement is
then defined with respect to the Hilbert spaces
of these two sources. A self-testing protocol for
entangling measurements whose eigenvectors are
partially entangled pairs of qubits, or GHZ states
is also presented in [RKB18]. In [BSS18] the tech-
niques for self-testing quantum channels is used
to self-test the BSM. For more details on these
techniques see section 9.1.
It is also worth mentioning here that in
[RHC
+
11], a protocol to device-independently
certify the existence of an entangling measure-
ment is given, which was later shown to be ro-
bust in [BNS
+
15] via the use of the numerical
Swap method (section 7.1.4). Both these works
however only prove the existence of some en-
tangling measurement but do not provide a self-
testing statement for a particular measurement.
In the context of witnessing irreducible dimen-
sion a way to certify entangling measurements is
also presented in [CCBS17] alongside with a fig-
ure of merit quantifying how entangled the mea-
surement operators are.
8.3 Robust measurement self-testing
In this section we give an overview of the ap-
proaches to robust self-testing of measurements.
As we saw in section 3.6 there are a number of
valid figures of merit one could consider when ro-
bustly self-testing a state. Given the increased
complexity of measurements compared to states,
the self-testing of measurements is even more di-
versified. In what follows we discuss a few ap-
proaches to quantify how close a physical mea-
surement {M
x
} is to some reference measurement
{M
0
x
}.
The straightforward approach, in accordance
with definition 4 uses the same methods as
in the self-testing of states. In robust
self-testing of states one defines a figure of
merit that captures the closeness between the
states Φ[ρ
AB
] and |ψ
0
ihψ
0
|
A
0
B
0
⊗ σ
¯
A
¯
B
. For
measurement self-testing, one can simply use
the same figure of merit between the sub-
normalised states Φ
h
M
a|x
⊗ ρ
AB
M
†
a|x
⊗
i
and
M
0
a|x
⊗ |ψ
0
ihψ
0
|
A
0
B
0
M
0
†
a|x
⊗
⊗σ
¯
A
¯
B
. This will
Accepted in Quantum 2020-23-08, click title to verify 32
mean that there will be a different value asso-
ciated to each of the measurement operators; a
single figure of merit can be obtained by, for ex-
ample, taking the average or maximum of these
values. Such an approach is used, for example, in
[MYS12, McK14, BP15, ŠASA16].
The Swap method [YVB
+
14, BNS
+
15] can also
be used to define a figure of merit for robust mea-
surement self-testing. Taking the CHSH example,
to estimate the closeness of Alice’s measurements
to the reference measurements, the Swap gate is
applied only on her system (i.e. only Alice’s local
branch of the Swap gate is used) and the ancilla
is initiated in one of the eigenstates
ϕ
A
0
i
±
E
of her
reference observable A
0
i
. In the ideal case, if Al-
ice measures the reference observable A
0
i
after the
Swap gate is applied she will deterministically ob-
tain the outcome ±1. The probability that this
measurement gives the result +1 is then used as
a figure of merit to asses the closeness of the mea-
surement, however it is not proven to give a dis-
tance measure. As with the estimation of the
fidelity with the reference state, one can use the
NPA hierarchy to lower bound this quantity with
respect to the given CHSH violation.
A different figure of merit, analogous to the
notion of state extractability (see section 3.6),
was suggested in different self-testing contexts
[TKV
+
18, RKB18, BSS18, TSV
+
20, WBSS18].
If D is a suitably chosen distance measure on the
set of measurements, one can define the distance
D between the physical measurement M
a|x
and
the reference one M
0
a|x
as
D(M
a|x
, M
0
a|x
) =
1
c
a
max
Λ
X
a
D(Λ(M
a|x
), M
0
a|x
),
(64)
where c
a
is a normalisation factor and the max-
imisation is taken over all completely positive
and unital maps Λ : L(H
A
) → L(H
A
0
). De-
pending on the type of reference measurements
D can be chosen to be the overlap (as in [RKB18,
TKV
+
18, TSV
+
20]), Uhlmann fidelity (as in
[BSS18, WBSS18]) or any other distance mea-
sure. While the physical state ρ does not explic-
itly appear in (64), the map Λ has to depend in
some way on it. In [WBSS18, BSS18] measure-
ments are self-tested through their action on the
maximally entangled pair of qudits and the state
appears explicitly in the distance measure.
9 Extensions of self-testing to other
scenarios
In this section we cover three extensions to the
standard scenario of self-testing. In section 9.1 we
cover works that self-test the action of a quantum
gate in a device-independent manner. In section
9.2 we focus on the so-called semi-device inde-
pendent approaches. In section 9.2.3 we focus on
self-testing via contextuality.
9.1 Self-testing of quantum gates and circuits
The paradigm of self-testing can be useful in sce-
narios going beyond the certification of states and
measurements. Anticipating usefulness in the
certification of devices for quantum computing,
one may ask if it is possible to certify quantum
gates, i.e. unitary transformations. First answers
to this question came already in the early days
of self-testing with two contributions devoted to
the task of self-testing of quantum gates or quan-
tum circuits, [vDMMS07] and [MMMO06]. Al-
though [vDMMS07] has the phrase ‘self-testing’
in its title, it does not correspond to the fully
device-independent scenario; the certification re-
lies on several assumptions such as knowledge of
the dimension of the system, which shifts it to the
landscape of semi-device-independent scenarios.
The first protocol providing a recipe to self-
test quantum gates acting on an arbitrary num-
ber of qubits is presented in [MMMO06]. Denote
the physical implementation of the gates Alice
and Bob use with G
A
and G
B
. For the protocol
to work, Alice and Bob must have access to the
same gate, that is, G
A
= G
B
. The core of the
protocol is the Mayers-Yao self-test of the max-
imally entangled pair of qubits. To self-test a
one-qubit unitary gate G
0
A
0
acting on her system,
Alice has to share a maximally entangled pair of
qubits with Bob. As usual, the state shared be-
tween Alice and Bob is |ψi and they perform mea-
surements {M
a|x
} and {N
b|y
}. The aim is to show
that there is a local isometry Φ such that
Φ ⊗
P
h
G
A
M
a|x
⊗ |ψi
ABP
i
=
= G
0
A
0
M
0
a|x
⊗
φ
+
A
0
B
0
⊗ |ξi
¯
A
¯
BP
where {M
0
a|x
} are the reference measurements for
the Mayers-Yao self-test. The protocol consists
of three parts:
Accepted in Quantum 2020-23-08, click title to verify 33
• The Mayers-Yao self-test on the input state
|ψi,
• The Mayers-Yao self-test on the output state
G
A
⊗ G
B
|ψi,
• A check that G
A
⊗
B
|ψi
AB
reproduces the
statistics of G
0
A
0
⊗
B
0
φ
+
A
0
B
0
with respect
to the Mayers-Yao measurements.
The first two steps serve for self-testing the un-
derlying state and ensure that G is a unitary
gate (at this step potentially the identity gate).
The third step can be seen as a tomography of
G, since the measurements and state are already
self-tested in the first two steps. Note that this
means one can only self-test gates having real co-
efficients with respect to the self-tested measure-
ments. The method is also extended to many-
qubit gates. Each of Alice’s qubit on which a gate
acts is maximally entangled with another qubit of
Bob, and the tensor product structure of Alice’s
and Bob’s Hilbert spaces is assumed. The pro-
cedure repeats as in the case of one-qubit gates,
with three steps involving the self-test of the in-
put and output states which are now tensor prod-
ucts of many maximally entangled pairs of qubits
and tomography of the corresponding multi-qubit
gate. By using this method one can self-test the
whole quantum circuit by self-testing each gate
in sequence according to the recipe given above.
This self-test is also proven to be robust.
Another self-test of quantum gates with sim-
pler structure and significantly better robustness
bounds is given in [SBWS18]. It is more general
than [MMMO06] since it provides a framework to
lower bound the fidelity with an arbitrary quan-
tum channel Γ
0
. For Alice to self-test the channel
Γ
0
she again has to share a maximally entangled
pair of qubits with Bob, but now Bob does not
perform any channel to his system. Let the phys-
ical implementation of the channel be denoted as
Γ. The aim is to find the fidelity between the
reference channel Γ
0
and the physical one Γ. The
protocol consists of two steps:
• The self-test of the input state |ψi. The
result provides a lower bound to the input
fidelity F
i
between Λ
i
A
⊗ Λ
B
|ψi and
φ
+
,
where Λ
i
A
and Λ
B
are the CPTP maps as
explained in 3.7.2.
• The appropriate self-test which finds a lower
bound of the output fidelity F
o
between the
state Λ
o
A
⊗ Λ
B
(Γ
A
⊗
B
|ψi
AB
) and the ref-
erence output state Γ
0
A
0
⊗
B
0
φ
+
A
0
B
0
.
The fidelity of the physical channel to the ref-
erence channel is proven to be lower bounded
by cos (arccos(F
i
) + arccos(F
o
)). The main chal-
lenge is to find the appropriate self-test necessary
for the second step. The paper gives the solu-
tion for unitary channels (i.e. quantum gates),
generalises the protocol for many-qubit channels
and provides the explicit solution for arbitrary
two-qubit controlled gate of the form CU
ϕ
=
|0ih0| ⊗ + |1ih1| ⊗ e
−iϕσ
x
. Since such gates
are necessary and sufficient for universal quantum
computing (together with a set of single qubit
gates) the toolkit represents an important contri-
bution to the self-testing of all the building-blocks
of a quantum computer.
Aside from this, the paper [SBWS18] is also
valuable for two contributions independent of
quantum channel self-testing. One is a method
to self-test multipartite states, described in 6.2
and the other a technique useful for robust self-
testing, described in 7.1.3. Furthermore, by defin-
ing the gates in terms of their Krauss represen-
tation, the techniques from [SBWS18] have been
generalised in [WBSS18] to allow for self-testing
of measurements other than rank-one POVMs.
9.2 Semi-device-independent scenarios
A number of works have investigated exten-
sions of self-testing to so-called semi-device-
independent (SDI) scenarios. In the device-
independent scenario, all devices are treated as
black boxes and one hence imposes minimal as-
sumptions on the states and measurements. In
the SDI scenario, some additional assumptions
are added, without assuming a full characteri-
sation of the entire set-up. As such, the SDI
scenario can be seen as a weaker version of the
DI scenario, intermediate between the scenarios
of full device independence and full characteri-
sation. Moving to the SDI scenario can be ad-
vantageous for at least three reasons. First, the
additional assumptions can overcome some of the
mathematical difficulties of the DI scenario and
make statements easier to prove and results more
tolerant to noise; second, for some scenarios it
may actually be necessary to move to SDI sce-
nario in order to make any non-trivial statements
(see 9.2.2), and third, the additional assumptions
Accepted in Quantum 2020-23-08, click title to verify 34
may be very natural given a particular experi-
mental set-up or level of trust in some devices. In
this section we give an overview of three exten-
sions of self-testing to the SDI scenario, namely
one-sided device-independent self-testing, com-
monly known as the EPR steering scenario (sec-
tion 9.2.1), self-testing in prepare-and-measure
scenarios (section 9.2.2), and self-testing based
on noncontextuality inequalities (section 9.2.3).
9.2.1 One sided device-independent self-testing
(EPR steering)
The one-sided device-independent scenario (also
commonly referred to as the EPR steering sce-
nario), is equivalent to the standard self-testing
scenario, with the additional assumption that
there is one trusted party (here Bob) whose de-
vice is fully characterised, that is, his measure-
ment operators are known. Thus, Bob is able to
apply any quantum measurement and can in prin-
ciple perform quantum state tomography of his
half of the state. Alice, as in the self-testing sce-
nario, receives classical input x to her device and
outputs classical output a. The subnormalised
state of Bob conditioned on Alice’s input x and
output a is given by
σ
a|x
= tr
A
h
M
a|x
⊗ ρ
AB
i
(65)
The set {σ
a|x
}
a,x
is called an assemblage. It is
said that the assemblage {σ
a|x
}
a,x
admits a local
hidden state model (LHS) if it admits a decom-
position
σ
a|x
=
Z
λ
dλ q(λ)p
a|x,λ
ρ
λ
, ∀a, x, (66)
where q(λ) is a normalised probability density
and ρ
λ
is a normalised density operator acting
on the local Hilbert space of Bob. If the assem-
blage {σ
a|x
}
a,x
is incompatible with a LHS model
one says that it demonstrates steering. The exis-
tence of a LHS model can be refuted by violation
of steering inequalities, which take into account
the correlations between Alice’s outputs and the
outputs of known measurements performed by
Bob. Another way to prove that the assemblage
{σ
a|x
}
a,x
demonstrates steering is by using simple
SDP optimisations [WJD07],[CS17a],[UCNG19].
The decomposition (66) captures the types of
assemblages that Bob can see if the two parties do
not share any entanglement. Thus, a violation of
Figure 9: Semi-device-independent scenarios. (a) The
one-sided device-independent scenario. One of the par-
ties (here Bob) is assumed to have a trusted measure-
ment device (i.e. his measurement operators are known).
(b) The prepare-and-measure scenario. Alice sends a
quantum state to Bob, conditioned on her input.
(66) demonstrates that the shared state must be
entangled. A natural question is in which cases
can we go beyond witnessing entanglement and
recover the shared state ρ
AB
. This task was in-
troduced in [ŠH16] and [GWK17] under the name
one-sided device-independent (1SDI) self-testing.
In [ŠH16] the authors are mostly interested in the
robustness of 1SDI self-testing and how it com-
pares to the robustness of standard self-testing,
while the authors of [GKW15] are also interested
in the application to delegated quantum com-
puting protocols (see Section 10.4). Two types
of 1SDI self-testing are introduced: correlation
based, which draws conclusions only from the vi-
olations of steering inequalities, and assemblage-
based, which works with the full assemblage. An
interesting conclusion is that in the case of the
simplest self-test of the maximally entangled pair
of qubits, the asymptotic behaviour of the self-
tested fidelity as a function of noise is the same in
both the 1SDI and DI scenarios. How general this
statement is remains as an open question. The
1SDI scenario is also very useful for self-testing a
tensor product of many EPR pairs. In the stan-
dard self-testing of such states the main difficulty
is establishing a tensor product structure, while
in 1SDI scenario this comes for free due to the
fact that Bob’s device is characterised. Numer-
ical techniques, similar to the Swap method, for
robust self-testing in the 1SDI scenario were also
presented in work [ŠH16]. 1SDI self-testing of all
pure two-qubit states is presented in [GBD
+
18].
Accepted in Quantum 2020-23-08, click title to verify 35
9.2.2 Self-testing in the prepare-and-measure sce-
nario
A recent series of works have adapted the self-
testing scenario to the prepare-and-measure sce-
nario. Here, Alice sends one of a number of states
|ψ
x
i, labelled by x, to Bob, who measures {M
b|y
}
conditioned on input y and obtains outcome b.
The statistics of the experiment are therefore
given by
p(b|x, y) = tr[|ψ
x
ihψ
x
|M
b|y
]. (67)
In analogy to the case of self-testing entangled
states and measurements, one aims to infer from
the statistics that the preparations and measure-
ments {|ψ
x
i, M
b|y
} are equal to some reference
set {|ψ
0
x
i, M
0
b|y
} up to some unknown isometry.
In contrast to the Bell scenario, non-trivial
statements can only be made if one places ad-
ditional assumptions on the experiment. To see
this, note that any statistics p(b|x, y) can be re-
produced by sending the label x to Bob, i.e. with
the preparations |ψ
x
i = |xi, and Bob simply out-
putting b with probability p(b|x, y). Thus, self-
testing any set of preparations that are not di-
agonal in the same basis is impossible without
some additional assumptions. As is common in
the prepare-and-measure scenario, a number of
recent works [TKV
+
18, TSV
+
20, MP19, FK19]
overcome this by assuming an upper bound on
the Hilbert space dimension of the preparations
and measurements. A similar assumption has
also been studied in the Bell scenario [BLM
+
09,
GBS16] where the state and measurements are
assumed to be qubit systems, making self-testing
statements significantly easier to prove.
A convenient figure of merit in prepare-and-
measure scenario is the success in a game called
a random access code (RAC) [ANTSV99, Nay99,
THMB15]. In a n
d
→ 1 RAC, Alice receives n
dits x = (x
1
, x
2
, ··· , x
n
) and Bob receives y =
1, 2, ··· , n. The aim is to maximise the average
probability that Bob correctly guesses the input
bit x
y
, i.e. to maximise the expression
A
n
d
→1
=
1
2d
n
X
x,y
p(b = x
y
|x, y). (68)
In [TKV
+
18], the authors study as a figure of
merit the 2
2
→ 1 RAC. It is proven that the value
A
2
2
→1
can be used to perform robust self-testing
of the preparations {|0i, |1i, |+i, |−i} and mea-
surements given by the qubit observables (σ
x
±
σ
z
)/
√
2 under the assumption of qubit prepara-
tions and measurements, using a technique based
on that in [Kan16] (similar statements also ap-
peared in [WLP13, BQB14] in the context of SDI
quantum key distribution protocols and dimen-
sion witnesses). This is then generalised to a
self-test of any pair of non-commuting qubit ob-
servables by adding an input-dependent bias to
(68). The authors also use A
2
2
→1
to self-test a
non-trivial set of qutrit preparations and mea-
surements, and implement an adaptation of the
numerical Swap method (see Section 7.1.4) to
deal with the prepare-and-measure scenario. In
[FK19], the authors study the 2
d
→ 1 RAC. It is
proven that this game provides a robust self-test
of a pair of measurements that correspond to two
mutually unbiased bases in dimension d. Further
to this, the authors show how the score of the
RAC can also be used to bound both the incom-
patibility robustness [HKR15] of the pair and the
randomness of the measurement outputs. It is
worth emphasising that self-testing claims in this
scenario can be made for arbitrary dimension-
bounded communication games, i.e. it is not re-
stricted to RACs.
Several works have investigated self-testing of
non-projective measurements in the dimension-
bounded prepare-and-measure scenario. In
[MP19] the authors start from the 3
2
→ 1 RAC
to develop a self-test of the extremal ‘tetrahedral’
qubit POVM. In [TSV
+
20], a general method
is given to self-test any extremal qubit POVM,
given a self-test of a set of preparations with op-
posite Bloch vectors to the POVM on the Bloch
sphere. In a similar fashion in [TRR19] the au-
thors provide a self-test of d-dimensional SIC
POVM (whenever it exists). To prove the ideal
self-testing all works use the same idea; if there
is one outcome of a measurement that never oc-
curs for a given preparation, it follows that the
corresponding POVM element must be opposite
to the preparation on the Bloch sphere. From
this one can use a self-test of preparations to ef-
fectively tomograph the POVM measurement in
a similar manner to that in Section 8.2.3. In
[TSV
+
20] the protocol is made robust by intro-
ducing a convenient distance measure which is
used in the experimental demonstration reported
therein. All three works also discuss certification
of the non-projective nature of a measurement,
which is a weaker form of certification than ro-
Accepted in Quantum 2020-23-08, click title to verify 36
bust self-testing since it does not discuss closeness
to any particular POVM.
The recent work [MBP19] investigates the
self-testing of non-projective measurements in
a prepare-and-measure scenario involving a se-
quence of measurements, using figures of merit
that are closely linked to the 2
2
→ 1 RAC. To
achieve the optimal success probability in the
game, one party needs to perform a so-called
Lüders instrument, which corresponds to a non-
projective measurement. The authors then de-
rive bounds on the maximal eigenvalue of the
corresponding measurement operators given an
observed score in the game, using a numerical
approach based on an adaptation of the NPA hi-
erarchy to the prepare-and-measure scenario.
9.2.3 Self-testing through noncontextuality in-
equalities
One of the important features of quantum the-
ory is contextuality, first noticed via the Kochen-
Specker theorem [KS67]. In general, in any de-
terministic hidden variable model reproducing
quantum correlations, the outcome of a measure-
ment M must depend on its context, i.e. the
set of compatible measurements one may per-
form alongside M. Contextuality can be de-
tected through the violation of noncontextual-
ity inequalities, of which the simplest is the
Klyshko-Can-Binicioğlu-Shumovsky (KCBS) in-
equality [KCBS08]. It corresponds to the sce-
nario in which five projective binary measure-
ments {M
i
= (M
0|i
, M
1|i
)}
5
i=1
can be performed
and each pair M
i
, M
i+1
is compatible, i.e.
[M
i
, M
i+1
] = 0 (69)
and exclusive i.e. tr(M
0|i
M
0|i+1
) = 0 (the labels
of measurements are taken modulo 5). If we de-
note p
i
= tr[M
0|i
ρ], where ρ is the measured state
the KCBS inequality reads:
5
X
i=1
p
i
≤ 2.
The inequality is satisfied in all outcome de-
terministic noncontextual theories, while quan-
tum measurements achieve the value c
q
=
5 cos(π/5)/(1 + cos(π/5)). The maximal quan-
tum value is achieved by measuring a pure state
ρ = |v
0
ihv
0
| and measurements M
0|i
= |v
i
ihv
i
|
where
|v
0
i = (1, 0, 0)
T
,
|v
i
i = (cos θ, sin θ sin φ
i
, sin θ cos φ
i
)
T
,
where cos θ = cos(π/5)/(1 + cos(π/5)) and φ
i
=
iπ(4/5).
Note that since this scenario consists of mak-
ing measurements on a single quantum system,
the statistics can always be simulated classi-
cally and it is thus impossible to achieve self-
testing in a fully device-independent manner. In
[BRV
+
19] the authors overcome this by assum-
ing compatibility relations such as (69), thus ren-
dering the scheme semi-device independent. Us-
ing this, they give a general scheme to self-test
quantum states using noncontextuality inequal-
ities and prove that the maximal quantum vio-
lation of the KCBS inequality robustly self-tests
the above strategy. In other words, for any state
ρ and measurements {|v
i
ihv
i
|
0
} which can be used
to achieve the violation c
q
− of KCBS inequality
there exist a unitary U such that
kU
v
0
i
− |v
i
ik ≤ O(
√
), ∀i.
The proof holds also for the generalisations of
KCBS inequality given in [BBC
+
11, AQB
+
13,
LSW11]. The crux of the proof is an equiva-
lence between the optimal strategy for violating
the KCBS inequality and the solution to a certain
type of SDP optimisation known as the Lovász
theta number of an odd cycle graph [Lov79].
10 Applications of self-testing
The birth of self-testing is usually associated to
the Mayers-Yao paper [MY04] from 2004. It set
the terminology and formalism, including the first
usage of the term self-testing in this context and
identifying local isometries as relevant transfor-
mations. A similar main result as in [MY04]
was presented in [MY98], although in the context
of whether untrusted sources can be pertinent
for cryptographic tasks. This earlier paper used
the term ‘self-checking’ instead of ‘self-testing’.
Moreover, [MY98] is at the same time one of the
pioneering works in device-independent cryptog-
raphy, indicating importance of self-testing for
the development of device-independent protocols.
Since then self-testing has been scrutinised as a
task of twofold significance:
Accepted in Quantum 2020-23-08, click title to verify 37
• purely theoretical, related to exploring the
conditions for a probability distribution to
determine a specific quantum state and/or
measurements, and proving that such state-
ments also hold approximately. This theo-
retical aspect was reviewed in Sections 4,5,
6, 7, 8 and 9.2. As a result, ideas and re-
sults from self-testing can lead to progress in
related theoretical areas.
• practical, relevant for creating new device-
independent or semi-device-independent
protocols for different tasks. In the Mayers-
Yao paper [MY04] the authors say ‘We hope
that it will have application in different
areas of quantum information processing’.
Fifteen years later we can observe that this
hope is fulfilled.
In this section we give an overview of the ap-
plications of self-testing during the first fourteen
years after the technique has been formally in-
troduced. On the practical side, we cover the
relation of self-testing with device-independent
randomness generation in section 10.1, device-
independent quantum cryptography in section
10.2, and device-independent entanglement cer-
tification in section 10.3. In section 10.4 we
describe the applications to delegated quantum
computing 10.4. Finally, from the theoretical
side, in section 10.5 we describe the influence self-
testing has had in understanding the structure of
the set of quantum correlations. In particular in
section 10.5.1 we highlight a link to the study of
quantum correlations produced by finite vs infi-
nite quantum systems, and in section 10.5.2 how
self-testing has been used to prove an inequiva-
lence between uncertainty relations and Bell non-
locality.
10.1 Device-independent randomness genera-
tion
The probabilistic nature of quantum mechanics
can be exploited for generation of random num-
bers. In the simplest example, measuring in the
computational basis a qubit in the state |+i re-
sults in a perfectly random output bit. However,
the certification of the random nature of bits ob-
tained this way relies on the exact characterisa-
tion of both the quantum state and the measure-
ment performed. The device-independent sce-
nario offers much less stringent requirements for
randomness certification, by qualitatively relat-
ing randomness with nonlocality. By treating her
devices as non-communicating black boxes Alice
can certify some amount of randomness by ob-
serving the violation of a Bell inequality. The
first results in this direction show that the maxi-
mal violation of the Mermin [Col06, CK11b] and
the CHSH inequality [PAM
+
10] can be used in
this way. For more information on certification of
quantum randomness see [AM16] and references
therein.
Here, we comment on the relation between
self-testing and randomness. A self-testing pro-
tocol proves the existence of a pure entangled
state and a certain set of measurements acting
on it. Once this conclusion is made, certified
random bits come for free since local measure-
ments on a pure entangled state necessarily pro-
duce random outcomes. As a result, ideas from
self-testing are often either implicitly present in
device-independent randomness works or are ex-
plicitly used as tools to prove randomness lower
bounds.
In the pioneering works of [Col06, CK11b] a
self-testing statement is implicitly present, where
the authors prove that only an orthogonal sum of
GHZ states can maximally violate the Mermin in-
equality. In [CY14], the sequential self-testing of
n EPR-pairs proven in [RUV13] is used as a sub-
protocol for infinite randomness expansion with
a constant number of devices.
Simple symmetry-based arguments are used in
[DPA13] to prove that the violation of some Bell
inequalities can be used to certify the presence
of genuine randomness. A necessary condition is
that there exists a unique probability distribution
maximally violating the Bell inequality. One way
to prove such uniqueness is through self-testing:
if the maximal violation of the Bell inequality
is a self-test the maximally violating probability
distribution has to be unique. Furthermore, in-
complete results from [DPA13], were proven to be
true by using self-testing techniques in [ŠASA16].
The results on self-testing properties of bi-
nary XOR games from [MS13] were expanded
in [MS16] and used to devise protocols for ex-
ponential randomness expansion. More recently,
the authors of [BMP18] directly use robust self-
testing bounds for the tilted-CHSH inequality
[BP15] to lower bound the randomness generated
in their protocol. Self-testing techniques are also
Accepted in Quantum 2020-23-08, click title to verify 38
used in [APVW16, ABDC18, WKB
+
19] to prove
that two bits of local randomness can be certified
from a two-qubit entangled state.
10.2 Device-independent quantum cryptogra-
phy
10.2.1 Quantum key distribution
Quantum key distribution (QKD) is the most
widely studied quantum cryptographic protocol
in which two parties, Alice and Bob, use quan-
tum resources to generate a shared private key
which can later be used for encryption and de-
cryption of messages. The security of a stan-
dard QKD protocol relies on the correct charac-
terisation of all devices, which can be difficult to
achieve in practice and far from ideal from a secu-
rity perspective. An alternative approach comes
from device-independent quantum key distribu-
tion (DIQKD), where security is based only on
the observation of the correlations, and can be
proven even if the constituent devices are treated
as black boxes. DIQKD is intimately related to
DI randomness generation; whereas in random-
ness generation one aims to have random out-
comes, in a DIQKD protocol one aims to have
random outcomes that are also correlated be-
tween Alice and Bob (thus ensuring a shared pri-
vate key). As with randomness generation, the
security of DIQKD is often measured against the
violation of some Bell inequality. For a concise
review on the topic see [ER14].
An indication of a close relation between
DIQKD and self-testing is their common root
in the Mayers-Yao work [MY98]. It discusses
self-testing as a protocol for the first time (un-
der the name self-checking) and recognises that
it can help to use untrusted devices in crypto-
graphic setting. [MY98] consider the BB84 pro-
tocol [BB84] in which Alice certifies an untrusted
source she wants to use. The source is supposed
to emit EPR pairs with Alice keeping one parti-
cle and measuring it and sending the other one
to Bob. The untrusted source can be self-tested
using the Mayers-Yao self-testing criterion, as ex-
plained in section 5. The protocol is later dis-
cussed in [MT02] in the context of the Ekert QKD
protocol [Eke91]. The second Mayers-Yao paper
[MY04], improving the first one by characterising
the measurements (and introducing the phrase
‘self-testing’) also discussed the relation of self-
testing with the BB84 protocol.
Ever since then self-testing and DIQKD have
been intertwined. A certification of some quan-
tum resource is implicitly present in every
DIQKD security proof, however in some works
the relation between self-testing, as the strongest
form of certification, and DIQKD was explicitly
examined. The effect of the inability to self-test
complex measurements on the security of cryp-
tographic tasks has been the subject of [MM11].
The authors prove that the 6-state QKD proto-
col [BBBW84, Bru98] can be secure even if the
devices are untrusted, despite the issue with com-
plex conjugation. Similarly like in the protocols
for randomness expansion, the self-testing prop-
erties of XOR binary nonlocal games explored in
[MS13] were used in [MS16] to prove the secu-
rity of certain class of QKD protocols. Finally,
the concept of parallel DIQKD developed analo-
gously to parallel self-testing was first introduced
in [JMS17] and the security proof relied on the
rigidity of the magic square game [WBMS16] al-
lowing for parallel self-testing of two singlets. A
simplified proof appeared in [Vid17].
10.2.2 Cryptography beyond quantum key distri-
bution
Bit commitment— Bit commitment is a crypto-
graphic primitive in which Alice chooses a bit b
that she wants to first commit, then later reveal,
to Bob. The protocol should be both binding (Al-
ice should not be able to change her choice of b af-
ter the commit step) and hiding (Bob should not
be able to know b until Alice chooses to reveal).
In a classical protocol, either Alice or Bob can
cheat with probability 1 without being caught,
i.e. either Alice can alter the bit after commit-
ting or Bob can learn it before it is revealed. Al-
though unconditionally secure bit commitment is
known to be impossible even using quantum re-
sources [May97, LC97], there exist quantum pro-
tocols in which either party’s probability to suc-
cessfully cheat is strictly smaller than 1 [CK11a].
[SCA
+
11] introduces the idea of a DI quantum bit
commitment protocol in which besides not trust-
ing each other, Alice and Bob do not trust their
equipment either. The security of the protocol
is based on the self-testing fact that the maxi-
mal violation of the Mermin inequality can only
be achieved by measuring the GHZ state. A ver-
sion of DI quantum bit commitment based on the
Accepted in Quantum 2020-23-08, click title to verify 39
violation of the CHSH inequality is presented in
[AMPS16], in which the security can be seen as
consequence of the self-testing properties of the
CHSH inequality. Similarly, the violation of the
CHSH inequality has been used to prove the secu-
rity of DI relativistic bit commitment in [AK15],
ruling out the possibility of location attacks in
which devices are able to track their own space
and time coordinates. None of the works explic-
itly relate their results to the corresponding self-
testing protocol however.
Weak string erasure— Weak string erasure
(WSE) [KWW12] is a primitive which can be
used in two-party cryptographic protocols in
which no large scale reliable quantum storage
is available to the cheating party. WSE pro-
vides a random bit string (b
1
, ··· , b
n
) to Al-
ice, while sending a randomly chosen substring
(b
i
1
, ··· , b
i
k
) to Bob, together with the set
(i
1
, ··· , i
k
) specifying the location of substring
bits. WSE is secure against Bob if he cannot
learn much about the full string given to Alice,
while it is secure against Alice if she cannot learn
the location of Bob’s bits. A DI version of this
protocol useful for bit commitment or oblivious
transfer is introduced in [KW16] and the secu-
rity is related to the self-testing properties of the
CHSH inequality.
Position verification— Finally, we briefly men-
tion the position verification primitive, useful in
position-based cryptography in which the parties
have to convince the (honest) verifiers that they
are located at a particular location. Protocols
for position verification that improve the secu-
rity by using quantum communication have been
proposed in [KMS11, Mal10]. The DI security of
position verification against adversaries with no
quantum memory is proven in [RTK
+
18], and can
also be traced to the self-testing properties of the
CHSH inequality.
10.3 Entanglement detection
One of the most basic tasks in quantum infor-
mation is that of detecting entanglement of a bi-
partite quantum system via local measurements
on its subsystems. Device-independent entan-
glement detection considers this problem in the
device-independent scenario, i.e. where all local
measurement devices are treated as black boxes.
Since the observation of Bell nonlocal correlations
necessarily implies that the underlying state is
entangled, the standard approach to DI entan-
glement detection involves violating a Bell in-
equality. However, since there exist entangled
mixed states that do not violate any Bell inequal-
ity
2
[Wer89, Bar02, ADA14, BHQB16, JHA
+
15,
BFF
+
16, HQB
+
16], this method cannot be used
for all entangled states. A partial solution to this
problem, allowing for the entanglement detection
of all entangled states, was given in [Bus12] (see
also [BRLG13]) using the concept of a ‘semi-
quantum game’. Here, the classical inputs in
a Bell test are replaced by ‘quantum inputs’
|ψ
x
i, |ψ
y
i, that is, a set of known quantum states
that are sent to the measurement device instead
of the classical labels x and y. This scenario is
semi-device-independent since although the mea-
surement devices are treated as black boxes, the
quantum input states must be trusted.
In [BŠCA18a, BŠCA18b], tools from self-
testing and semi-quantum games were used to
construct fully DI protocols for the entanglement
detection of all entangled mixed states. The idea
is as follows. If one achieves a self-test of a par-
ticular state and local measurements for Alice,
then this certifies (up to a local isometry) the
reduced states of Bob conditioned on a particu-
lar choice of input/output for Alice. In this way
one can certify an ensemble of state preparations
(conditioned on Alice’s input/output) on Bob’s
local Hilbert space. These preparations can then
be used as quantum inputs in a semi-quantum
game. Since (i) the quantum inputs are now certi-
fied device-independently, (ii) the semi-quantum
games scenario can be applied to all entangled
states, the two can be combined to construct a
fully device-independent protocol that works for
all entangled states. Specifically, one needs to
consider a network scenario in which the state of
interest is augmented with two auxiliary bipar-
tite states that are used to prepare the quantum
inputs. Here, tools from parallel self-testing as
well as the issue of complex conjugation become
important for the general proof.
2
At least in the original Bell scenario in which Alice
and Bob can perform any number of non-sequential lo-
cal measurements on a single copy of the state. In more
complex measurement scenarios (see [CASA11, SDSB
+
05,
Pal12, TRC19, BRGP12, HQBB13, Pop95]) it is generally
unknown if such states exists.
Accepted in Quantum 2020-23-08, click title to verify 40
10.4 Delegated quantum computing
Delegated computation is a protocol in which a
party, usually called a verifier, delegates a com-
putational task to another party, usually called a
prover. The verifier aims to solve difficult com-
putational tasks, but does not have enough com-
putational resources. The prover, on the other
side, has a very powerful computer and is able to
solve any task the verifier is interested in. When
one talks about delegated quantum computation
(DQC) the prover possesses a quantum computer,
while the verifier has either only classical comput-
ing resources or limited quantum resources but
wants to solve a problem intractable for classical
computing devices. For a concise review on the
existing approaches in DQC see [GKK18].
There are two desirable properties of a DQC
protocol: verifiability and blindness. The proto-
col is said to be verifiable if the verifier can be con-
vinced that the solution provided by the prover(s)
is correct. This is non-trivial, since the verifier is
unable to solve the problem. Blindness of the pro-
tocol is related to the secrecy of the computation.
It is ensured when the prover(s) cannot learn any-
thing about the computational task the verifier
wants to perform. It is very difficult to construct
a DQC protocol with a fully classical verifier and
a single prover which is verifiable and blind. The
first protocol that achieves this, under computa-
tional assumptions, is [Mah18]. In principle, it
is easier to achieve both verifiable and blind pro-
tocol with a classical verifier when there is more
than one prover. In this case the provers are en-
tangled and forbidden to communicate. Verifia-
bility is proven if the verifier can be convinced
that the two or more non-communicating provers
are performing the prescribed sequence of mea-
surements. This, of course, requires that the ver-
ifier be able to test that the provers perform mea-
surements from a set that is universal for quan-
tum computing. The latter is exactly a self-
testing task. Thus it is no surprise, that self-
testing is useful for such a delegation protocol, yet
orchestrating such a computation in a verifiable
fashion is a delicate task. Delegation protocols
with two or more provers based on self-testing
typically achieve information theoretic security.
The first such protocol was presented by Re-
ichardt, Unger and Vazirani (RUV) in [RUV13].
The protocol involves two provers sharing a ten-
sor product of many EPR pairs and the compu-
tation model is quantum computation by telepor-
tation [GC99]. The provers are able to convince
the verifer that they posses N EPR pairs by ob-
taining the optimal score in sequential playing of
the CHSH game. The complexity of RUV proto-
col in terms of time and the number of EPR pairs
needed is extremely large. The protocol was sub-
sequently improved in [JMS20] where each Bell
pair is shared by two provers, making the num-
ber of provers increase significantly at the expense
of reducing the overall complexity. The advan-
tage of parallel self-testing of N EPR pairs in-
stead of sequential was exploited in [NV17] and
[CGJV17]. The latter obtains a protocol with
an almost optimal overhead (in the size of the
computation) in terms or resources used, by ex-
ploiting parallel self-tests with robustness inde-
pendent of the number of EPR pairs tested, dis-
cussed previously in subsection 5.3. The work
[GKW15] shows that RUV protocol can be signif-
icantly improved if the verifier is actually quan-
tum and wants to be convinced that they share
with the prover n EPR pairs. In this case the self-
testing of n EPR pairs through steering becomes
a relevant sub-protocol.
Another example of self-testing incorporated
into a delegated quantum computing protocol is
[McK16a]. The DQC protocol involves many
provers which share a graph state. The model
of computation is measurement-based quantum
computing (MBQC) [RB01, RBB03]. The proto-
col is made verifiable by using the self-testing of
graph-states mostly based on [McK14]. Based on
a similar idea a significantly simplified protocol
appeared in [HH18]: since the triangular lattice
graph state is universal for MBQC [MP13] the
number of provers can be reduced to three and
the number of necessary copies of the graph states
is also considerably smaller than in [McK16a].
10.5 Structure of the set of quantum correla-
tions
Here we highlight two ways in which self-testing
as furthered our understanding of the set of quan-
tum correlations.
10.5.1 Correlations from finite vs infinite dimen-
sional quantum strategies
It is said that the correlations {p(a, b|x, y)}
a,b,x,y
admit a quantum strategy if there exists a state
Accepted in Quantum 2020-23-08, click title to verify 41
|ψi
AB
∈ H
A
⊗ H
B
and projective measurements
{M
a|x
}
a,x
, {N
b|y
}
b,y
such that
p(a, b|x, y) = hψ|M
a|x
⊗ N
b|y
|ψi.
This strategy is also called a tensor-product
strategy due to the tensor product between Al-
ice’s and Bob’s spaces. Different sets of quan-
tum correlations arise when certain conditions
are imposed on the state |ψi and/or measure-
ments {M
a|x
}
a,x
, {N
b|y
}
b,y
. The correlations ob-
tained through a tensor-product strategy on
finite-dimensional Hilbert spaces H
A
and H
B
con-
stitute the set denoted as C
q
. If the Hilbert spaces
H
A
and H
B
can also be infinite dimensional the
set is denoted as C
qs
. The closure of the set C
q
is called C
qa
. Additionally, a commuting-operator
strategy is one in which no tensor product struc-
ture is imposed but instead all measurement op-
erators of Alice commute with all those of Bob,
i.e.
p(a, b|x, y) = hψ|M
a|x
N
b|y
|ψi.
with [M
a|x
, N
b|y
] = 0. The commuting-operator
model is used in algebraic quantum field theory
and all such correlations are denoted by the set
C
qc
. The set inclusion relation defines a hierarchy
among these sets [PT15]:
C
q
⊆ C
qs
⊆ C
qa
⊆ C
qc
. (70)
Whether C
qc
is equivalent to either C
qs
or C
qa
are
problems known as Tsirelson’s problems [Tsi93].
Recently, it has been proven that C
qs
6= C
qc
[Slo20], and also that C
qs
6= C
qa
in [Slo19].
Whether C
qa
⊂ C
qc
remains an open problem.
Self-testing techniques have also inspired prov-
ing a strict inclusion of C
q
in C
qs
. The in-
equivalence between these two sets was proven
in [CS17c] in cases when either the number of in-
puts or the number of outputs is infinite. The
separation between C
q
and C
qs
for finite input or
output alphabets was proven by the same authors
in [CS18]. The separation is demonstrated by ex-
plicitly giving the correlations which can be ob-
tained with infinite-dimensional quantum strate-
gies, but not with any finite-dimensional ones.
The proof is inspired by the protocol for self-
testing infinite-dimensional bipartite pure states,
described in section 5.2.1. In the finite case, the
self-test of a bipartite state can be interpreted as
involving one of two different protocols depend-
ing on the parity of the dimension. An infinite
dimensional state does not have a defined par-
ity and can thus be self-tested by either protocol.
The authors use this fact as the theoretical basis
to prove the separation.
Following Slofstra’s proof of non-closure of the
quantum set of correlations [Slo19] alternative
proofs appeared in [DPP19] and [MR19]. All
these proofs rely on the representation theory
of C
∗
-algebras. A relatively simpler proof, us-
ing embezzling entanglement [vDH03] and self-
testing, is presented in [Col20].
10.5.2 Uncertainty relations and Bell nonlocality
Self-testing has also been used as a tool to under-
stand the relationship between uncertainty rela-
tions, steering and Bell nonlocality. In [OW10] it
was shown that in the optimal quantum strategy
for a large class of Bell inequalities called XOR
games, the steered states of Bob after a measure-
ment by Alice are always such that they saturate
a fine-grained uncertainty relation defined by the
Bell inequality itself. This was then extended to
more Bell inequalties in [ZGZ
+
16]. Whether such
a link was true for generic Bell inequalities re-
mained an open question.
In [RM
+
18] (building on [ZGZ
+
16]) the au-
thors answer this in the negative by providing
specific examples of Bell inequalities for which the
relationship does not hold. To achieve this, self-
testing statements for the Bell inequalities are
proven, from which the form of the steered states
can be determined and checked against the cor-
responding uncertainty relations. The self-testing
statements are proven using Jordan’s lemma (see
section 7.1.2) to reduce the problem to essentially
a two-qubit strategy, which simplifies the analysis
significantly.
11 Experiments
The bulk of self-testing procedures are still only
theoretical recipes. This is understandable since
the majority of robust self-testing protocols have
fidelity bounds that decrease rapidly with noise
and serve only as a proof of principle. However,
in the last years there has been an increasing
number of self-testing protocols robust to real-
istic amounts of noise (see section 7). Here we
mention the few experimental realisations of such
protocols.
Accepted in Quantum 2020-23-08, click title to verify 42
The biggest experimental hurdle towards
fully device-independent protocols is simultane-
ously closing detection and locality loopholes.
[BRS
+
18] reports the self-testing of a Bell state
distributed over 398 meters through the viola-
tion of the CHSH inequality free of both detec-
tion and locality loopholes and furthermore free
of the i.i.d. assumption. In this light it is the
first fully device-independent self-testing proto-
col to be implemented in practice. Entanglement
between the distant atoms is generated by entan-
gling the spin of each atom with polarisation of a
single photon. The obtained fidelity is 55.54% at
a confidence level of 99%. By applying the same
theoretical tools to analyse the data obtained in
the loophole-free Bell test presented in [HKB
+
16]
no fidelity higher than the trivial 50% could be
found.
The remainder of the experimental self-testing
contributions are not based on a completely
loophole-free Bell tests. [TWE
+
17] reports a
high violation of the CHSH inequality by a pair
of
9
Be
+
ions. The violation is used to make
a self-testing statement, based on [Kan16]. At
the 95% confidence level the pair of
9
Be
+
ions
has 0.958 fidelity with the maximally entangled
pair of qubits. An overview of the inferred self-
testing bounds from some previous works re-
porting CHSH violations is also presented. In
[ZCP
+
18] the operator inequalities for robust
self-testing from [Kan16] are tested for a large
number of bipartite and tripartite qubit states
encoded in photon polarisation degrees of free-
dom.
Experimental robust self-testing of partially
entangled pairs of qubits using the Swap method
is presented in [ZCY
+
19]. The systems under
consideration are polarisation entangled photons.
Self-testing of partially entangled qubit pairs is
also used to heuristically estimate the fidelity of
a produced ququart state with a given reference
state. Robust self-testing of partially entangled
pairs of qubits encoded in photon polarisation
degrees of freedom was reported in [GMM
+
19].
The self-testing was done through the violation
of the tilted CHSH inequality and the robust-
ness bounds were estimated by using the numer-
ical results from [CKS19]. In [GPL
+
19] the au-
thors explore the certification of partially entan-
gled pairs of photons encoded in the polarisation
degree of freedom. The fidelity of the entangled
pair with the corresponding partially entangled
pair of qubits is estimated in two ways: by using
standard tomographic methods and self-testing.
The obtained fidelities have ratio ≈ 0.998, imply-
ing that for the case of qubit states self-testing
can be used to achieve almost the same conclu-
sions as tomography. It is argued that self-testing
may have an advantage over tomography even
when the detection and locality loopholes are not
closed since it avoids characterisation of measure-
ments and assumptions about dimension and, in
principle, requires estimating fewer average val-
ues.
In [WPD
+
18] various two-qutrit entangled
states are self-tested using photons entangled in
the mode degree of freedom of the waveguides
in a silicon based integrated optical chip. Fi-
delity bounds were obtained numerically via the
Swap method. The self-tested states are the max-
imally entangled pair of qutrits (estimated fi-
delity 0.799), the state maximally violating the
CGLMP inequality [CGL
+
02] (estimated fidelity
0.68) and a state maximally violating an exten-
sion of one of the SATWAP inequalities [SAT
+
17]
(estimated fidelity 0.832).
Experimental self-testing in the steering sce-
nario is the subject of [LLW
+
19]. The fidelity
of the underlying physical state with the GHZ
state is estimated based on the violation of the
Mermin’s steering inequality [CHRW11]. The fi-
delity lower bound is estimated to be 0, 7866,
while the tomographically retrieved fidelity is
0.8725 ± 0.0034.
Finally, in [ZCP
+
19] the authors report an ex-
perimental realisation of a robust self-test of a
Bell state measurement based on the entangle-
ment swapping protocols of [SBWS18, RKB18]
(see section 9.1). Photon pairs that are hyper-
entangled in the spatial and polarisation degrees
of freedom are used to encode the two maximally
entangled pairs of qubits that are needed for the
entanglement swapping protocol.
12 Concluding remarks and open ques-
tions
Recent years have seen an increased interest in
device-independent self-testing, accompanied by
the plethora of self-testing protocols and meth-
ods presented in this review. However, there
are still many important unresolved questions.
Accepted in Quantum 2020-23-08, click title to verify 43
Without aiming to exhaust the list, we name
some open questions and research directions
which we believe worthy of attention.
Analytic methods for dimension larger than
2 —The majority of known self-testing protocols
are either built to self-test multi-qubit states and
measurements, or apply existing qubit protocols
to the self-testing of higher dimensional systems.
The self-testing of states and measurements us-
ing methods that exploit the genuine d dimen-
sional nature of quantum systems, is however still
a very unexplored area. For instance, it has been
known for a long time that via the numerical
Swap method, the CGLMP inequality self-tests
the two-qutrit state of equation (43). However,
a corresponding analytic proof of this statement
is still lacking, despite the relative simplicity of
the inequality. Similarly desirable are analytic
proofs for the self-testing of the maximally entan-
gled states in dimension d using the SATWAP in-
equalities [SAT
+
17]. With respect to high dimen-
sional measurement self-testing, one important
open question is to extend the analytic self-test
of a set of qutrit mutually unbiased basis mea-
surements and the maximally entangled state of
[KŠT
+
19] to higher dimensions. Moving beyond
high dimensional systems to continuous variable
systems, essentially nothing is known and there
exist no protocols to self-test such states.
Multipartite methods—In a similar vein, tech-
niques for self-testing general multipartite states
are also needed, since current methods are only
known for restricted classes such as graph states.
One potential line of research in this direction
would be to develop methods to self-test multi-
qubit hypergraph states [RHBM13], which ex-
hibit a richer structure than graph states but
still admit a useful description in terms of Clif-
ford group stabiliser operators. One would thus
need a general method to construct Bell inequal-
ities for such states, as was done for graph states
[GTHB05, BAŠ
+
20]. This appears more com-
plicated for hypergraph states however since the
nonlocal nature of the hypergraph stabilizer op-
erators means they do not have an obvious inter-
pretation as local measurement observables (al-
though some progress has been made [GBG16]).
Identifying the set of undetectable transforma-
tions—Part of the challenge in going beyond two
qubit methods is to identify the set of local trans-
formations defining the equivalence classes of self-
testable states and measurements in higher di-
mensions. As we have seen, the standard defini-
tions presented in section 3 need to be adapted
in order to self-test complex valued measure-
ments, stemming from the invariance of quan-
tum correlations under complex conjugation of
the state and measurement operators. In higher
dimensions, it is still unknown whether there ex-
ist more state and measurement transformations
that leave correlations invariant. If such trans-
formations exist, an all-encompassing definition
of what it means to self-test a state and mea-
surements in general dimension is therefore still
missing.
Self-testing of a state or measurements only—
In many self-testing works, a self-testing state-
ment for the state is accompanied by an ad-
ditional self-testing statement for the measure-
ments. One interesting problem would be to
find situations where the correlations allow one
to identify the state, but not the measurements,
even allowing for the freedom of complex conju-
gation of the measurements in the definition of
measurement self-testing. Similarly, it would be
interesting to find correlations that allow one to
self-test the measurements, but not the state.
Self-testing in non-i.i.d. scenarios—
Throughout this review we have made the
assumption that each round of the experiment
is independent and identical to all others. It
would be interesting and practically relevant
to attempt to remove this assumption from
self-testing protocols. One way to achieve
this would be to build sequential self-testing
protocols akin one in [RUV13] or by lever-
aging some recently introduced techniques
[BRS
+
18, DFR16, AFRV19, AF18] for such
scenarios. Methods could be borrowed also
from the protocols for delegated quantum
computation, i.e. [CGJV17]. Since the state
and measurements can now depend on settings
and outcomes in previous rounds, one would
also need to adapt the definition of self-testing
to apply to such scenarios, as well as derive
corresponding confidence bounds from finite
statistics. For example, the aim would be to
show that, to high statistical confidence, the
source is producing something close (by some
measure) to n independent copies of |ψ
0
i, where
n is the number of experimental rounds.
Accepted in Quantum 2020-23-08, click title to verify 44
Improved robustness methods—Finally, meth-
ods to improve the robustness bounds of gen-
eral self-testing protocols are much in need. In
practice, the applicability of the majority of self-
testing protocols is hindered by very poor tol-
erance to noise. Significant improvements have
been achieved for simple scenarios [Kan16], how-
ever it is not clear if these methods can be ex-
tended to scenarios with more inputs and out-
puts due to their dependence on Jordan’s lemma.
Finding a good robustness bound involves a dif-
ficult maximisation over all local isometries, and
as a result nearly all methods use one of the few
Swap isometries that are known to give good re-
sults in the well-explored simple cases. Thus,
knowing more useful isometries and understand-
ing which work well for particular classes of states
would likely lead to improved robustness bounds.
Acknowledgements
We are grateful to Alexia Salavrakos, Jean-Daniel
Bancal, Andrea Coladangelo, Jed Kaniewski,
Armin Tavakoli, Erik Woodhead, Alejandro
Pozas-Kerstjens, Aleksandra Dimić, Marc-Olivier
Renou, Felix Hüber, Sébastien Designolle, Nico-
las Brunner, Yeong Cherng Liang, Nikolai Mik-
lin and Rotem Arnon-Friedman for suggestions
while preparing the manuscript. We acknowledge
funding from the ERC CoG QITBOX, the Span-
ish MINECO (QIBEQI FIS2016-80773-P, Severo
Ochoa SEV-2015-0522), Fundacio Cellex, Gener-
alitat de Catalunya (SGR 1381 and CERCA Pro-
gramme). IŠ acknowledges funding from SNSF
(Starting grant DIAQ). JB acknowleges funding
from the Juan de la Cierva-formación grant the
AXA chair in quantum information science.
A Appendix
A.1 Self-testing complex measurements
Here we give a possible definition of self-testing
of states and complex valued measurements.
Definition 6. (self-testing of states and complex
measurements)
We say that the correlations p(a, b|x, y)
self-test the state and measurements
|ψ
0
i
A
0
B
0
, {M
0
a|x
}, {N
0
b|y
} if for all states and
measurements ρ
AB
, {M
a|x
}, {N
b|y
} compatible
with p(a, b|x, y) there exists a local isometry
Φ = Φ
A
⊗ Φ
B
such that for any purification
|ψi
ABP
of ρ
AB
there exists some state |ξi
¯
A
¯
BP
such that
Φ ⊗
P
h
M
a|x
⊗ N
b|y
⊗
P
|ψi
ABP
i
=
˜
M
a|x
⊗
˜
N
b|y
⊗
P
ψ
0
A
0
B
0
⊗ |ξi
¯
A
¯
BP
,
for all a, x, b, y, and where
˜
M
a|x
= M
0
a|x
⊗ S
¯
A
0
+ (M
0
a|x
)
∗
⊗ S
¯
A
1
˜
N
b|y
= N
0
b|y
⊗ T
¯
B
0
+ (N
0
b|y
)
∗
⊗ T
¯
B
1
S
0
+ S
1
¯
A
, T
0
+ T
1
=
¯
B
,
hξ|(S
0
⊗ T
0
+ S
1
⊗ T
1
) ⊗
P
|ξi = 1.
Here, the S
i
and T
i
part of the measurements
are acting as effective controlled complex conju-
gations of the reference measurements. The final
condition ensures that this conjugation is per-
formed in a correlated fashion as implied from
(14). The probability that the conjugation is per-
formed depends on the (unknown) junk state and
is thus unknown. Note that if one traces out all
but the
¯
A
¯
B space, one obtains some convex com-
bination of the reference and conjugated measure-
ments acting on the reference state.
A.2 Regularisation trick
In this appendix we give more details about the
so-called regularisation trick. It refers to the case
when one of the operators used to build the Swap
gate (see figure 4) is not unitary. This hap-
pens already in the case described in Chapter 4
where the operators Z
A
and X
A
from equation
(32) might have some zero eigenvalues. Let us
focus on Z
A
= (A
0
+ A
1
)/
√
2. The first step in
the regularisation procedure is to change all the
zero eigenvalues of Z
A
to 1, resulting in a new
operator Z
∗
A
. In the second step, all eigenvalues
are normalized, i.e. the new operator defined as
ˆ
Z
A
= Z
∗
A
/|Z
∗
A
| is unitary by construction. How-
ever, one has to prove that
ˆ
Z
A
acts on the phys-
ical state in the same way as Z
A
. For that the
following series of inequalities can be used (Note
that Z
∗
A
acts on |ψi in the same way as Z
A
since
it can be seen as Z
A
+P where P is the projector
Accepted in Quantum 2020-23-08, click title to verify 45
on the kernel of Z
A
):
k(
ˆ
Z
A
− Z
A
) |ψik = k( −
ˆ
Z
†
A
Z
A
) |ψik
= k( − |Z
A
|) |ψik
= k( − |Z
A
Z
B
|) |ψik
≤ k( − Z
A
Z
B
) |ψik
= 0
The first line is the consequence of the unitarity
of
ˆ
Z
A
and the second uses the definition of
ˆ
Z
A
.
To get the third line we used the fact that Z
B
is unitary. The inequality follows from the op-
erator inequality A ≤ |A|. The last line stems
from equation (31). The key ingredient neces-
sary for regularisation is exactly equation (31).
In general, the regularisation of any operator A
can be done if there is a unitary U such that
A ⊗ |ψi = ⊗ U |ψi.
A.3 Swap isometries
In this appendix we provide further comments
on the different Swap isometries used in the self-
testing protocols. In section 4.3 we mentioned
that the partial Swap gate given on figure 4 is
appropriate only if the ancillas are initiated in
the state |0i. In the case that the ancillas are in
a different state the correct isometry to use is the
full Swap gate, given in figure 10.
A generalisation of the Swap gate, given on
figure 11, can be used for self-testing of bipar-
tite qudit states |ψi =
P
d−1
j=0
λ
j
|jji, where λ
j
are
positive real numbers. The gate F is the Fourier
transform defined as:
F |ji =
1
√
d
d−1
X
k=0
ω
jk
|ki,
where d is the local dimension of the reference
state, and ω is the d-th root of the unity. The
controlled gates C
¯
Z and C
¯
X are defined as fol-
lows
3
:
C
¯
X |ji|ψi = |ji
¯
X
(j)
|ψi
C
¯
Z |ji|ψi = |ji
¯
Z
j
|ψi.
For the gate on figure 11 to work as an effective
Swap gate the operators
¯
X and
¯
Z have to satisfy
certain conditions, mimicking anticommutativity
3
Note that {X
(j)
} are j different operators, while {Z
j
}
are j-th powers of the operator Z.
ρ
A
0
τ
B
0
|ψi
AB
X
A
X
B
H
H
Z
A
Z
B
H
H
X
A
X
B
Figure 10: The full Swap gate used in some robust self-
testing protocols. If the ancillas are initiated in the state
|0i the gate reduces to the partial Swap gate, given on
figure 4.
|0i
A
0
|0i
B
0
|ψi
AB
F
F
¯
Z
A
¯
Z
B
F
†
F
†
¯
X
A
¯
X
B
Figure 11: The partial Swap gate used in some protocols
for robust self-testing of qudit entangled states.
from the qubit case. In [YN13] the authors give
the recipe: operators
¯
Z
A
and
¯
Z
B
have to satisfy
d−1
X
j=0
ω
ja
¯
Z
j
A
⊗ |ψi = ⊗
d−1
X
j=0
ω
ja
Z
j
B
|ψi, (71)
for all a ∈ {1, ··· , d}. In addition, operators
¯
X
(k)
A
and
¯
X
(k)
B
must satisfy
λ
0
¯
X
(k)
A
⊗
d−1
X
j=0
ω
jk
¯
Z
j
B
|ψi
= λ
k
d−1
X
j=0
ω
jd
¯
Z
j
A
⊗
¯
X
(k)
B
†
|ψi, ∀k. (72)
It can be proven that the output state of the isom-
etry Φ
d
given on figure 11 and built from the op-
erators satisfying the conditions (71) and (72) has
the form
P
k
λ
k
|kki
AB
⊗|ξi
AB
where |ξi is some
normalised state. For more details about the qu-
dit Swap isometry see [YN13, CGS17, ŠCAA18].
A.4 Localising matrices in the Swap method
In section 7.1.4 we presented the numerical Swap
method used in robust self-testing protocols. The
Accepted in Quantum 2020-23-08, click title to verify 46
isometry used in the Swap method is the Swap
gate and as we discussed in Appendix A.2 when
one of the operators X
A
, X
B
, Z
A
or Z
B
is de-
fined as a sum or difference of physical measure-
ment observables, the Swap isometry might not
be unitary. Let us, for simplicity, focus the CHSH
case and the operator Z
A
= (A
0
+ A
1
)/
√
2 and
X
A
= (A
0
−A
1
)/
√
2 used to build the Swap isom-
etry. In Appendix A.2 we showed how to regu-
larize such operators for the purposes of the ideal
self-testing.
The procedure to solve this problem when us-
ing the Swap isometry in robust self-testing pro-
tocols is introduced in [BNS
+
15, YVB
+
14]. In
the context of the Swap method one solves the
problem by introducing two new operators A
2
and A
3
which are unitary and not too different
from (A
0
+ A
1
)/
√
2 and (A
0
− A
1
)/
√
2, respec-
tively. That way the isometry built by defining
Z
A
= A
2
and X
A
= A
3
is necessarily unitary.
One way to impose the proximity of A
2
and A
3
to (A
0
+ A
1
)/
√
2 and (A
0
−A
1
)/
√
2, respectively,
is by imposing the relaxation
A
2
(A
0
+ A
1
)/
√
2 ≥ 0 (73a)
A
3
(A
0
− A
1
)/
√
2 ≥ 0. (73b)
This can be enforced by introducing two new mo-
ment matrices called the localizing matrices. The
condition (73a) can be imposed as a requirement
that the moment matrix defined as
Γ
k
i,j
(A
2
) = hψ|S
(i)†
A
2
A
0
+ A
1
√
2
S
(j)
|ψi, (74)
where S = { , A
0
, A
1
, A
2
}, is positive semi-
definite. An analogous constraint can be made
to enforce the condition (73b).
B State and measurement assump-
tions
Here we overview the assumptions made about
the state and measurements in the definitions of
self-testing.
B.1 State
In the definitions of section 3 we use a purifi-
cation |ψi
ABP
of the potentially mixed physical
state ρ
AB
. One does not assume that the state
shared between Alice and Bob is pure however,
since it is given by
ρ
AB
= tr
P
[|ψ
ABP
ihψ
ABP
|]. (75)
Suppose we have a self-testing statement in the
form of definition 1:
Φ
A
⊗ Φ
B
⊗
P
[|ψi
ABP
] =
ψ
0
A
0
B
0
⊗ |ξi
¯
A
¯
BP
.
If we trace over the purification space on both
sides of the above we obtain
Φ
A
⊗ Φ
B
[ρ
AB
] = |ψ
0
ihψ
0
|
A
0
B
0
⊗ σ
¯
A
¯
B
.
where σ = tr
P
[|ξihξ|]. Thus a self-testing state-
ment of the form of definition 1, implies that the
same isometry maps ρ
AB
to the reference state.
This is possible because the isometry map acts
trivially on the purification space. This follows
from the fact that the isometry is constructed
from the measurement operators, which by as-
sumption themselves act non-trivially only on ρ.
B.2 Measurements
We do, however, assume that the physical mea-
surements are projective. This can lead to some
confusion, since naturally one may want to repeat
an analogous argument to the above that would
allow one to treat the measurements as general
POVM measurements, and simply make use of a
Naimark dilation for mathematical convenience.
We elaborate on this difficulty in doing this be-
low.
Let us drop the projective assumption, so that
the physical measurements are in general POVM
measurements. Imagine we aim to self-test a set
of measurements M
0
a|x
for Alice and the refer-
ence state |ψ
0
i in the sense of definition 2. This
immediately poses a problem for the majority
of self-testing proofs. For example in (30), one
needs the physical measurements to be projec-
tive to guarantee A
2
x
= in order to prove anti-
commutativity of the observables. In order to
proceed to use the standard proofs of measure-
ment self-testing, of which almost all assume the
projective nature of the measurements, one has
three options:
1. Prove that even if treating the physical mea-
surements as POVM measurements, the only
physical measurements that are compatible
with the observed correlations are projective
measurements.
Accepted in Quantum 2020-23-08, click title to verify 47
2. Prove a general theorem that states that if
one has an isometry mapping any Naimark
dilation of the physical measurements to the
reference measurements, this implies a (pos-
sibly different) isometry mapping the POVM
physical measurements to the reference mea-
surements in the sense of definition 2.
3. Argue that projective measurements are the
only fundamental measurements in quantum
theory. That is, POVM measurements can
only be physically realised via a projective
measurement on a dilated space.
Option 1 has been done very rarely; perhaps
the only example can be found in [Kan16] when
self-testing qubit Pauli measurements. Option 2
would be an analogue for the ability to use a pu-
rification of the state as explained above. How-
ever, such a theorem has not been proven to the
best of our knowledge, and my may not be possi-
ble. We comment on option 3 at the end of this
section.
Let us focus further on option 2. Define the
physical measurements for Alice as usual by M
a|x
,
which may now be POVM. Furthermore, define a
Naimark dilation of these measurements by
˜
M
a|x
.
The vast majority of self-testing works prove an
isometry mapping the dilated measurements
˜
M
a|x
to the reference measurements:
Φ
h
˜
M
a|x
⊗
B
|ψi ⊗
˜
0
i
= M
0
a|x
⊗
B
0
ψ
0
⊗ |ξi,
(76)
where we have explicitly written the ancilla state
˜
0
∈ H
˜
A
used for the dilation, and the purifica-
tion space of the state is left implicit as in section
4. Now consider the physical (potentially POVM)
measurements acting on the physical state.
M
a|x
⊗
B
|ψi. (77)
At this point, it is tempting to define an isometry
Ω that maps this measurement to it’s Naimark
dilation:
Ω[M
a|x
⊗
B
|ψi] =
˜
M
a|x
⊗
B
0
|ψi ⊗
˜
0
, (78)
One could then use the standard proof of self-
testing by concatenating isometries. Note, how-
ever, that fixing an input x, the vectors on the
right hand side of (78) are orthogonal for differ-
ent a since for a 6= a
0
hψ| ⊗ h
˜
0|
˜
M
a
0
|x
⊗
˜
M
a|x
⊗ |ψi ⊗
˜
0
=hψ| ⊗ h
˜
0|
˜
M
a
0
|x
˜
M
a|x
⊗ |ψi ⊗
˜
0
= 0. (79)
The corresponding inner product between the
vectors inside the isometry in (78) is
hψ|M
a
0
|x
M
a|x
⊗ |ψi, (80)
which is generally not equal to zero if the mea-
surements are POVM. Thus, (78) is generally
not valid, since isometry maps conserve the in-
ner product between vectors. This means that
one cannot simply consider a Naimark dilation
‘for free’ since the map which takes the POVM
measurements to the dilation is not an isometry.
Nevertheless, one may hope to prove that given
a self-test of the dilated measurements (76), one
could recover the corresponding map for the phys-
ical measurements by somehow discarding the an-
cilla degrees of freedom used in the dilation, as
was done for the purification space of the state.
The difficulty here however is that—unlike the
purification space— the dilated space is explic-
itly used by the isometry, since it is constructed
from the dilated measurements.
Consider a general state ρ on H
A
. For any
Naimark dilation
˜
M
a|x
of M
a|x
one has
tr[
˜
M
a|x
ρ ⊗ |
˜
0ih
˜
0|] = tr[M
a|x
ρ]. (81)
Tracing over the ancilla space only we have
tr[ ⊗ h
˜
0|(
˜
M
a|x
) ⊗
˜
0
ρ] = tr[M
a|x
ρ] (82)
and since this holds for all ρ we must have
M
a|x
= ⊗ h
˜
0|(
˜
M
a|x
) ⊗
˜
0
. (83)
Returning to (76), we insert the identity
A
⊗
P
k
|
˜
kih
˜
k|
˜
A
inside the isometry, giving
Φ
"
(
A
⊗
X
k
|
˜
kih
˜
k|)
˜
M
a|x
) ⊗
B
|ψi ⊗
˜
0
#
= M
0
a|x
⊗
B
0
ψ
0
⊗ |ξi. (84)
Taking the sum outside we find
X
k
Φ
h
˜
M
k
a|x
⊗
˜
A
⊗
B
|ψi ⊗
˜
k
Ei
= M
0
a|x
⊗
B
0
ψ
0
⊗ |ξi. (85)
Where
˜
M
k
a|x
=
A
⊗ h
˜
k|(
˜
M
a|x
)
A
⊗
˜
0
. (86)
From (83) we have
˜
M
0
a|x
= M
a|x
and so
Φ
M
a|x
⊗
˜
A
⊗
B
|ψi ⊗
˜
0
(87)
= M
0
a|x
⊗ |ψ
0
i ⊗ |ξi −
X
k>0
Φ
h
˜
M
k
a|x
⊗
B
|ψi ⊗
˜
k
i
.
Accepted in Quantum 2020-23-08, click title to verify 48
If the sum in the right hand side of the above
is zero then we have proven that the isometry Φ
indeed maps the physical measurements to the
reference measurements. However, it is not clear
that there exists a Naimark extension such that
this is always the case. Showing whether this is or
is not possible would be a valuable contribution
to the field.
Finally we comment on option 3. This posi-
tion can be justified on the basis that all other
operations in quantum theory (CPTP maps, non-
projective measurements) can be seen as the re-
sult of combining unitary evolution and projec-
tive measurement, and so there is no logical prob-
lem that arises with this stance. For some, this
argument is not convincing however, since they
consider POVMs to be as ‘real’ as projective mea-
surements, and thus deserve a place in the ontol-
ogy of the theory.
An alternative way to argue the projective-only
assumption is from a information theoretic per-
spective. More precisely, the measurement up-
date rule in quantum theory can be understood
as a process by which an observer updates their
description of a quantum state given new infor-
mation. This new information is given by the
outcome of a measurement, and is classical in-
formation from the perspective of the observer
(i.e. it is the classical information that the ob-
server reads from the macroscopic degrees of free-
dom of the measurement device). Thus it is onto
these degrees of freedom (which may have become
entangled with the quantum system during the
measurement procedure) that the measurement
operators act. Since these degrees of freedom are
perfectly distinguishable, the measurement oper-
ators are described by orthogonal projectors. If
one performs another measurement, this amounts
to applying a different unitary Schrodinger evo-
lution to the state before observing the mea-
surement procedure, thus mapping the measure-
ment to a different projective measurement. The
POVM update rule can then be understood as
the effective measurement operator applied to the
state that is induced by the observation of this
classical information. In this sense, any measure-
ment operator—which by definition describes the
effect of obtaining classical information—is de-
scribed by a projector.
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The paragraph after Equation (51) looks out of place, maybe it should be put before (51)?