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