Semi-device-independent certification of indefinite causal
order
Jessica Bavaresco
1
, Mateus Araújo
2
, Časlav Brukner
1,3
, and Marco Túlio Quintino
4
1
Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090
Vienna, Austria
2
Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany
3
Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics,
University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
4
Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033,
Japan
When transforming pairs of independent
quantum operations according to the fun-
damental rules of quantum theory, an in-
triguing phenomenon emerges: some such
higher-order operations may act on the in-
put operations in an indefinite causal or-
der. Recently, the formalism of process
matrices has been developed to investig-
ate these noncausal properties of higher-
order operations. This formalism pre-
dicts, in principle, statistics that ensure
indefinite causal order even in a device-
independent scenario, where the involved
operations are not characterised. Nev-
ertheless, all physical implementations of
process matrices proposed so far require
full characterisation of the involved oper-
ations in order to certify such phenom-
ena. Here we consider a semi-device-
independent scenario, which does not re-
quire all operations to be characterised.
We introduce a framework for certifying
noncausal properties of process matrices
in this intermediate regime and use it to
analyse the quantum switch, a well-known
higher-order operation, to show that, al-
though it can only lead to causal statistics
in a device-independent scenario, it can ex-
hibit noncausal properties in semi-device-
independent scenarios. This proves that
the quantum switch generates stronger
noncausal correlations than it was previ-
ously known.
Jessica Bavaresco: jessica.bavaresco@oeaw.ac.at
Marco Túlio Quintino: quintino@eve.phys.s.u-tokyo.ac.jp
A common quantum information task consists
in certifying that some uncharacterised source is
preparing a system with some features. By mak-
ing the assumption that the measurement devices
are completely characterised, that is, that they
are known exactly, it is possible to infer proper-
ties of the system. In this device-dependent scen-
ario, fidelity of a quantum state with respect to a
target state can be estimated, entanglement wit-
nesses can be evaluated [1], and even complete
characterisation of the source via state tomo-
graphy is possible [2].
Remarkably, it is possible to certify properties
of systems even without fully characterizing the
measurement devices [3, 4]. In such a device-
independent scenario it is only assumed that the
measurements are done by separated parties and
compose under a tensor product, which is justi-
fied by implementing them with a space-like sep-
aration. Under these circumstances, Bell scen-
arios can be used to certify properties like en-
tanglement of quantum states [4], incompatib-
ility of quantum measurements [5], or to per-
form device-independent state estimation via self-
testing [6, 7].
Since the assumptions are weaker, demonstra-
tions of device-independent certification are usu-
ally experimentally challenging. For instance, al-
though experimental device-independent certific-
ation of entanglement has been reported [8, 9, 10,
11], its experimental difficulty has so far preven-
ted its use in practical applications such as device-
independent quantum key distribution [12] and
randomness certification [13].
An interesting middle ground is the semi-
device-independent scenario, where assumptions
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arXiv:1903.10526v3 [quant-ph] 9 Aug 2019
are made about some parties but not others.
Semi-device-independent schemes have been de-
veloped and extensively studied for the certific-
ation of entanglement [14] and measurement in-
compatibility [15, 16], known as EPR-steering,
and applied to quantum key distribution proto-
cols where some but not all parties can be trusted
[17].
A close analogy can be developed with regard
to the certification of indefinite causal order, as
encoded in a process matrix [18]. A process mat-
rix is a higher-order operation [19, 20, 21] – i.e.
a transformation of quantum operations – that
acts on independent sets of operations. Funda-
mental laws of quantum theory predict the exist-
ence of process matrices that act on these opera-
tions in a such a way that a well-defined causal
order cannot be established among them. Pro-
cess matrices with indefinite causal order were
proven to be a powerful resource, outperforming
causally ordered ones in tasks such as quantum
channel discrimination [22], communication com-
plexity [23, 24], quantum computation [25], and
inverting unknown unitary operations [26].
To certify that a process matrix in fact does
not act in a causally ordered way, there are two
standard methods available in the literature. The
first is to evaluate a causal witness [27, 28]. Ana-
logous to the evaluation of an entanglement wit-
ness, this method relies on detailed knowledge of
the quantum operations being implemented, and
as such it allows for a device-dependent certifica-
tion. All experimental certifications of indefinite
causal order to date either measure a causal wit-
ness [29, 30] or rely on similar device-dependent
assumptions [31, 32, 33]. The second method is
the violation of a causal inequality, phenomenon
which is also predicted by quantum mechanics
[18, 34]. Analogous to the violation of a Bell in-
equality, this method does not rely on detailed
knowledge of the quantum operations implemen-
ted by the parties, but rather only that they com-
pose under a tensor product. As such, it allows
for a device-independent certification. When a
causal inequality is violated, it is verified that
least one of the principles used to derived it is
not respected. Moreover, this claim holds true in-
dependently of the physical theory that supports
the experiment that led to the violation. Hence,
although we focus on quantum theory, indefinite
causal order could in principle be certified even
without relying on the laws of quantum mechan-
ics. Although it would be highly desirable to per-
form such device-independent certification of in-
definite causal order, no physical implementation
of process matrices that would violate a causal
inequality is currently known.
In this work, we introduce a semi-device-
independent framework for certifying noncausal
properties of process matrices that allows for an
experimental certification of indefinite causal or-
der that relies on fewer assumptions than previ-
ous ones. In our semi-device-independent scen-
ario, the operations of some parties are fully char-
acterized while no assumptions are made about
the others.
We begin by considering the bipartite case,
for which we construct a general framework
for certifying indefinite causal order in a semi-
device-independent scenario, while contextualiz-
ing previously developed device-dependent and -
independent ones. We then extend our framework
to a tripartite case in which the third party is al-
ways in the future of the other two, and provide
an extensive machinery that may be generalized
to other multipartite scenarios. We apply our
methods to the notorious quantum switch [22, 35],
a process matrix that, despite having an indefin-
ite causal order, only leads to causal correlations
in a device-independent scenario. We show that
the noncausal properties of the quantum switch
can be certified in a semi-device-independent way,
proving that it generates stronger noncausal cor-
relations than it was previously known.
1 Preliminaries
In our certification scheme, we will deal with
statistical data in the form of behaviours.
A general bipartite behaviour {p
(
ab|xy
)
} is a set
of joint probability distributions, that is, a set in
which each element p
(
ab|xy
)
is a real non-negative
number such that
P
a,b
p
(
ab|xy
) = 1
for all x, y,
where a ∈ {
1
, . . . , O
A
} and b ∈ {
1
. . . , O
B
} are
labels for outcomes and x ∈ {
1
, . . . , I
A
} and y ∈
{
1
, . . . , I
B
} are labels for inputs, for parties Alice
and Bob, respectively.
The most general operation allowed by
quantum theory is modelled by a quantum instru-
ment, which is a set of completely-positive (CP)
maps that sum to a completely-positive trace-
preserving (CPTP) map [36].
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The Choi-Jamiołkowski (CJ) isomorphism [37,
38, 39] allows us to represent every linear map
1
M
:
L
(
H
I
)
→ L
(
H
O
)
by a linear operator M ∈
L
(
H
I
⊗H
O
)
acting on the joint input and output
Hilbert spaces. In this representation, a set of
instruments is a set of operators {I
a|x
}, I
a|x
∈
L(H
I
⊗ H
O
), that satisfies
I
a|x
≥ 0, ∀ a, x (1)
Tr
O
X
a
I
a|x
= 1
I
, ∀ x, (2)
where x ∈ {
1
, . . . , I} labels the instrument in the
set and a ∈ {
1
, . . . , O} its outcomes, and 1
I
is
the identity operator on H
I
.
Now let us consider the most general set of
behaviours which respects quantum theory. For
that we follow the steps of ref. [18] to analyse
behaviours that can be extracted by pairs of
independent quantum instruments. Let {A
a|x
},
A
a|x
∈ L
(
H
A
I
⊗ H
A
O
)
and {B
b|y
}, B
b|y
∈
L
(
H
B
I
⊗ H
B
O
)
be the Choi operators of Alice’s
and Bob’s local instruments. We then seek a
function which assigns probabilities to a pair of
instrument elements A
a|x
and B
b|y
. In order
to preserve the structure of quantum mechan-
ics, we assume that this function is linear in
both arguments. It follows from the Riesz rep-
resentation lemma [40] that this general linear
function necessarily has the form of p
(
ab|xy
) =
Tr
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
) W
i
for some linear oper-
ator W ∈ L
(
H
A
I
⊗ H
A
O
⊗ H
B
I
⊗ H
B
O
)
. In or-
der to be consistent with extended quantum scen-
arios, we also consider the case where Alice and
Bob may share a (potentially entangled) auxili-
ary quantum state ρ ∈ L
(
H
A
I
0
⊗H
B
I
0
)
, and have
instruments {A
0
a|x
}, A
0
a|x
∈ L
(
H
A
I
0
A
I
A
O
)
, {B
0
b|y
},
B
0
b|y
∈ L
(
H
B
I
0
B
I
B
O
)
which acts on the space of
the operator W and the auxiliary state ρ. A pro-
cess matrix is then defined as the most general
linear operator W such that
p(ab|xy) =
Tr
h
(A
0A
I
0
A
I
A
O
a|x
⊗ B
0B
I
0
B
I
B
O
b|y
)(W
A
I
A
O
B
I
B
O
⊗ ρ
A
I
0
B
I
0
)
i
(3)
represents
2
elements of valid probability distribu-
tions for every state ρ and sets of instruments
{A
0
a|x
} and {B
0
b|y
}.
1
In this paper we only consider finite dimensional com-
plex linear spaces. That is, all linear spaces are isomorphic
to C
d
for some natural number d.
2
In eq. (3), as well as in some other equations, we add
superscripts on the operators to indicate the Hilbert spaces
in which they act, for sake of clarity.
It was shown in ref. [27] that a linear operator
W is a process matrix if and only if it respects
W ≥ 0 (4)
Tr W = d
A
O
d
B
O
(5)
A
I
A
O
W =
A
I
A
O
B
O
W (6)
B
I
B
O
W =
A
O
B
I
B
O
W (7)
W =
A
O
W +
B
O
W −
A
O
B
O
W, (8)
where
X
W
:
=
Tr
X
W ⊗
1
X
d
X
is the trace-and-replace
operation and d
X
= dim(H
X
).
We then define process behaviours {p
Q
(
ab|xy
)
}
as behaviours which can be obtained by process
matrices according to
p
Q
(ab|xy) = Tr
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
) W
A
I
A
O
B
I
B
O
i
,
(9)
for all a, b, x, y.
Now we begin to discuss the causal properties
of behaviours and process matrices.
A behaviour is considered causally ordered
when it can be established that one party acted
before the other because the marginal probability
distributions of one party do not depend on the
inputs of the other. Formally, we have that a be-
haviour {p
A≺B
(
ab|xy
)
} is causally ordered from
Alice to Bob if it satisfies
X
b
p
A≺B
(ab|xy) =
X
b
p
A≺B
(ab|xy
0
), (10)
for all a, x, y, y
0
, and equivalently from Bob to
Alice.
Behaviours that are within the convex hull of
causally ordered behaviours are also considered
causal, as they can be interpreted as a classical
mixture of causally ordered behaviours. Hence,
a causal behaviour {p
causal
(
ab|xy
)
} is a behaviour
that can be expressed as a convex combination of
causally ordered behaviours, i.e.,
p
causal
(ab|xy)
:
= qp
A≺B
(ab|xy) + (1 − q)p
B≺A
(ab|xy),
(11)
for all a, b, x, y, where
0
≤ q ≤
1
is a real number.
Behaviours that do not satisfy eq. (11) are called
noncausal behaviours.
Following the same reasoning as the one in the
definition of a general process matrix, in order
to associate causal properties to process matrices
we now define causally ordered process matrices
as the most general operator that takes pairs of
local instruments to causally ordered behaviours,
that is,
p
A≺B
(ab|xy) = Tr
h
(A
a|x
⊗ B
b|y
) W
A≺B
i
, (12)
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for all a, b, x, y. This definition is equivalent to
the one in refs. [20, 18], which states that a bi-
partite process matrix W
A≺B
∈ L
(
H
A
I
⊗H
A
O
⊗
H
B
I
⊗H
B
O
)
is causally ordered from Alice to Bob
if it satisfies
W
A≺B
=
B
O
W
A≺B
, (13)
and equivalently from Bob to Alice.
In line with the definition of a causal beha-
viour, a causally separable process matrix W
sep
∈
L
(
H
A
I
⊗H
A
O
⊗H
B
I
⊗H
B
O
)
is a process matrix
that can be expressed as a convex combination of
causally ordered process matrices, i.e.,
W
sep
:
= qW
A≺B
+ (1 − q)W
B≺A
, (14)
where
0
≤ q ≤
1
is a real number. Process
matrices that do not satisfy eq. (14) are called
causally nonseparable process matrices.
2 Certification
Let us consider the following task: we are
given a behaviour that describes the statistics of a
quantum experiment. We analyse this behaviour
in the process matrix formalism, that is, we as-
sume that there exists a process matrix W and
sets of local instruments that give rise to this be-
haviour according to the rules of quantum theory.
Without any information about W – i.e., without
direct assumptions about the process matrix – the
goal is to verify whether it is causally nonsepar-
able. Additionally, information about the instru-
ments which were performed may or may not be
given.
The assumptions about the instruments
can be split in three: device-dependent, -
independent, and semi-device-independent. A
device-dependent certification scenario is one in
which the operations of all parties are fully char-
acterised, i.e., the whole matrix description of
the elements of all applied instruments is known.
A device-independent certification scenario is the
opposite, no knowledge or assumption is made re-
garding the operations performed by any parties,
not even the dimension of the linear spaces
used to describe them. Finally, a semi-device-
independent certification scenario is one in which
at least one party is device-dependent, which is
often called trusted, and at least one is device-
independent, often called untrusted.
In the following we formalise our notions of cer-
tification for bipartite process matrices. We refer
to appendix E and to section 3 for a discussion of
more general scenarios.
Definition 1
(Device-dependent certification)
.
Given a process behaviour
{p
Q
(
ab|xy
)
}
, that arises
from known instruments
{A
a|x
}
and
{B
b|y
}
and
an unknown bipartite process matrix, one certifies
that this process matrix is causally nonseparable
in a device-dependent way if
p
Q
(ab|A
a|x
, B
b|y
) 6= Tr
h
(A
a|x
⊗ B
b|y
)W
sep
i
,
(15)
for all
a, b, x, y
, and for all causally separable
process matrices W
sep
.
Definition 2
(Device-independent certification)
.
Given a process behaviour
{p
Q
(
ab|xy
)
}
, that arises
from unknown instruments and an unknown bi-
partite process matrix, one certifies that this pro-
cess matrix is causally nonseparable in a device-
independent way if
p
Q
(ab|xy) 6= Tr
h
(A
a|x
⊗ B
b|y
)W
sep
i
(16)
for all
a, b, x, y
, and for all causally separable
process matrices
W
sep
and all general instruments
{A
a|x
} and {B
b|y
}.
Definition 3
(Semi-device-independent certific-
ation)
.
Given a process behaviour
{p
Q
(
ab|xy
)
}
,
that arises from unknown instruments on Alice’s
side, known instruments
{B
b|y
}
on Bob’s side,
and an unknown bipartite process matrix, one
certifies that this process matrix is causally non-
separable in a semi-device-independent way if
p
Q
(ab|x , B
b|y
) 6= Tr
h
(A
a|x
⊗ B
b|y
)W
sep
i
(17)
for all
a, b, x, y
, and for all causally separable
process matrices
W
sep
and all general instruments
{A
a|x
}.
On eqs. (15), (16), and (17) one can identify
the quantities that are given – the behaviour and
the trusted instruments that belong to the device-
dependent parties – and the variables in the certi-
fication problem – the unknown instruments that
belong to the device-independent parties and any
causally separable process matrix. If one can
guarantee that, for any sets of instruments for the
device-independent parties and any causally sep-
arable process matrix, the given behaviour can-
not be described by the left-hand side of eqs. (15),
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Device-dependent
Given quantities Variables
{p
Q
(ab|xy)} W
{A
a|x
}, {B
b|y
}
Device-independent
Given quantities Variables
{p
Q
(ab|xy)} d
A
I
, d
A
O
, d
B
I
, d
B
O
{A
a|x
}, {B
b|y
}
W
Semi-device-independent
Given quantities Variables
{p
Q
(ab|xy)} d
A
I
, d
A
O
{B
b|y
} {A
a|x
}
W
Table 1: Comparison between the given (known) quant-
ities and the variables (unknown quantities) in each scen-
ario with different levels of assumptions about the op-
erations of each party involved in the task of certifying
noncausal properties of a process matrix.
(16), or (17), then the fact that the process matrix
that generated this behaviour is causally nonsep-
arable is certified. A summary of the given quant-
ities and variables in each certification scenario is
provided by table 1.
Before proceeding we remark an analogy with
the entanglement certification problem in which
behaviours are assumed to arise from quantum
measurements performed on a quantum state.
In the entanglement certification case, device-
dependent scenarios are related to entangle-
ment witnesses [1], device-independent scenarios
to Bell nonlocality [4], and the semi-device-
independent ones to EPR-steering [14].
Device-dependent
We start be analysing the device-dependent
scenario and definition 1. This scenario has been
thoroughly studied before using the concept of
causal witnesses [27], analogous to entanglement
witnesses [1], and now we formulate it under our
certification paradigm.
In this scenario, the behaviour and sets of in-
struments of both parties are given and we aim to
check whether the given process behaviour is con-
sistent with performing these exact instruments
on a causally separable process matrix. This
problem can be solved by semidefinite program-
ming (SDP) with the following formulation:
given {p
Q
(ab|xy)}, {A
a|x
}, {B
b|y
}
find W
subject to p
Q
(ab|xy) = Tr
(A
a|x
⊗ B
b|y
)W
∀ a, b, x, y
W ∈ SEP,
(18)
where SEP denotes the set of causally separable
matrices (i.e. W is constrained to eq. (14)) which
can be characterized by SDP.
If the problem is infeasible, that is, if there does
not exist a process matrix W that satisfies the
constraints of eq. (18), then the process matrix
that generated {p
Q
(
ab|xy
)
} is certainly causally
nonseparable. Consequently, there exists a causal
witness that can certify it, without the need of
performing full tomography of the process mat-
rix [27]. On the contrary, if the problem is feas-
ible, then the solution provides a causally separ-
able process matrix that could have generated the
given behaviour.
We now show that all causally nonseparable
process matrices can be certified to be so in a
device-dependent way.
Theorem 1.
All noncasual process matrices can
be certified in a device-dependent way for some
choice of instruments.
The proof of the above theorem follows from
the fact that one can always consider a scenario
where Alice and Bob have access to tomograph-
ically complete instruments, which allows for the
complete characterization of the process matrix
[41, 42].
Device-independent
A device-independent approach for the process
matrix formalism has been previously studied in
terms of causal inequalities [34], analogous to Bell
inequalities [4], and now we formulate it under
our certification paradigm, exploring definition 2.
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In a device-independent scenario, besides as-
suming that the given behaviour is a process be-
haviour following eq. (9), no extra assumptions
are made. It is then necessary to check whether
the given behaviour is consistent with probability
distributions that come from any tensor product
of pair of sets of instruments, with fixed number
of inputs and outputs, performed on any causally
separable process matrix of any dimension.
Before characterizing the problem of whether
a certain behaviour can be obtained by a caus-
ally separable process matrix, we ask the more
fundamental question of whether any behaviour
can be obtained by a general process matrix on
which instruments are performed locally. That is,
whether all general (valid) behaviours are process
behaviours.
By definition, every process matrix leads to
valid general behaviours. However, it is shown
in ref. [43] that in the scenario where all parties
have dichotomic inputs and outputs, the determ-
inistic two-way signalling behaviour defined by
p
2WS
(
ab|xy
) :=
δ
a,y
δ
b,x
, where δ
i,j
= 1
if i
=
j
and δ
i,j
= 0
otherwise, cannot be obtained ex-
actly by any process matrix. Here we show that
one cannot obtain this two-way signalling beha-
viour even approximately for finite-dimensional
process matrices. The proof can be found in ap-
pendix A.
Theorem 2.
All process behaviours are valid be-
haviours, however, not all valid behaviours are
process behaviours.
In particular, in the scenario where all parties
have dichotomic inputs and outputs, any beha-
viour {p(ab|xy)} such that
1
4
X
a,b,x,y
δ
a,y
δ
b,x
p(ab|xy) > 1 −
1
d + 1
is not a process behaviour for process matrices
with total dimension d
A
I
d
A
O
d
B
I
d
B
O
= d.
Here we can make a parallel with Bell non-
locality, where the Popescu-Rohrlich behaviour
is known to respect the non-signalling conditions
which arise naturally in Bell scenarios but cannot
be obtained by performing local measurements on
entangled states [44, 45, 46].
As for causal behaviours, the analogous ques-
tion is also pertinent. Can all causal behaviours
be obtained by pair of sets of instruments and
causally separable process matrices? We answer
this question positively, which allows us to re-
late the properties of a behaviour directly to the
properties of the process matrices that could have
given rise to it.
Lemma 1.
A general behaviour is causal if and
only if it is a process behaviour that can be ob-
tained by a causally separable process matrix.
The proof is made by explicitly constructing
the instruments and causally separable process
matrix that can recover any causal behaviour. It
can be found in appendix B.
This result allows us to identify which causally
nonseparable process matrices can be certified in
a device-independent way:
Theorem 3.
A process matrix can be certified to
be causally nonseparable in a device-independent
way if and only if it can generate a noncausal
behaviour for some choice of instruments for Alice
and Bob.
Proof.
If a process matrix is causally separable
then its behaviours will be causal. If a behaviour
is causal, even though it could in principle have
been generated by a causally nonseparable process
matrix, according to lemma 1 it can always be
reproduced by a causally separable process matrix.
Hence, no causal properties of the process matrix
can be inferred.
From the formulation of the device-
independent certification problem in definition 2,
it is not clear whether one could obtain a simple
characterisation to solve it. In particular, be-
cause there are no constraints on the dimension
of the linear spaces and there is a product of
variables (that represent the unknown instru-
ments and process matrix). Interestingly, we
can explore the above theorem to present simple
necessary and sufficient conditions for a general
behaviour to allow for device-independent certi-
fication of indefinite causal order. This follows
from the fact that a behaviour can be checked
to be noncausal by linear programming [34].
More explicitly, the certification problem can be
formulated as follows:
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 6
given {p
Q
(ab|xy)}
find q
1
(λ), q
2
(λ)
s.t. p
Q
(ab|xy) =
X
λ
h
q
1
(λ)D
A≺B
λ
(ab|xy)+
+ q
2
(λ)D
B≺A
λ
(ab|xy)
i
, ∀ a, b, x, y
q
1
(λ) ≥ 0, q
2
(λ) ≥ 0, ∀ λ,
(19)
where {D
A≺B
λ
(
ab|xy
)
} and {D
B≺A
λ
(
ab|xy
)
} are
the finite set of deterministic causal distributions
described in ref. [34].
If the problem is infeasible, then the process
matrix that was used to generate the process be-
haviour {p
Q
(
ab|xy
)
} is certainly causally nonsep-
arable and there exists a causal inequality that
can witness it [34]. If the problem is feasible, then
one can use the results presented in appendix B to
explicitly find a causally separable process matrix
W
sep
and sets of instruments {A
a|x
} and {B
b|y
}
such that p
Q
(ab|xy) = Tr
h
(A
a|x
⊗ B
b|y
)W
sep
i
.
Differently from the device-dependent scenario,
it is known that some causally nonseparable pro-
cess matrices cannot be certified in a device-
independent way [27, 47, 48]. In particular,
there exist causally nonseparable bipartite pro-
cess matrices that, for any choice of instruments
of Alice and Bob, will always lead to causal beha-
viours. This result was first presented in ref. [48]
and we rephrase it here:
Proposition 1
(Device-dependent certifiable,
device-independent noncertifiable process matrix)
.
There exist causally nonseparable process matrices
that, for any sets of instruments, always give rise
to causal behaviours. That is, a causally nonsep-
arable process matrix that cannot be certified in a
device-independent way.
In particular, let
W ∈ L
(
H
A
I
A
O
B
I
B
O
) be a pro-
cess matrix and
W
T
B
be the partial transposition
of
W
with respect to some basis in
L
(
H
B
I
B
O
) for
Bob. If
W
T
B
is causally separable, the behaviour
generated by
p
Q
(
ab|xy
) =
Tr
h
(A
a|x
⊗ B
b|y
) W
i
is
causal for every sets of instruments
{A
a|x
}
and
{B
b|y
}.
We would like to remark that this phenomenon
can be seen as a consequence of the choice of
definition of causally separable process matrices.
Recalling section 1, a causally ordered process
matrix is defined as the most general operator
W
A≺B
that takes any pairs of sets of instru-
ments to causally ordered behaviours according
to p
A≺B
(
ab|xy
) =
Tr
[(
A
a|x
⊗ B
b|y
)
W
A≺B
]
. On
the other hand, the definition of a causally sep-
arable process matrix W
sep
as a convex combin-
ation of process matrices with definite causal or-
ders, instead of focusing on the behaviours, has
an arguably more physical motivation of a clas-
sical mixture of causal orders. If the definition
were to, alternatively, focus on the behaviours,
then a natural choice would be to define a ‘caus-
ally separable’ process matrix
˜
W
sep
as the most
general operator that takes any pairs of sets of
instruments to causal behaviours according to
p
causal
(
ab|xy
) =
Tr
[(
A
a|x
⊗ B
b|y
)
˜
W
sep
]
. With
this alternative, inequivalent definition, ‘caus-
ally nonseparable’ process matrices would always
lead to noncausal behaviours, for some choice
of instruments, by definition. We observe the
phenomenon of causally nonseparable process
matrices leading exclusively to causal behaviours,
presented in proposition 1, when we take the phys-
ically motivated definition, and it exposes an in-
trinsic difference between these two kinds of reas-
oning. In the next sections, we show how this
phenomenon manifests itself in the semi-device-
independent scenario.
Semi-device-independent
In this final scenario, which has not been ex-
plored for process matrices before, we have the
information of the behaviour and the instruments
of one party, in this case, Bob. According to defin-
ition 3, one needs to check whether the given be-
haviour can be reproduced by performing these
exact given instruments for Bob, and any set of
instruments for Alice with fixed number of inputs
and outputs, on any causally separable process
matrix that has a fixed dimension on Bob’s side.
Since both the process matrix and Alice’s instru-
ments are variables in eq. (17), it is not clear
whether this problem can be solved by SDP.
Our approach contrasts a previous one which
exploits communication complexity tasks to cer-
tify indefinite causal order in process matrices as-
suming an upper bound for communication capa-
city between parties and the dimension of their
local systems [23, 24].
However, consider the following expression for
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 7
a process behaviour,
p(ab|xy) = Tr
h
(A
a|x
⊗ B
b|y
)W
i
(20)
= Tr
h
B
b|y
Tr
A
(A
a|x
⊗ 1
B
W )
i
(21)
= Tr
B
b|y
w
Q
a|x
, ∀ a, b, x, y, (22)
which motivates us to define w
Q
a|x
.
Definition 4
(Process assemblage)
.
A process
assemblage
{w
Q
a|x
}
is a set of operators
w
Q
a|x
∈
L
(
H
B
I
⊗ H
B
O
) for which there exist a process
matrix
W
A
I
A
O
B
I
B
O
and a set of instruments
{A
A
I
A
O
a|x
} such that
w
Q
a|x
= Tr
A
I
A
O
h
(A
A
I
A
O
a|x
⊗ 1
B
I
B
O
)W
A
I
A
O
B
I
B
O
i
,
(23)
for all a, x.
By defining the process assemblage, we gather
all the variables in the certification problem in
one object and can start to relate properties of
this object to properties of the process matrix.
We remark that the process assemblage gener-
alizes the notion of assemblage in EPR-steering
[14, 49], which is recovered when both Alice’s and
Bob’s output spaces have d
A
O
=
d
B
O
= 1
. Con-
sequently, {A
a|x
} becomes a set of POVMs, W
becomes a bipartite quantum state, and the pro-
cess assemblage recovers the steering assemblage
σ
a|x
= Tr
A
(A
a|x
⊗ 1
B
ρ
AB
) [49].
Let us first examine the equation below more
closely:
p(ab|xy) = Tr
B
b|y
w
a|x
. (24)
In the same way that a process matrix was
defined as the most general operator that takes
sets of local instruments to a behaviour, we can
define a general assemblage to be the most gen-
eral object that takes a set of instruments to a
valid behaviour and respects linearity. In the ap-
pendix C, we prove that this definition is equival-
ent to:
Definition 5
(General assemblage)
.
A general
assemblage
{w
a|x
}
is a set of operators
w
a|x
∈
L(H
B
I
⊗ H
B
O
) that satisfies
w
a|x
≥ 0 ∀ a, x (25)
Tr
X
a
w
a|x
= d
B
O
∀ x (26)
X
a
w
a|x
=
B
O
X
a
w
a|x
∀ x. (27)
By defining the general assemblage as the most
general set of operators that takes a set of instru-
ments to a behaviour and respects linearity, we
are no longer considering its relation with a pro-
cess matrix or requiring that it is a process as-
semblage.
If one compares the set of all general as-
semblages to the set of all process assemblages,
it is clear that the set of general assemblages con-
tains the set of process assemblages, since one
can see from eq. (21) that all process assemblages
lead to valid behaviours. But the former set is in
principle larger, an outer approximation with a
simpler characterisation. We show that, indeed,
the set of general assemblages is larger than the
set of process assemblages, because just like gen-
eral behaviours, not all general assemblages can
be realised by process matrices.
Theorem 4.
All process assemblages are valid
assemblages, however, not all valid assemblages
are process assemblages.
In particular, in the scenario where Alice has
dichotomic inputs and outputs, the general as-
semblage
{w
a|x
}
given by
w
a|x
=
|xihx|⊗|aiha|
is
not a process assemblage.
The proof presented in appendix A is based on
the fact that this assemblage can lead to a de-
terministic two-way signalling behaviour, which
we know not to be attainable by process matrices
from theorem 2.
Although the process assemblage can be re-
garded as a generalisation of the steering as-
semblage that arises in EPR-steering scenarios,
here we point out an important difference
between semi-device-independent certification of
indefinite causal order and entanglement. A fun-
damental result on EPR-steering theory is that
all steering assemblages admit a quantum realisa-
tion by performing POVMs on a quantum state
[50, 51, 52]. On the other hand, theorem 4 shows
that some process assemblages do not admit a
quantum realisation by performing a set of instru-
ments on a process matrix.
The next step is to assign causal properties to
assemblages. A natural approach is to define that
an assemblage {w
Q, A≺B
a|x
} is causally ordered from
Alice to Bob if it is a process assemblage that can
be obtained from a process matrix that is causally
ordered from Alice to Bob, namely,
w
Q, A≺B
a|x
= Tr
A
[(A
a|x
⊗1
B
) W
A≺B
] ∀ a, x, (28)
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for some set of instruments {A
a|x
} and some caus-
ally ordered process matrix W
A≺B
, and equival-
ently from Bob to Alice.
Since the above definition depends on an un-
known set of instruments {A
a|x
}, it is not easy to
check whether a given general assemblage {w
a|x
}
is causally ordered. We therefore derive a simpler
characterisation of causally ordered assemblages
that is equivalent to the one above.
Analogously to how we defined a general as-
semblage, we characterise the most general set
of operators {w
A≺B
a|x
} that give rise to a caus-
ally ordered behaviour {p
A≺B
(
ab|xy
)
} according
to the equation
p
A≺B
(ab|xy) = Tr(B
b|y
w
A≺B
a|x
), (29)
for any set of instruments {B
b|y
} for Bob. We
then prove its equivalence to the definition below
in appendix C.
Definition 6
(Causally ordered assemblages)
.
An assemblage
{w
A≺B
a|x
}
is causally ordered from
Alice to Bob if it satisfies
w
A≺B
a|x
=
B
O
w
A≺B
a|x
∀ a, x, (30)
while an assemblage
{w
B≺A
a|x
}
is causally ordered
from Bob to Alice if it satisfies
X
a
w
B≺A
a|x
=
X
a
w
B≺A
a|x
0
∀ x, x
0
. (31)
In appendix B, we show that all causally
ordered assemblages {w
A≺B
a|x
} and {w
B≺A
a|x
} can
be realized by some set of instruments {A
a|x
} and
some causally ordered process matrix W
A≺B
and
W
B≺A
, respectively. That is, we show that all
{w
A≺B
a|x
} satisfy eq. (28), and analogously for the
causal order B ≺ A.
We now contrast the statement made in the
previous paragraph with general assemblages. As
stated before, the technique of defining the gen-
eral assemblage as the most general set of linear
operators that takes instruments to general be-
haviours results in an object that cannot always
be described by process matrices. On the other
hand, in the case of causally ordered assemblages,
this technique yielded an object that can always
be described by (causal) process matrices. The
main point to be taken here is that to character-
ize the most general set of linear operators that
takes instruments to some kind of behaviour is a
mathematical artifice to find an outer approxim-
ation to the set of assemblages that are described
by process matrices. The goal is to find an ap-
proximation of this set with a potentially sim-
pler characterization. This approximation may
be tight, as in the case of causal assemblages, or
may not be tight, as in the case of general as-
semblages. We explore this further in appendix E
for assemblages in tripartite scenarios.
We now define a causal assemblage by taking
the elements of the convex hull of causally ordered
assemblages.
Definition 7
(Causal assemblage)
.
An as-
semblage
{w
causal
a|x
}
is causal if it can be expressed
as a convex combination of causally ordered as-
semblages, i.e.,
w
causal
a|x
:
= qw
A≺B
a|x
+ (1 − q)w
B≺A
a|x
, (32)
for all
a, x
, where 0
≤ q ≤
1 is a real number. An
assemblage that does not satisfy eq. (32) is called
a noncausal assemblage.
We can now express our result in terms of the
following lemma, proved in appendix B.
Lemma 2.
A general assemblage is causal if and
only if it is a process assemblage that can be ob-
tained from a causally separable process matrix.
This result allows us to identify which causally
nonseparable process matrices can be certified in
a semi-device-independent way:
Theorem 5.
A process matrix is certified
to be causally nonseparable in a semi-device-
independent way if and only if it can generate
a noncausal assemblage for some choice of instru-
ments for Alice.
Proof.
If a process matrix is causally separable
then its assemblages will be causal. If the as-
semblage is causal, even though it could in prin-
ciple have been generated by a causally nonsep-
arable process matrix, according to lemma 2 it
can always be reproduced by a causally separable
process matrix and hence this property cannot be
certified.
Note that all requirements for an assemblage
to be causal are linear and positive semidefinite
constraints, hence one can check whether an as-
semblage is causal via SDP.
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However, in our semi-device-independent certi-
fication scenario, the only information available
is the process behaviour and Bob’s instruments,
not the assemblage itself. If it were the case that
Bob’s instruments are tomographically complete,
he could obtain full information about the as-
semblage, and check whether it is causal via SDP.
Nevertheless, we show that it is possible to check
whether a given behaviour can certify indefin-
ite causal order in a semi-device-independent
scenario using SDP even without the knowledge
of the assemblage. We do this by rephrasing
our certification task in terms of an unknown
assemblage.
Definition 3’ (Semi-device-independent certific-
ation, with assemblages). Given a behaviour
{p
Q
(
ab|xy
)
}, that arises from unknown instru-
ments on Alice’s side, known instruments {B
b|y
}
on Bob’s side, and an unknown bipartite pro-
cess matrix, one certifies that this process mat-
rix is causally nonseparable in a semi-device-
independent way if
p
Q
(ab|x , B
b|y
) 6= Tr(B
b|y
w
causal
a|x
) (33)
for all a, b, x, y, and for all causal assemblages
{w
causal
a|x
}.
Now we are able to formulate the semi-device-
independent certification problem in terms of
SDP:
given {p
Q
(ab|xy)}, {B
b|y
}
find {w
a|x
}
s.t. p
Q
(ab|xy) = Tr(B
b|y
w
a|x
) ∀a, b, x, y
{w
a|x
} ∈ CAUSAL,
(34)
where CAUSAL denotes the set of causal as-
semblages, that is, {w
a|x
} is constrained to
eq. (32).
As in the previous cases, if the problem is in-
feasible, then the process matrix that was used
to generate the process behaviour {p
Q
(
ab|xy
)
} is
certainly causally nonseparable. If the problem
is feasible, then one can use the results presented
in appendix B to explicitly find a causally separ-
able process matrix W
sep
and sets of instruments
{A
a|x
} such that w
a|x
= Tr
A
[(A
a|x
⊗ 1
B
) W
sep
].
All three SDP formulations we presented in
eqs. (18), (19) and (34) are feasibility problems
which can be turned into optimisation problems
that allow for a robust certification of indefinite
causal order. We discuss this further in section 3.
We now show that not all process matrices can
be certified in a semi-device-independent way, as
some process matrices cannot lead to noncausal
assemblages. The proof is in appendix D.
Theorem 6
(Device-dependent certifiable,
semi-device-independent noncertifiable process
matrix)
.
There exist causally nonseparable pro-
cess matrices that, for any sets of instruments
on Alice’s side, always give rise to causal as-
semblages. That is, causally nonseparable process
matrices that cannot be certified in a semi-device-
independent way.
In particular, let
W ∈ L
(
H
A
I
A
O
B
I
B
O
) be a pro-
cess matrix and
W
T
A
be the partial transposition
of
W
with respect to some basis in
L
(
H
A
I
A
O
)
for Alice. If
W
T
A
is causally separable, the as-
semblages generated by
w
a|x
=
Tr
A
[(
A
a|x
⊗1
B
)
W
]
are causal for every set of instruments {A
a|x
}.
We remark that the above theorem strictly ex-
tends proposition 1 which was first proved in
ref. [48]. That is, with the same hypothesis –
that W
T
A
is causally separable – we can make a
stronger claim – that W cannot be certified as
causally nonseparable even if Bob is treated in a
device-dependent way (is trusted).
In ref. [48], the authors show that, when ex-
tended with an entangled state, the resulting pro-
cess matrix can violate a causal inequality and
can therefore be certified in a device-independent
way. This implies that this extended process
matrix can also be certified in a semi-device-
independent way. Ref. [48] leaves as an open
question the existence of a causally nonseparable
bipartite process matrix that cannot be certified
in a device-independent way even when extended
by entanglement. We remark that this open ques-
tion is also relevant in the context of semi-device-
independent certification.
Another natural question also emerges: is there
a bipartite process matrix that can be certified
to be causally nonseparable in a semi-device-
independent scenario but that cannot be certi-
fied to be causally nonseparable in any device-
independent scenario? Although we believe such
process matrix exists, no example is currently
known.
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3 The Quantum Switch
The concepts of certification presented in the
previous section have a natural generalisation to
different multipartite scenarios. We are now go-
ing to illustrate one particular tripartite case by
discussing and presenting some results involving
the quantum switch [22, 35]. In appendix E, we
present a detailed extension of the concepts and
results from the bipartite case, introduced in sec-
tion 2, to the tripartite case in which the quantum
switch is defined, and pave the way to future more
general tripartite and multipartite extensions.
On its first appearance, the quantum switch
was defined as a higher-order transformation that
maps quantum channels into quantum channels
and it can be defined as the following. Let U
A
and
U
B
be two unitary operators that act on the same
space of a target state |ψi
t
. Let |ci
c
:=
α|
0
i
+
β|
1
i,
|α|
2
+
|β|
2
= 1
, be a ‘control’ state that is able
to coherently control the order in which the op-
erations U
A
and U
B
are applied. The quantum
switch acts as following:
switch(U
A
, U
B
) = |0ih0|
c
⊗ U
A
U
B
+ |1ih1|
c
⊗ U
B
U
A
.
(35)
When applied to the state |ci
c
⊗ |ψi
t
we have
switch(U
A
, U
B
)|ci ⊗ |ψi = α |0i ⊗ U
A
U
B
|ψi
+ β |1i ⊗ U
B
U
A
|ψi.
(36)
Physically, the equation above can be under-
stood as the control qubit determining which unit-
ary is going to be applied first on the target
state |ψi. If the control qubit is in the state |
0
i
(α
= 1
, β
= 0
), the unitary U
B
is performed be-
fore the unitary U
A
. If the control qubit is in
the state |
1
i (α
= 0
, β
= 1
), the unitary U
B
is
performed before the unitary U
A
. In general, if
the control qubit is in the state |ci
=
α|
0
i
+
β|
1
i,
α 6
= 0
, β 6
= 0
, the output state will be in a coher-
ent superposition of two different causal orders.
In the process matrix formalism, ref. [21] has
analysed the quantum switch as a four-partite
process matrix of which the first party has only
an output space, which defines the input target
and control states, one party inputs U
A
, another
party inputs U
B
, and a final party obtains the
output control and target states. Here we follow
instead the steps of ref. [27] to associate tripart-
ite process matrices to the quantum switch. This
can be done by absorbing the input target and
control states into the process matrix and setting
the third party with output space of dimension
equal to one. Hence, the quantum switch is de-
scribed by a family of tripartite process matrices
that is shared among three parties, Alice, Bob,
and Charlie, for which Charlie is always in the
future of Alice and Bob, and the causal order
between Alice and Bob may or may not be well
defined. A consequence of the fact that Charlie is
last and his output space H
C
O
has d
= 1
is that
the most general instrument Charlie can perform
is a POVM.
Formally, we define a family of tripartite pro-
cess matrices associated to the quantum switch
according to:
Definition 8
(Quantum switch processes)
.
Let
|w(ψ, α, β)i ∈ H
A
I
A
O
B
I
B
O
C
t
I
C
c
I
be
|w (ψ, α, β)i = α |ψi
A
I
|Φ
+
i
A
O
B
I
|Φ
+
i
B
O
C
t
I
|0i
C
c
I
+ β |ψi
B
I
|Φ
+
i
B
O
A
I
|Φ
+
i
A
O
C
t
I
|1i
C
c
I
,
(37)
where
|ψi
is a
d
-dimensional pure state,
α, β
are
complex numbers such that
|α|
2
+
|β|
2
= 1, and
|
Φ
+
i
=
P
d
i=1
|iii
is a maximally entangled unnor-
malised bipartite qudit state, the Choi representa-
tion of the identity channel.
Then, the pure quantum switch processes are a
family of tripartite process matrices given by
W
switch
(ψ, α, β) = |w (ψ, α, β)ihw (ψ, α, β)|.
(38)
When the control state is in a nontrivial su-
perposition of |
0
i and |
1
i, the quantum switch
processes have been shown to have some interest-
ing properties [27, 47]. They are causally non-
separable process matrices, meaning they cannot
be expressed as a convex combination of tripart-
ite process matrices with definite causal ordered
between Alice and Bob, with Charlie in their com-
mon future. Namely, when α 6= 0 and β 6= 0,
W
switch
(ψ, α, β) 6= qW
A≺B≺C
+ (1 − q)W
B≺A≺C
,
(39)
for all |ψi, and all real numbers
0
≤ q ≤
1
. The
exact definitions of causally ordered and general
tripartite process matrices can be found in ap-
pendix E.
However, when Charlie is traced out, the res-
ulting bipartite process matrices shared by Alice
and Bob are causally separable, namely,
Tr
C
c
I
C
t
I
[W
switch
(ψ, α, β)] = qW
A≺B
+ (1 − q)W
B≺A
,
(40)
for all |ψi, α, and β, where q = |α
2
|.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 11
These causally nonseparable tripartite process
matrices can be certified in a device-dependent
way, since theorem 1 also holds for tripartite
process matrices. Yet, it has been shown in
refs. [27, 47] that the quantum switch processes
cannot be certified in a device-independent way,
as they always lead to causal behaviours for any
choice of instruments of Alice, Bob, and Charlie.
It remains to find out whether these processes can
be certified in semi-device-independent scenarios.
For this purpose, we extend all concepts and
methods from bipartite semi-device-independent
certification. Much like in the bipartite case,
we make different assumptions about the know-
ledge of the operations performed by each party.
We call untrusted (U) a party that is treated
in a device-independent way and trusted (T) a
party that is treated in a device-dependent way,
and we use the convention Alice Bob Charlie
for denoting the parties. For example, a scen-
ario TTU means Alice = T (device-dependent),
Bob = T (device-dependent), and Charlie = U
(device-independent). The four inequivalent semi-
device-independent tripartite scenarios are TTU,
TUU, UTT, and UUT.
The core idea of the certification task remains
the same. For a given process behaviour and
given sets of instruments for the trusted parties,
one needs to check whether it is possible that
this behaviour comes from performing this instru-
ments on a causally separable tripartite process
matrix. We also derive the concepts of general,
process, causally ordered, and causal assemblages
for each scenario. In appendix E, we provide
all details and calculations, including for the tri-
partite device-dependent TTT and -independent
UUU scenarios.
Our next theorem strengthens the previous res-
ult [27] that showed that the quantum switch
processes cannot be certified in a full device-
independent scenario, i.e., in the UUU scenario.
We show that when the instruments of Alice
and Bob are unknown, even if the measurements
performed by Charlie are known, the quantum
switch processes can never be proven to be caus-
ally nonseparable, for any pairs of sets of instru-
ments for Alice and Bob. In other words, we
prove that the quantum switch processes cannot
be certified to be causally nonseparable in the
UUT scenario. The previous result of the full
device-independent scenario can now be seen as
a particular case of the theorem we now present,
whose proof is in appendix F.
Theorem 7.
The quantum switch processes can-
not be certified to be causally nonseparable on a
semi-device-independent scenario where Alice and
Bob are untrusted and Charlie is trusted (UUT).
Moreover, any tripartite process matrix
W ∈
L
(
H
A
I
A
O
B
I
B
O
C
I
), with Charlie in the future of
Alice and Bob, that satisfies
Tr[(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
⊗ 1
C
I
)W
A
I
A
O
B
I
B
O
C
I
] =
qp
A≺B
(ab|xy) + (1 − q)p
B≺A
(ab|xy),
(41)
for all
a, b, x, y
, where 0
≤ q ≤
1 is a real number,
cannot be certified to be causally nonseparable in
a UUT scenario.
We now show that in the three remaining semi-
device-independent scenarios, TTU, TUU, and
UTT, the quantum switch processes can be cer-
tified to be causally nonseparable, proving that
they can demonstrate stronger noncausal proper-
ties than it was previously known.
For our remaining calculations we use the re-
duced quantum switch process
W
red
:
= Tr
C
t
I
W
switch
|0i,
1
√
2
,
1
√
2
. (42)
By choosing a reduced, mixed switch process in
these scenarios, in which the space of the tar-
get state is traced out, we guarantee that the
pure switch process version can also be certified
without performing any measurements on the out-
put target space.
We show that the reduced quantum switch pro-
cess W
red
can be certified in the TTU, TUU, and
UTT scenarios by providing sets of instruments
for the device-independent (untrusted) parties
that, when applied to W
red
, generate TTU-,
TUU-, and UTT-assemblages that are noncausal.
We prove these assemblages to be noncausal by
means of SDP.
To calculate the assemblages, we choose the fol-
lowing instruments for each untrusted party:
A
A
I
A
O
0|0
= B
B
I
B
O
0|0
= |0ih0| ⊗ |0ih0|, M
C
c
I
0|0
= |+ih+|,
A
A
I
A
O
1|0
= B
B
I
B
O
1|0
= |1ih1| ⊗ |1ih1|, M
C
c
I
1|0
= |−ih−|,
A
A
I
A
O
0|1
= B
B
I
B
O
0|1
= |+ih+| ⊗ |+ih+|,
A
A
I
A
O
1|1
= B
B
I
B
O
1|1
= |−ih−| ⊗ |−ih−|,
(43)
where |±i =
1
√
2
(|0i ± |1i).
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 12
Let us illustrate with the TUU case. To con-
struct the TUU-assemblage, we use the instru-
ments from eq. (43) for the untrusted parties, Bob
and Charlie, according to
w
switch
bc|y z
= Tr
BC
[(1
A
⊗ B
b|y
⊗ M
c|z
) W
red
] ∀ b, c, y, z.
(44)
Then, we show via SDP that
w
switch
bc|y z
6= qw
A≺B≺C
bc|y z
+ (1 − q)w
B≺A≺C
bc|y z
, ∀ b, c, y, z,
(45)
proving that the switch process can be certified
in this scenario. This is possible due to the SDP
characterization of causal assemblages presented
in appendix E (for instance, see definition 20 for
the TUU scenario). It follows analogously for the
scenarios TTU and UTT.
To be able to compare and quantify the causal
properties of the quantum switch across different
certification scenarios, from full device-dependent
to full device-independent, we introduce a robust-
ness measure. We start by defining the noisy
version of the switch, a mixture of the reduced
quantum switch process with a ‘trivial’ process
matrix (the normalised identity):
W
η
red
:
= η
1
d
I
+ (1 − η)W
red
, (46)
where d
I
=
d
A
I
d
B
I
d
C
c
I
is the dimension of the
joint input spaces.
We then estimate the minimum value of η for
which W
η
red
has only causal properties in a given
scenario. For example, in the device-dependent
scenario, this is the minimum value of η for which
W
η
red
is causally separable. In a semi-device-
dependent scenario, this is the minimum value
of η for which W
η
red
only generates causal as-
semblages. Finally, for the device-independent
scenario, this is the minimum value of η for which
W
η
red
only generates causal behaviours.
It is immediate to see that in the UUU (device-
independent) scenario, η
∗
= 0
, since the switch
process always generates causal behaviours [27].
Equivalently for the UUT scenario, as a direct
consequence of our theorem 7. For the TTT scen-
ario, we evaluate via SDP a value of η
∗
= 0
.
6118
for which W
η
∗
red
is causally separable and bellow
which it is causally nonseparable.
In the remaining scenarios, TTU, TUU, and
UTT, in order to calculate the exact value of
η
∗
, one should consider every possible assemblage
that could be generated from W
η
red
, by optim-
izing over the set of instruments of the device-
independent (untrusted) parties. Since we only
consider the fixed instruments of eq. (43), we cal-
culate lower bounds for η
∗
. Additionally, as de-
tailed in appendix E, in some of these scenarios
our SDP characterization of the set of causal as-
semblages only constitutes an outer approxima-
tion of the set of assemblages that can be de-
scribed by causal process matrices. Since with
SDP we calculate the minimum η for the as-
semblages to be inside this outer approximation,
this again gives only a lower bound for η
∗
.
Let us illustrate again with the TUU case. We
construct a noisy TUU-assemblage using the in-
struments from eq. (43) and W
η
red
according to
w
η,switch
bc|yz
= Tr
BC
[(1
A
⊗ B
b|y
⊗ M
c|z
) W
η
red
] ∀ b, c, y, z.
(47)
We then calculate via SDP the minimum value
of η for which w
η,switch
bc|yz
is causal and below
which it is noncausal. This value constitutes a
lower bound for η
∗
. In the TUU scenario, we
evaluate η
∗
≥
0
.
1621
for the reduced switch
process. Analogously, we evaluate η
∗
≥
0
.
1802
in
the UTT scenario. Finally, in the TTU scenario,
η
∗
≥
0
.
5687
. All these values are summarized
in table 2. All code used to obtain these results
is freely available in an online repository [53].
We remark that the set of instruments required
to obtain a robust certification of noncausal
separability of the quantum switch is relatively
simple and that Charlie can be restricted to
perform a single POVM.
The quantum switch has motivated several ex-
periments that explore optical interferometers to
certify indefinite causal order of process matrices
[31, 29, 30, 33]. Up to now, all experimental
results rely on, among other assumptions, com-
plete knowledge of the instruments to certify of
causal nonseparability, i.e., they are fully device-
dependent. As shown in this section, one can also
certify the causal nonseparability of the quantum
switch without trusting some of the instruments
and measurement apparatuses, in a semi-device-
inpedendent way. We have used the machinery
developed in this paper to analyse the experi-
ments of refs. [29] and [30] and concluded that,
the instruments used in these experiments could
allow us to make a stronger claim than what was
reported. More precisely, the instruments used
to certify that the quantum switch is causally
nonseparable on refs. [29] and [30] can lead to a
semi-device-independent certification of the non-
causal properties of the quantum switch in the
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 13
TTT
η
∗
= 0.6118
Noncausal
UTT TTU
η
∗
≥ 0.1802 η
∗
≥ 0.5687
Noncausal Noncausal
UUT TUU
η
∗
= 0 η
∗
≥ 0.1621
Causal Noncausal
UUU
η
∗
= 0
Causal
Table 2: The quantum switch can be certified to be
causally nonseparable in scenarios TTT, UTT, TTU,
and TUU and cannot be certified in scenarios UUT and
UUU, where T stands for trusted (device-dependent), U
for untrusted (device-independent) and we have chosen
the order Alice, Bob, and Charlie (for instance, TTU
represents the scenario where Alice and Bob are treated
in a device-dependent and Charlie in a device-independent
way). The values and bounds for
η
∗
concern the critical
value of the mixing parameter 0
≤ η ≤
1 in eq. (46) for
which the quantum switch cannot be certified to causally
nonseparable on each scenario, and below which, it can
be certified. All non-zero values were obtained via SDP.
All code is publicly available in an online repository [
53
].
TTU scenario. We discuss this results further in
appendix G.
Conclusions
We developed a framework for certifying indef-
inite causal order in the process matrix formalism
under different sets of assumptions about the op-
erations of the involved parties. In particular, we
constructed a semi-device-independent approach
to certification of causally nonseparable process
matrices, and unified previously explored device-
dependent and -independent approaches.
We showed that the sets of causally nonsep-
arable process matrices that can be certified in
each scenario are different. More specifically,
we proved that some bipartite process matrices
can be certified to be causally nonseparable in a
device-dependent way but not in a semi-device-
independent way, and that some tripartite pro-
cess matrices can be certified to be causally non-
separable in a semi-device-independent way but
not in a device-independent way.
In our framework, we formulated the prob-
lem of certifying causally nonseparable process
matrices in the device-dependent, semi-device-
independent, and device-independent scenarios in
terms of semidefinite programming (SDP), imply-
ing they can be efficiently solved.
We also showed that some noncausal beha-
viours and some noncausal assemblages can-
not be obtained by process matrices according
to the rules of quantum mechanics. For the
device-independent case, we presented non-trivial
bounds that relate the dimension of a process
matrix with its maximal attainable violation of
a causal game inequality. Concerning bipartite
causal behaviours and causal assemblages, we ex-
plicitly showed how to obtain them from causally
separable process matrices.
Concerning the quantum switch, we proved
that it can produce noncausal correlations, that
is, can be certified to be causally nonseparable, in
three out of four semi-device-independent scen-
arios, and proved that its noncausal properties
cannot be certified in the remaining one.
Finally, we showed that previous experiments
that claim to have certified causal nonseparability
with the quantum switch under device-dependent
assumptions [29, 30] could have, in principle,
dropped some assumptions to achieve a stronger
form of certification. Our results provide the
theoretical basis for a future experimental demon-
stration of stronger noncausal phenomena that
will rely on weaker assumptions than previous
ones.
Acknowledgements. We are grateful to Cyril
Branciard for insightful discussions, particularly
regarding theorem 7, and to Costantino Budroni,
for sharing a preliminary version of ref. [43].
This work was supported by the Austrian Sci-
ence Fund (FWF) through the START project
Y879-N27, the Excellence Initiative of the Ger-
man Federal and State Governments (Grant ZUK
81), the Japan Society for the Promotion of Sci-
ence (JSPS) by KAKENHI grant No. 16F16769,
and the Q-LEAP project of the MEXT, Japan.
CB acknowledges the support of the FQXi and
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 14
the Austrian Science Fund (FWF) through the
SFB project “BeyondC”, the projects I-2526-N27
and I-2906. This publication was made possible
through the support of a grant from the John
Templeton Foundation. The opinions expressed
in this publication are those of the authors and
do not necessarily reflect the views of the John
Templeton Foundation.
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ciard. Genuinely multipartite noncausality.
Quantum 1, 39 (2017). [arXiv: 1708.07663].
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 17
Appendix
In these appendices we provide material to complement the main text. In appendix A, we present the
proofs of theorems 2 and 4, regarding behaviours and assemblages that cannot be expressed in terms of
process matrices. In appendix B, we present the proofs of lemmas 1 and 2, regarding the realization of
causal behaviours and assemblages in terms of causally separable process matrices. In appendix C, we
show how to obtain the characterization of general and causal assemblages presented in definitions 5
and 6. In appendix D, we present the proof of theorem 6, showing a class of causally nonseparable
process matrices that cannot be certified in a semi-device-independent way. In appendix E, we re-derive
all concepts and results of the main text concerning certification of bipartite process matrices for the
tripartite process matrices whose third party is always in the future of the other two. We start by
defining all notions of certification for these tripartite process matrices and then explore each scenario
(TTT, UUU, TTU, TUU, UTT, and UUT) in detail. This appendix contains technical results not used
nor mentioned in the main text. In appendix F, we present the proof of theorem 7, i.e., we prove that
the quantum switch cannot be certified to be causally nonseparable in the UUT scenario. Finally, in
appendix G, we present our theoretical analysis of the quantum switch experiments of refs. [29] and
[30].
A Behaviours and assemblages unattainable by process matrices
In this appendix, we prove theorem 2 and theorem 4, concerning behaviours and assemblages which
cannot be obtained by process matrices. We start by presenting and proving the following lemma,
which will be necessary for the proof of theorem 2.
Lemma 3. Let A, B ∈ H
1
⊗ H
2
be positive semidefinite operators. Then
Tr(AB) ≤ d
1
d
2
Tr(
2
AB). (48)
Proof. First notice that if
A ≤ d
1
d
2 2
A (49)
then for any positive semidefinite B it is true that
Tr(AB) ≤ d
1
d
2
Tr(
2
AB), (50)
so the problem reduces to proving inequality (49).
Since
A
and
2
A
commute, they are both diagonal in a common orthonormal basis
{|ψ
i
i}
i
, so the
inequality (49) is equivalent to showing that for all eigenvectors |ψ
i
i we have
hψ
i
|A|ψ
i
i ≤ d
1
d
2
hψ
i
|
2
A|ψ
i
i. (51)
Let
ψ
i,1
:=
Tr
2
(
|ψ
i
ihψ
i
|
) be the partial trace of the operator
|ψ
i
ihψ
i
|
over the space
H
2
. Substituting
A =
P
j
λ
j
|ψ
j
ihψ
j
| on the right hand side of equation (51) we get
d
1
d
2
hψ
i
|
2
A|ψ
i
i = d
1
d
2
X
j
λ
j
hψ
i
|
Tr
2
(|ψ
j
ihψ
j
|) ⊗
1
d
2
|ψ
i
i (52)
≥ d
1
d
2
λ
i
hψ
i
|
Tr
2
(|ψ
i
ihψ
i
|) ⊗
1
d
2
|ψ
i
i (53)
= d
1
λ
i
hψ
i
|(ψ
i,1
⊗ 1) |ψ
i
i (54)
= d
1
λ
i
Tr[(ψ
i,1
⊗ 1) (|ψ
i
ihψ
i
|)] (55)
= d
1
λ
i
Tr(ψ
2
i,1
). (56)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 18
Notice now that, since
ψ
i,1
is a positive operator of trace one,
Tr(ψ
2
i,1
) represents the purity of a
d
1
-dimension quantum state, which lower bounded by 1/d
1
. It then follows that
d
1
λ
i
Tr(ψ
2
i,1
) ≥ d
1
λ
i
1
d
1
(57)
= λ
i
(58)
= hψ
i
|A|ψ
i
i, (59)
which concludes the proof.
Theorem 2.
All process behaviours are valid behaviours, however, not all valid behaviours are process
behaviours.
In particular, in the scenario where all parties have dichotomic inputs and outputs, any behaviour
{p(ab|xy)} such that
1
4
X
a,b,x,y
δ
a,y
δ
b,x
p(ab|xy) > 1 −
1
d + 1
is not a process behaviour for process matrices with total dimension d
A
I
d
A
O
d
B
I
d
B
O
= d.
Proof.
It follows by definition that all process behaviours are valid behaviours, we now show that there
exist valid behaviours that are not process behaviours for any finite dimension. Assume that there
exists a process matrix
W
with total dimension
d
A
I
d
A
O
d
B
I
d
B
O
=
d
and instruments
{A
a|x
}, {B
b|y
}
such that
p(ab|xy) = Tr
W (A
a|x
⊗ B
b|y
)
(60)
and
p
succ
GYNI
=
1
4
X
abxy
δ
ay
δ
bx
p(ab|xy) > 1 −
1
d + 1
, (61)
where
δ
ij
is the Kronecker’s delta function and
p
succ
GYNI
is the probability of success achieved by this
behaviour in the GYNI causal game (defined in ref. [34]).
Let then we define the GYNI operator
M :=
1
4
X
abxy
δ
ay
δ
bx
A
a|x
⊗ B
b|y
, (62)
so that
p
succ
GYNI
= Tr(W M ). (63)
Since W is a valid process matrix, it admits the decomposition
W =
A
O
W +
B
O
W −
A
O
B
O
W, (64)
and by linearity we have
p
succ
GYNI
= Tr(
A
O
W M) + Tr(
B
O
W M) − Tr(
A
O
B
O
W M). (65)
Since
A
O
W
and
B
O
W
are causally ordered process matrices, they cannot violate the GYNI causal
inequality and respect [34]
Tr(MW
sep
) ≤
1
2
. (66)
Also, it follows from lemma 3 that
− Tr(
A
O
B
O
W M) ≤ −
1
d
Tr(W M ). (67)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 19
We than have
p
succ
GYNI
= Tr(W M ) = Tr(
A
O
W M) + Tr(
B
O
W M) − Tr(
A
O
B
O
W M) (68)
≤
1
2
+
1
2
−
1
d
Tr(W M ), (69)
which implies
p
succ
GYNI
= Tr(W M ) ≤ 1 −
1
d + 1
, (70)
and bounds the maximal attainable value for process behaviours on the GYNI causal game, which can
attain p
succ
GYNI
= 1 for general behaviours.
Note that this robust bound only holds for finite dimensional process matrices. In ref. [
43
], the
authors have shown that a process matrix
W
cannot attain the exact maximal success probability in
the GYNI causal game i.e., p
succ
GYNI
= Tr(W M ) = 1, even with infinite dimension.
Theorem 4.
All process assemblages are valid assemblages, however, not all valid assemblages are
process assemblages.
In particular, in the scenario where Alice has dichotomic inputs and outputs, the general assemblage
{w
a|x
} given by w
a|x
= |xihx| ⊗ |aiha| is not a process assemblage.
Proof.
First we see that
w
a|x
=
|xihx| ⊗ |aiha|
is a valid assemblage, since
|xihx| ⊗ |aiha| ≥
0 for all
a, x, and
X
a
w
a|x
= |xihx| ⊗ 1 ∀ x. (71)
Assume then that
{w
a|x
}
is a process assemblage. Then, there must exist a process matrix
W
and
instruments {A
a|x
} such that
w
a|x
= Tr
A
I
A
O
h
A
a|x
⊗ 1
W
i
, (72)
and that for any set of instruments {B
b|y
},
p(ab|xy) = Tr(B
b|y
w
a|x
) = Tr
h
A
a|x
⊗ B
b|y
W
i
. (73)
Consider now the valid set of instruments given by
B
b|y
= |bihb| ⊗ |yihy|. (74)
Behaviours obtained by this set of instruments and the assemblage {w
a|x
} would be
p(ab|xy) = Tr(B
b|y
w
a|x
) (75)
= δ
b,x
δ
a,y
∀ a, b, x, y, (76)
which according to theorem 2 cannot be obtained by any finite dimensional process matrix, contradicting
the hypothesis that {w
a|x
} is a process assemblage.
B Behaviours and assemblages attainable by causally separable process matrices
In this appendix we prove lemma 1 and lemma 2, which we reestate for the convenience of the
reader.
Lemma 1.
A general behaviour is causal if and only if it is a process behaviour that can be obtained
by a causally separable process matrix.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 20
Proof.
To prove the if part we show that causally separable process matrices can only give rise to
causal behaviours, regardless of the instruments performed on them.
We start by showing that a behaviour
{p
(
ab|xy
)
}
that comes from acting with any sets of instruments
{A
a|x
}
and
{B
b|y
}
on a process matrix that is causally ordered from Alice to Bob
W
A≺B
is also causally
ordered from Alice to Bob. Given {p(ab|xy)} that arises from W
A≺B
according to
p(ab|xy) = Tr
h
(A
a|x
⊗ B
b|y
) W
A≺B
i
(77)
= Tr
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
) W
A
I
A
O
B
I
⊗
1
B
O
d
B
O
i
, (78)
one can check that Alice’s marginal probability distributions,
X
b
p(ab|xy) =
X
b
Tr
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
) W
A
I
A
O
B
I
⊗
1
B
O
d
B
O
i
(79)
= Tr
h
(A
A
I
A
O
a|x
⊗ 1
B
I
)(W
A
I
A
O
B
I
) Tr
B
O
(1
A
I
A
O
⊗
1
d
B
O
X
b
B
B
I
B
O
b|y
)
i
(80)
=
1
d
B
O
Tr
h
(A
A
I
A
O
a|x
⊗ 1
B
I
)W
A
I
A
O
B
I
i
, (81)
are independent of
y
for all
a, x
. Hence,
p
(
ab|xy
) =
p
A≺B
(
ab|xy
) is causally ordered from Alice to Bob.
Equivalently, W
B≺A
implies {p
B≺A
(ab|xy)}.
Hence, behaviours that come from a causally separable process matrix according to
p(ab|xy) = Tr
h
(A
a|x
⊗ B
b|y
) W
causal
i
(82)
= Tr
h
(A
a|x
⊗ B
b|y
) (qW
A≺B
+ (1 − q)W
B≺A
)
i
(83)
= qp
A≺B
(ab|xy) + (1 − q)p
B≺A
(ab|xy) (84)
are causal by definition. In other words, causally separable process matrices can only generate causal
behaviours.
To prove the only if part we show that all behaviours that are causal can be reproduced by performing
some instruments on some causally separable process matrix.
Given a causal behaviour
{p
causal
(
ab|xy
)
}
, according to Bayes’ rule and the non-signalling principle,
one can decompose it in the following form:
p
causal
(ab|xy) =q p
A≺B
(ab|xy) + (1 − q)p
B≺A
(ab|xy) (85)
=q p
A≺B
A
(a|x)p
A≺B
B
(b|axy) + (1 − q)p
B≺A
B
(b|y)p
B≺A
A
(a|bxy), (86)
so that {p
A≺B
A
(a|x)}, {p
A≺B
B
(b|axy)}, {p
B≺A
B
(b|y)}, {p
B≺A
A
(a|bxy)}, and q are given quantities.
First, for the contribution that is causally ordered from Alice to Bob,
p
A≺B
(
ab|xy
), we construct
instruments {A
A≺B
a|x
} and {B
A≺B
b|y
} according to
A
A≺B
a|x
= 1
A
I
⊗ p
A≺B
A
(a|x)|axihax|
A
O
(87)
B
A≺B
b|y
=
X
a,x
p
A≺B
B
(b|axy)|axihax|
B
I
⊗
1
B
O
d
B
O
, (88)
and process matrix
W
A≺B
=
1
A
I
d
A
I
⊗ |Φ
+
ihΦ
+
|
A
O
B
I
⊗ 1
B
O
, (89)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 21
where
|
Φ
+
ih
Φ
+
|
is the Choi operator of the identity channel, with
|
Φ
+
i
=
P
d
i=1
|iii
. It can be checked
that
Tr
h
(A
A≺B
a|x
⊗ B
A≺B
b|y
) W
A≺B
i
= p
A≺B
A
(a|x)p
A≺B
B
(b|axy) (90)
= p
A≺B
(ab|xy), (91)
for every
a, b, x, y
. Hence, such construction can recover any behaviour that is causally ordered from
Alice to Bob. Analogously, for {p
B≺A
(ab|xy)}, one has
A
B≺A
a|x
=
X
b,y
p
B≺A
A
(a|bxy)|byihby|
A
I
⊗
1
A
O
d
A
O
, (92)
B
B≺A
b|y
= 1
B
I
⊗ p
B≺A
B
(b|y)|byihby|
B
O
(93)
W
B≺A
=
1
B
I
d
B
I
⊗ |Φ
+
ihΦ
+
|
B
O
A
I
⊗ 1
A
O
. (94)
Now, for causal behaviours, which are convex combinations of causally ordered behaviours, we
construct instruments and process matrices in such a way that each causal order acts on a complementary
subspace. That is, we define the valid process matrix
W
causal
= qW
A≺B
⊗ |00ih00|
A
0
I
B
0
I
+ (1 − q)W
B≺A
⊗ |11ih11|
A
0
I
B
0
I
, (95)
by extending Alice’s and Bob’s input spaces according to
H
A
I
(B
I
)
→ H
A
I
(B
I
)
⊗H
A
0
I
(B
0
I
)
, and we define
the valid instruments
A
a|x
= A
A≺B
a|x
⊗ |0ih0|
A
0
I
+ A
B≺A
a|x
⊗ |1ih1|
A
0
I
(96)
B
b|y
= B
A≺B
b|y
⊗ |0ih0|
B
0
I
+ B
B≺A
b|y
⊗ |1ih1|
B
0
I
. (97)
In this way, it is easy to check that orthogonal terms cancel out and we arrive at
Tr
h
(A
a|x
⊗ B
b|y
) W
causal
i
= qp
A≺B
(ab|xy) + (1 − q)p
B≺A
(ab|xy) (98)
= p
causal
(ab|xy), (99)
recovering any causal behaviour.
Lemma 2.
A general assemblage is causal if and only if it is a process assemblage that can be obtained
from a causally separable process matrix.
Proof.
To prove the if part we show that a causally separable process matrix can only give rise to a
causal assemblage regardless of Alice’s sets of instruments. First we show that an assemblage that is
generated by a process matrix that is ordered from Alice to Bob is also ordered from Alice to Bob.
B
O
w
a|x
=
B
O
Tr
A
h
(A
a|x
⊗ 1)W
A≺B
i
(100)
= Tr
A
h
(A
a|x
⊗ 1)
B
O
W
A≺B
i
(101)
= Tr
A
h
(A
a|x
⊗ 1)W
A≺B
i
(102)
= w
a|x
, (103)
since W
A≺B
=
B
O
W
A≺B
. Hence, {w
a|x
} = {w
A≺B
a|x
} is causally ordered from Alice to Bob.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 22
Next, we show that an assemblage that is generated by a process matrix that is ordered from Bob to
Alice is also ordered from Bob to Alice.
X
a
w
a|x
=
X
a
Tr
A
h
(A
a|x
⊗ 1)W
B≺A
i
(104)
= Tr
A
h
(
X
a
A
a|x
⊗ 1)
A
O
W
B≺A
i
(105)
= Tr
A
I
h
W
B
I
B
O
A
I
(Tr
A
O
X
a
A
a|x
⊗ 1
B
I
B
O
)
i
(106)
= Tr
A
I
h
W
B
I
B
O
A
I
(1
A
I
⊗ 1
B
I
B
O
)
i
, (107)
which is independent of x. Hence, {w
a|x
} = {w
B≺A
a|x
} is causally ordered from Bob to Alice.
Consequently, assemblages that come from causally separable process matrices according to
w
a|x
= Tr
A
h
(A
a|x
⊗ 1)W
causal
i
(108)
= Tr
A
h
(A
a|x
⊗ 1) (qW
A≺B
+ (1 − q)W
B≺A
)
i
(109)
= qw
A≺B
a|x
+ (1 − q)w
B≺A
a|x
(110)
are causal by definition. In other words, causally separable process matrices can only generate causal
assemblages.
To prove the only if part we show that all assemblages that are causal can be reproduced by
performing some instruments on some causally separable process matrix. Given a causal assemblage, it
can be decomposed in {w
A≺B
a|x
} and {w
B≺A
a|x
} with some convex weight q.
With the contribution that is causally ordered from Alice to Bob,
{w
A≺B
a|x
}
, one can write its elements
as w
A≺B
a|x
= σ
B
I
a|x
⊗ 1
B
O
, where Tr
P
a
σ
a|x
= 1. We show that it can be recovered by instruments
A
A≺B
a|x
= 1
A
I
⊗ σ
T A
O
a|x
, (111)
where
T
is the transposition in the computational basis of
H
A
O
, and a causally ordered process matrix
from Alice to Bob
W
A≺B
=
1
A
I
d
A
I
⊗ |Φ
+
ihΦ
+
|
A
O
B
I
⊗ 1
B
O
, (112)
where |Φ
+
ihΦ
+
| is the Choi operator of the identity channel, with |Φ
+
i =
P
d
i=1
|iii. Indeed,
Tr
A
I
A
O
h
(A
A≺B
a|x
⊗ 1
B
I
B
O
) W
A≺B
i
= Tr
A
O
h
(σ
T A
O
a|x
⊗ 1
B
I
B
O
)(|Φ
+
ihΦ
+
|
A
O
B
I
⊗ 1
B
O
)
i
(113)
= σ
B
I
a|x
⊗ 1
B
O
(114)
= w
A≺B
a|x
. (115)
On the other hand, with the contribution that is causally ordered from Bob to Alice,
{w
B≺A
a|x
}
, one
can write
P
a
w
B≺A
a|x
=
ρ
B
I
⊗ 1
B
O
, where
Trρ
B
I
= 1. Since
ρ ≥
0, it can be written as
ρ
=
P
i
µ
i
|iihi|
,
which is purified by
|ψi
=
P
i
√
µ
i
|iii
. We show that
{w
B≺A
a|x
}
can be recovered by instruments
{A
a|x
}
,
A
B≺A
a|x
= (ρ
−
1
2
A
0
I
⊗ 1
A
00
I
) w
T A
I
a|x
(ρ
−
1
2
A
0
I
⊗ 1
A
00
I
) ⊗
1
A
O
d
A
O
, (116)
where
ρ
−1
is the inverse of
ρ
on its support,
H
A
I
=
H
A
0
I
⊗ H
A
00
I
,
T
is the transposition in the
{|ii}
i
basis, and a causally ordered process matrix from Bob to Alice
W
B≺A
= |ψihψ|
B
I
A
0
I
⊗ |Φ
+
ihΦ
+
|
B
O
A
00
I
⊗ 1
A
O
. (117)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 23
Indeed,
Tr
A
I
A
O
h
(A
B≺A
a|x
⊗ 1
B
I
B
O
) W
B≺A
i
= w
B≺A
a|x
. (118)
Finally, just like in the proof of lemma 1, by allowing the different causal orders to act in comple-
mentary subspaces, we can recover any convex combinations of causally ordered assemblages, i.e.,
causal assemblages, from causally separable process matrices.
C Characterization theorems for general and causal assemblages
In this appendix, we prove the equivalence between the definition induced by eq. (24) and definition 5,
that is, we show that the most general set of operators {w
a|x
} that satisfies
p(ab|xy) = Tr(B
b|y
w
a|x
) ∀ a, b, x, y, (119)
where {p
(
ab|xy
)
} is a general behaviour and {B
b|y
} is a valid set of instruments, is a general assemblage
of definition 5.
Next, we prove this equivalence to hold also between the definition induced by eq. (29) and defini-
tion 6, that is, we show that the most general set of operators {w
A≺B
a|x
} that satisfies
p
A≺B
(ab|xy) = Tr(B
b|y
w
A≺B
a|x
) ∀ a, b, x, y, (120)
where {p
A≺B
(
ab|xy
)
} is a behaviour that is causally ordered from Alice to Bob and {B
b|y
} is a valid
set of instruments, is an assemblage that is causally ordered from Alice to Bob, of definition 6, end
equivalently from Bob to Alice.
Let us begin with the conditions of a general assemblage.
Non-negativity yields
p(ab|xy) ≥ 0 ∀B
b|y
≥ 0 ⇐⇒ w
a|x
≥ 0 ∀ a, x, (121)
while normalization yields
X
a,b
p(ab|xy) =
X
a,b
Tr(B
b|y
w
a|x
) = 1. (122)
Let B
y
=
P
b
B
b|y
be the Choi operator of a CPTP map. Then, following ref. [27], we can parametrize
it according to
B
y
=
[1−B
O
]
Y +
1
d
B
O
, (123)
where Y is an hermitian operator and we define the map
[1−B
O
]
M = M −
B
O
M.
Applying the parametrization, it follows that
X
a
Tr
[1−B
O
]
Y +
1
d
B
O
w
a|x
= 1. (124)
If Y = 0, we have
Tr
1
d
B
O
X
a
w
a|x
= 1 ⇐⇒ Tr
X
a
w
a|x
= d
B
O
. (125)
If Y 6= 0, and using eq. (125), we have
Tr
[1−B
O
]
Y
X
a
w
a|x
= 0. (126)
Due to the self-duality of the map
[1−B
O
]
·,
Tr
[1−B
O
]
Y
X
a
w
a|x
= Tr
Y
[1−B
O
]
X
a
w
a|x
= 0 ∀Y ⇐⇒
[1−B
O
]
X
a
w
a|x
= 0. (127)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 24
Hence,
X
a
w
a|x
=
B
O
X
a
w
a|x
. (128)
Together, eqs. (121), (125) and (128) define a general assemblage. It is simple to verify that general
assemblages lead to valid probability distributions for all sets of instruments.
For assemblages that are causally ordered from Alice to Bob, we require first that they satisfy
the conditions of a general assemblage, in order to lead to valid behaviours, and second that Alice’s
marginal probability distributions
X
b
p
A≺B
(ab|xy) = Tr
X
b
B
b|y
w
A≺B
a|x
(129)
are independent of y, so that the resulting behaviour is causally ordered from Alice to Bob. The
implication is that
X
b
p
A≺B
(ab|xy) = Tr
[1−B
O
]
Y +
1
d
B
O
w
A≺B
a|x
(130)
= Tr
[1−B
O
]
Y w
A≺B
a|x
+
1
d
B
O
Tr
w
A≺B
a|x
. (131)
will be independent of y if and only if
Tr
[1−B
O
]
Y w
A≺B
a|x
= Tr
Y
[1−B
O
]
w
A≺B
a|x
= 0 ∀ Y ⇐⇒
[1−B
O
]
w
A≺B
a|x
= 0. (132)
Hence,
w
A≺B
a|x
=
B
O
w
A≺B
a|x
∀ a, x. (133)
Finally, for an assemblage that is causally ordered from Bob to Alice, it is also required that they
satisfy the conditions of a general assemblage. To guarantee that Bob’s marginal probability distribu-
tions
X
a
p
A≺B
(ab|xy) = Tr
B
b|y
X
a
w
A≺B
a|x
(134)
are independent of x, it is necessary and sufficient that
X
a
w
B≺A
a|x
=
X
a
w
B≺A
a|x
0
∀ b, x, x
0
, y. (135)
Together, eqs. (133) and (135) and the conditions of general assemblages define causally ordered
assemblages. It is simple to verify that causal assemblages lead to causal probability distributions for
all sets of instruments.
The technique used here to arrive at the definitions of general and causal assemblages is the same
that will be used in all tripartite scenarios to define their respective general and causal assemblages.
D Causally nonseparable process matrices that cannot be certified
in a semi-device-independent way
In this section we present the proof of theorem 6 from the main text.
Theorem 6
(Device-dependent certifiable, semi-device-independent noncertifiable process matrix)
.
There exist causally nonseparable process matrices that, for any sets of instruments on Alice’s side,
always give rise to causal assemblages. That is, causally nonseparable process matrices that cannot be
certified in a semi-device-independent way.
In particular, let
W ∈ L
(
H
A
I
A
O
B
I
B
O
) be a process matrix and
W
T
A
be the partial transposition of
W
with respect to some basis in
L
(
H
A
I
A
O
) for Alice. If
W
T
A
is causally separable, the assemblages
generated by w
a|x
= Tr
A
[(A
a|x
⊗ 1
B
) W ] are causal for every set of instruments {A
a|x
}.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 25
Proof.
We start our proof by showing that if
W
T
A
is causally separable then the assemblages generated
by
w
a|x
=
Tr
A
[(
A
a|x
⊗ 1
B
)
W
] are causal for every set of instruments
{A
a|x
}
. Straightforward
calculations shows that the transposition map is self-adjoint (i.e.
Tr
[
A
T
B
] =
Tr
[
A B
T
]
, ∀ A, B
), in
particular, it is true that
Tr
A
h
(A
a|x
⊗ 1) W
T
A
i
= Tr
A
h
(A
T
a|x
⊗ 1) W
i
∀ A
a|x
. (136)
Now notice that every instrument can be written as a transposition of another instrument and
we can define new valid instruments via
A
0
a|x
:=
A
T
a|x
, which allows us to recover any assemblages
generated by
W
and instruments
{A
a|x
}
with a causally separable process matrix. More precisely,
since (A
T
a|x
)
T
= A
a|x
, the mathematical identities
w
a|x
= Tr
A
[(A
a|x
⊗ 1) W ] (137)
= Tr
A
((A
T
a|x
)
T
⊗ 1) W
(138)
= Tr
A
(A
T
a|x
⊗ 1) W
T
A
(139)
= Tr
A
[(A
0
a|x
⊗ 1) W
sep
] (140)
provide an explicit decomposition for the assemblage
{w
a|x
}
in terms of a causally separable process
matrix and valid instruments.
We finish our proof by referring to ref. [
48
], which presents several examples of process matrices
W
which are causally nonseparable but that W
T
A
is causally separable.
E Certification of tripartite process matrices
Here we generalize our approach to certification of indefinite causal order for a tripartite scenario
where a process matrix is shared between parties Alice, Bob, and Charlie. In this particular tripartite
scenario, Charlie is always in the future of Alice and Bob, so we only study tripartite process matrices
that have the causal order
(
A, B
)
≺ C, however, the causal order between Alice and Bob may or may
not be well defined. We will only consider this kind of tripartite scenario, the one which is appropriate
for the study of the quantum switch processes. A semi-device-independent approach to more general
tripartite scenarios, or multipartite scenarios, may be derived, in principle, from a straightforward
generalization of our particular tripartite scenario. For more general multipartite scenarios under a
device-dependent approach, we refer the reader to ref. [54], and for device-independent, ref. [55].
In the following, we explicitly extend all concepts, definitions, and results from the bipartite case
presented in the main text to the tripartite case. We start by defining certification in all
6
inequivalent
scenarios that arise from making different assumptions about the operations of each party: TTT (device-
dependent), UUU (device-independent), TTU, TUU, UTT, and UUT (semi-device-independent).
Definition 9
(TTT (device-dependent) certification)
.
Given a tripartite behaviour
{p
(
abc|xyz
)
}
that
arises from known instruments
{A
a|x
}
and
{B
b|y
}
, known POVMs
{M
c|z
}
, and an unknown tripartite
process matrix, one certifies that this process matrix is causally nonseparable when
p(abc|A
a|x
, B
b|y
, M
c|z
) 6= Tr
h
(A
a|x
⊗ B
b|y
⊗ M
c|z
)W
sep
i
∀ a, b, c, x, y, z, (141)
for all causally separable tripartite process matrices W
sep
.
Definition 10
(UUU (device-independent) certification)
.
Given a tripartite behaviour
{p
(
abc|xyz
)
}
that arises from unknown instruments and an unknown tripartite process matrix, one certifies that this
process matrix is causally nonseparable when
p(abc|xyz) 6= Tr
h
(A
a|x
⊗ B
b|y
⊗ M
c|z
)W
sep
i
∀ a, b, c, x, y, z, (142)
for all causally separable tripartite process matrices
W
sep
, all general instruments
{A
a|x
}
and
{B
b|y
}
,
and all general POVMs
{M
c|z
}
. A process matrix certified in such way is called UUU-noncausal, or
device-independent noncausal.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 26
Definition 11
(TTU (semi-device-dependent) certification)
.
Given a tripartite behaviour
{p
(
abc|xyz
)
}
that arises from known instruments
{A
a|x
}
and
{B
b|y
}
on Alice’s and Bob’s side, unknown POVMs
on Charlie’s side, and an unknown tripartite process matrix, one certifies that this process matrix is
causally nonseparable when
p(abc|A
a|x
, B
b|y
, z) 6= Tr
h
(A
a|x
⊗ B
b|y
⊗ M
c|z
)W
sep
i
∀ a, b, c, x, y, z, (143)
for all causally separable tripartite process matrices
W
sep
and all general POVMs
{M
c|z
}
. A process
matrix certified in such way is called TTU-noncausal.
Definition 12
(TUU (semi-device-dependent) certification)
.
Given a tripartite behaviour
{p
(
abc|xyz
)
}
that arises from known instruments
{A
a|x
}
on Alice’s, unknown instruments on Bob’s and Charlie’s side,
and an unknown tripartite process matrix, one certifies that this process matrix is causally nonseparable
when
p(abc|A
a|x
, y, z) 6= Tr
h
(A
a|x
⊗ B
b|y
⊗ M
c|z
)W
sep
i
∀ a, b, c, x, y, z, (144)
for all causally separable tripartite process matrices
W
sep
, all general instruments
{B
b|y
}
, and all
general POVMs {M
c|z
}. A process matrix certified in such way is called TUU-noncausal.
Definition 13
(UTT (semi-device-dependent) certification)
.
Given a tripartite behaviour
{p
(
abc|xyz
)
}
that arises from unknown instruments on Alice’s side, known instruments
{B
b|y
}
and
{M
c|z
}
on Bob’s
and Charlie’s side, and an unknown tripartite process matrix, one certifies that this process matrix is
causally nonseparable when
p(abc|x , B
b|y
, M
c|z
) 6= Tr
h
(A
a|x
⊗ B
b|y
⊗ M
c|z
)W
sep
i
∀ a, b, c, x, y, z, (145)
for all causally separable tripartite process matrices
W
sep
and all general instruments
{A
a|x
}
. A process
matrix certified in such way is called UTT-noncausal.
Definition 14
(UUT (semi-device-dependent) certification)
.
Given a tripartite behaviour
{p
(
abc|xyz
)
}
that arises from unknown instruments on Alice’s and Bob’s side, known POVMs
{M
c|z
}
on Charlie’s
side, and an unknown tripartite process matrix, one certifies that this process matrix is causally
nonseparable when
p(abc|x, y, M
c|z
) 6= Tr
h
(A
a|x
⊗ B
b|y
⊗ M
c|z
)W
sep
i
∀ a, b, c, x, y, z, (146)
for all causally separable tripartite process matrices
W
sep
and all general instruments
{A
a|x
}
and
{B
b|y
}. A process matrix certified in such way is called UUT-noncausal.
E.1 Device-dependent – TTT
Following ref. [27], a tripartite process matrix, with Charlie in the future of Alice and Bob, is an
operator W ∈ L(H
A
I
⊗ H
A
O
⊗ H
B
I
⊗ H
B
O
⊗ H
C
I
) that satisfies
W ≥ 0 (147)
Tr W = d
A
O
d
B
O
(148)
A
I
A
O
C
I
W =
A
I
A
O
B
O
C
I
W (149)
B
I
B
O
C
I
W =
A
O
B
I
B
O
C
I
W (150)
C
I
W =
A
O
C
I
W +
B
O
C
I
W −
A
O
B
O
C
I
W. (151)
A tripartite process matrix W
A≺B≺C
is causally ordered from Alice to Bob to Charlie if it satisfies
C
I
W
A≺B≺C
=
B
O
C
I
W
A≺B≺C
(152)
B
I
B
O
C
I
W
A≺B≺C
=
A
O
B
I
B
O
C
I
W
A≺B≺C
, (153)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 27
and a tripartite process matrix W
B≺A≺C
is causally ordered from Bob to Alice to Charlie if it satisfies
C
I
W
B≺A≺C
=
A
O
C
I
W
B≺A≺C
(154)
A
I
A
O
C
I
W
B≺A≺C
=
A
I
A
O
B
O
C
I
W
B≺A≺C
. (155)
A tripartite process matrix W
sep
is causally separable if it can be expressed as a convex combination
of causally ordered process matrices of the kind W
A≺B≺C
and W
B≺A≺C
, i.e.,
W
sep
:
= qW
A≺B≺C
+ (1 − q)W
B≺A≺C
(156)
where
0
≤ q ≤
1
is a real number. A tripartite process matrix that does not satisfy eq. (156) is called
causally nonseparable.
Just as in the bipartite case, all causally nonseparable tripartite process matrices can be certified in
a device-dependent way for some choice of instruments, since tomographically complete instruments
allow for the full characterization of the process matrix. Hence, theorem 1 also holds in the tripartite
case.
E.2 Device-independent – UUU
A tripartite behaviour {p
(
abc|xyz
)
} is a set of joint probability distributions, that is, a set in which
each element p(abc|xyz) is a real number, such that
p(abc|xyz) ≥ 0 ∀ a, b, c, x, y, z (157)
X
a,b,c
p(abc|xyz) = 1 ∀ x, y, z, (158)
where a ∈ {
1
, . . . , O
A
}, b ∈ {
1
. . . , O
B
}, and c ∈ {
1
. . . , O
C
} label the outcomes and x ∈ {
1
, . . . , I
A
},
y ∈ {1, . . . , I
B
}, and z ∈ {1, . . . , I
C
} label the inputs.
A tripartite behaviour in which Charlie is in the future of Alice and Bob is a behaviour {p
(
abc|xyz
)
}
that satisfies
X
c
p(abc|xyz) =
X
c
p(abc|xyz
0
) ∀ a, b, x, y, z, z
0
, (159)
that is, a behaviour whose Alice and Bob’s joint marginal does not depend on Charlie’s inputs. We
will only consider this type of behaviours and will refer to them as simply tripartite behaviours.
A tripartite behaviour is called a tripartite process behaviour if there exist a tripartite process matrix
W
A
I
A
O
B
I
B
O
C
I
, sets of instruments {A
A
I
A
O
a|x
} and {B
B
I
B
O
b|y
}, and a set of POVMs {M
C
I
c|z
} such that
p
Q
(abc|xyz) = Tr
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
⊗ M
C
I
c|z
) W
A
I
A
O
B
I
B
O
C
I
i
∀ a, b, c, x, y, z. (160)
Just like in the bipartite case, it can be shown that the tripartite process matrix is the most general
operator that leads to valid tripartite behaviours when taken the trace with product instruments.
Additionally, since not all bipartite behaviours can be realized by process matrices and the bipartite
case is a particular case of the tripartite case, not all tripartite behaviours can be realized by process
matrices as well. Hence, theorem 2 also holds in the tripartite case.
A tripartite behaviour {p
A≺B≺C
(
abc|xyz
)
} is causally ordered from Alice to Bob to Charlie if Alice’s
marginals do not depend on the inputs of Bob and Charlie, that is,
X
b,c
p
A≺B≺C
(abc|xyz) =
X
b,c
p
A≺B≺C
(abc|xy
0
z
0
) ∀ a, x, y, y
0
, z, z
0
, (161)
and a tripartite behaviour {p
B≺A≺C
(
abc|xyz
)
} is causally ordered from Bob to Alice to Charlie if Bob’s
marginals do not depend on the inputs of Alice and Charlie, that is,
X
a,c
p
B≺A≺C
(abc|xyz) =
X
a,c
p
B≺A≺C
(abc|x
0
yz
0
) ∀ a, x, x
0
, y, z, z
0
. (162)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 28
A tripartite behaviour {p
causal
(
abc|xyz
)
} is causal if it can be written as a convex combination of
causally ordered behaviours of the kind {p
A≺B≺C
(abc|xyz)} and {p
B≺A≺C
(abc|xyz)}, i.e.,
p
causal
(abc|xyz)
:
= qp
A≺B≺C
(abc|xyz) + (1 − q)p
B≺A≺C
(abc|xyz), (163)
where
0
≤ q ≤
1
is a real number. A tripartite behaviour that does not satisfy eq. (163) is called
noncausal.
Lemma 4.
A tripartite behaviour is causal if and only if it is a tripartite process behaviour that can
be obtained by a causally separable tripartite process matrix.
Proof.
It can be straightforwardly checked that a behaviour that comes from a causally separable
process matrix is causal, the tripartite case being analogous to the bipartite case (see lemma 1). To
prove that all causal tripartite behaviours can be reproduced by a causally separable tripartite process
matrix, we provide the explicit construction of instruments and process matrix below. Given a causal
tripartite behaviour
{p
causal
(
abc|xyz
)
}
, one can write its decompostion into definite causal orders
using
{p
A≺B≺C
(
abc|xyz
)
}
,
{p
B≺A≺C
(
abc|xyz
)
}
, and
q
. According to Bayes’ rule and the nonsignaling
principle we can calculate the quantities
p
A≺B≺C
(abc|xyz) = p
A≺B≺C
A
(a|xyz)p
A≺B≺C
B
(b|axyz)p
A≺B≺C
C
(c|abxyz) (164)
= p
A≺B≺C
A
(a|x)p
A≺B≺C
B
(b|axy)p
A≺B≺C
C
(c|abxyz). (165)
We use them to define instruments
A
A≺B≺C
a|x
= 1
A
I
⊗ p
A≺B≺C
A
(a|x)|axihax|
A
O
(166)
B
A≺B≺C
b|y
=
X
a,x
p
A≺B≺C
B
(b|axy)|axihax|
B
I
⊗ |abxyihabxy|
B
O
(167)
M
A≺B≺C
c|z
=
X
a,b,x,y
p
A≺B≺C
C
(c|abxyz)|abxyihabxy|
C
I
, (168)
and the process matrix
W
A≺B≺C
=
1
A
I
d
A
I
⊗ |Φ
+
ihΦ
+
|
A
O
B
I
⊗ |Φ
+
ihΦ
+
|
B
O
C
I
. (169)
One can check that
Tr
h
(A
A≺B≺C
a|x
⊗ B
A≺B≺C
b|y
⊗ M
A≺B≺C
c|z
) W
A≺B≺C
i
= (170)
= p
A≺B≺C
A
(a|x)p
A≺B≺C
B
(b|axy)p
A≺B≺C
C
(c|abxyz) (171)
= p
A≺B≺C
(abc|xyz). (172)
Equivalently, the instruments and process matrix for the causal order
B ≺ A ≺ C
can be constructed,
and by allowing each order to act on a complementary input subspace, just like in the proof of lemma 1,
all causal tripartite behaviours can be recovered.
Theorem 8.
A tripartite process matrix is certified to be causally nonseparable in a device-independent
way if and only if it can generate a noncausal tripartite behaviour for some choice of instruments for
Alice and Bob and some choice of POVMs for Charlie.
The proof is analogous to theorem 3.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 29
E.3 Semi-device-independent – TTU
We start the first semi-device-independent tripartite scenario by defining assemblages in the TTU
scenario using the same reasoning and motivation as the bipartite case, explained in the main text.
Definition 15
(Process TTU-assemblage)
.
A process TTU-assemblage is a set of operators
{w
Q
c|z
}
,
w
Q
c|z
∈ L
(
H
A
I
A
O
⊗ H
B
I
B
O
), for which there exists a tripartite process matrix
W
A
I
A
O
B
I
B
O
C
I
and a set
of POVMs {M
C
I
c|z
} such that
w
Q
c|z
= Tr
C
I
h
(1
A
I
A
O
⊗ 1
B
I
B
O
⊗ M
C
I
c|z
)W
A
I
A
O
B
I
B
O
C
I
i
, (173)
for all c, z.
Definition 16
(General TTU-assemblage)
.
A general TTU-assemblage is a set of operators
{w
c|z
}
,
w
c|z
∈ L(H
A
I
A
O
⊗ H
B
I
B
O
), that satisfies
w
c|z
≥ 0 ∀ c, z (174)
X
c
w
c|z
= W
A
I
A
O
B
I
B
O
∀ z, (175)
where W
A
I
A
O
B
I
B
O
is a valid bipartite process matrix.
Intuitively, behaviours are extracted from TTU-assemblages according to
p(abc|xyz) = Tr
h
(A
a|x
⊗ B
b|y
)w
c|z
i
∀ a, b, c, x, y, z. (176)
Contrarily to the bipartite case, for which we proved that not all general assemblages can be realized
by process matrices (theorem 4), for the TTU scenario, we prove that general and process TTU-
assemblages are actually equivalent.
Theorem 9. A general TTU-assemblage is valid if and only if it is a process TTU-assemblage.
Proof.
By substituting the definition of a tripartite process matrix and POVMs into eq. (173) it is easy
to check that the resulting assemblage satisfies definition 16. To show that any valid TTU-assemblage
can be obtained with process matrices and instruments, we give the following explicit construction.
Given a general TTU-assemblage
{w
c|z
}
and the general bipartite process matrix
W
=
P
c
w
c|z
, we
construct Charlie’s POVMs {M
c|z
} according to
M
c|z
= W
−
1
2
w
T
c|z
W
−
1
2
, ∀ c, z, (177)
where the transpose
T
is taken in the basis in which
W
is diagonal. Notice that this implies
dim
(
H
C
I
) =
dim
(
H
A
I
A
O
⊗ H
B
I
B
O
) for this particular construction. The sum
P
c
M
c|z
=
W
−
1
2
P
c
w
T
c|z
W
−
1
2
=
W
−
1
2
W W
−
1
2
= 1 for all z guarantees it is a valid set of POVMs.
Now, by writing the process matrix
W
in its diagonal basis,
W
=
P
ij
µ
ij
|ijihij|
, we can define its
purification
|W
ABC
i =
X
ij
√
µ
ij
|ij iji. (178)
The object
W
ABC
=
|W
ABC
ihW
ABC
|
is a well defined tripartite process matrix. This is true particularly
because the dimension of Charlie’s output space is 1, that is, because it is in the future of Alice and
Bob, and follows from the fact that Tr
C
W
ABC
= W . Hence,
Tr
C
h
(1
A
⊗ 1
B
⊗ M
C
c|z
)W
ABC
i
= Tr
C
h
(1
AB
⊗ W
−
1
2
w
T
c|z
W
−
1
2
)W
ABC
i
(179)
= w
c|z
, (180)
for all
c, z
. This concludes the proof that all TTU-assemblages can be realized with valid tripartite
process matrices and a set of POVMs for Charlie.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 30
We now define our notion of causality for TTU-assemblages.
Definition 17
(Causal TTU-assemblage)
.
A TTU-assemblage is causally ordered from Alice to Bob
to Charlie if it satisfies
X
c
w
A≺B≺C
c|z
= W
A≺B
∀ z, (181)
where
W
A≺B
is a bipartite process matrix causally ordered from Alice to Bob, and equivalently from
Bob to Alice.
A TTU-assemblage
{w
causal
c|z
}
is causal if it can be expressed as a convex combination of causally
ordered TTU-assemblages, i.e.,
w
causal
c|z
:
= qw
A≺B≺C
c|z
+ (1 − q)w
B≺A≺C
c|z
∀ c, z, (182)
where 0
≤ q ≤
1 is a real number. A TTU-assemblage that does not satisfy eq. (182) is called a
noncausal TTU-assemblage.
Notice that one can decide whether a TTU-assemblage is causal by means of SDP.
Lemma 5.
A TTU-assemblage is causal if and only if it is a process TTU-assemblage that can be
obtained from a causal tripartite process matrix.
The proof is analogous to the one in theorem 9. Notice that in this case W
ABC
will be the purification
of a causally ordered process matrix, and therefore, also causally ordered.
Theorem 10.
A tripartite process matrix is certified to be causally nonseparable in a semi-device-
independent TTU way if and only if it can generate a noncausal TTU-assemblage for some choice of
POVMs for Charlie.
The proof is analogous to theorem 5.
E.4 Semi-device-independent – TUU
The other semi-device-independent cases follow analogously. We continue with the TUU scenario.
Definition 18
(Process TUU-assemblage)
.
A process TTU-assemblage is a set of operators
{w
bc|yz
}
,
w
bc|yz
∈ L
(
H
A
I
A
O
), for which there exist a tripartite process matrix
W
A
I
A
O
B
I
B
O
C
I
, a set of instruments
{B
B
I
B
O
b|y
}, and a set of POVMs {M
C
I
c|z
} such that
w
Q
bc|yz
= Tr
B
I
B
O
C
I
h
(1
A
I
A
O
⊗ B
B
I
B
O
b|y
⊗ M
C
I
c|z
)W
A
I
A
O
B
I
B
O
C
I
i
, ∀ b, c, y, z. (183)
Definition 19
(General TUU-assemblage)
.
A general TTU-assemblage is a set of operators
{w
bc|yz
}
,
w
bc|yz
∈ L(H
A
I
A
O
), that satisfies
w
bc|yz
≥ 0 ∀ b, c, y, z (184)
Tr
X
b,c
w
bc|yz
= d
A
O
∀ y, z (185)
X
c
w
bc|yz
=
X
c
w
bc|yz
0
∀ b, y, z, z
0
(186)
X
b,c
w
bc|yz
=
A
O
X
b,c
w
bc|yz
∀ y, z. (187)
Intuitively, behaviours are extracted from TUU-assemblages according to
p(abc|xyz) = Tr
A
a|x
w
bc|yz
∀ a, b, c, x, y, z. (188)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 31
In the bipartite case, we have proven that not all general bipartite assemblages can be realized by
process matrices (are process bipartite assemblages), while in the previous tripartite case, TTU, we
have proven that all general TTU-assemblages can indeed be realized by process matrices (are process
TTU-assemblages). However, in this tripartite scenario, TUU, as well as in the remaining cases, UTT
and UUT, it is not clear whether all general TUU-, UTT-, and UUT-assemblages can be realized by
process matrices. We leave this problem as an open question.
Nevertheless, all process TUU-assemblages are valid general TUU-assemblages.
We now define our notion of causality for TUU-assemblages.
Definition 20
(Causal TUU-assemblage)
.
A TUU-assemblage is causally ordered from Alice to Bob
to Charlie if it satisfies
X
b,c
w
A≺B≺C
bc|yz
=
X
b,c
w
A≺B≺C
bc|y
0
z
0
∀ y, y
0
, z, z
0
, (189)
and from Bob to Alice to Charlie if it satisfies
X
c
w
B≺A≺C
bc|yz
=
A
O
X
c
w
B≺A≺C
bc|yz
∀ b, y, z. (190)
A TUU-assemblage
{w
causal
bc|yz
}
is causal if it can be expressed as a convex combination of causally
ordered TUU-assemblages, i.e.,
w
causal
bc|yz
:
= qw
A≺B≺C
bc|yz
+ (1 − q)w
B≺A≺C
bc|yz
∀ b, c, y, z, (191)
where 0
≤ q ≤
1 is a real number. A TUU-assemblage that does not satisfy eq. (191) is called a
noncausal TUU-assemblage.
Notice that one can decide whether a TUU-assemblage is causal by means of an SDP.
All causally separable tripartite process matrices lead to causal TUU-assemblages, for whatever
choice of instruments. Whether all causal TUU-assemblages can be written in terms of causally sep-
arable process matrices is not clear, although for the particular case of assemblages that are causally
ordered from Alice to Bob to Charlie, we can show this to be the case.
Here we present our explicit construction of any TUU-assemblage that is causally ordered from Alice
to Bob to Charlie by a tripartite process matrix that is causally ordered from Alice to Bob to Charlie.
Given a TUU-assemblage {w
A≺B≺C
bc|yz
}, we can define the state ρ such that
P
b,c
w
A≺B≺C
bc|yz
=
ρ
A
I
⊗1
A
O
.
Since ρ ≥
0
it can be written as ρ
=
P
i
µ
i
|iihi|, which can be purified by |psii
=
P
i
√
µ
i
|iii. Then, let
{B
A≺B≺C
b|y
} and {M
A≺B≺C
c|z
} be instruments
B
A≺B≺C
b|y
= 1
B
I
⊗ |byihby|
B
O
(192)
M
A≺B≺C
c|z
=
X
b,y
ρ
−
1
2
C
0
I
⊗ 1
C
00
I
w
T A≺B≺C C
0
I
C
00
I
bc|yz
ρ
−
1
2
C
0
I
⊗ 1
C
00
I
⊗ |byihby|
C
000
I
, (193)
where ρ
−1
be the inverse of ρ on its support, the transpose
T
is taken on the {|ii}
i
basis and H
C
I
=
H
C
0
I
⊗ H
C
00
I
⊗ H
C
00
I
. Let
W
A≺B≺C
= |ψihψ|
C
0
I
A
I
⊗ |Φ
+
ihΦ
+
|
C
00
I
A
O
⊗
1
B
I
d
B
I
⊗ |Φ
+
ihΦ
+
|
C
000
I
B
O
(194)
be a tripartite process matrix that is causally ordered from Alice to Bob to Charlie. Then, it is true
that the assemblage {w
A≺B
bc|yz
} can be recovered by
Tr
B
I
B
O
C
I
h
(1
A
⊗ B
A≺B≺C
b|y
⊗ M
A≺B≺C
c|z
)W
A≺B≺C
i
= w
A≺B≺C
bc|yz
. (195)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 32
E.5 Semi-device-independent – UTT
Here we detail our concepts and definitions for the UTT scenario.
Definition 21
(Process UTT-assemblage)
.
A process UTT-assemblage is a set of operators
{w
a|x
}
,
w
a|x
∈ L(H
B
I
B
O
⊗ H
C
I
), for which there exist a tripartite process matrix W
A
I
A
O
B
I
B
O
C
I
and a set of
instruments {A
A
I
A
O
a|x
} such that
w
Q
a|x
= Tr
A
I
A
O
h
(A
A
I
A
O
a|x
⊗ 1
B
I
B
O
⊗ 1
C
I
)W
A
I
A
O
B
I
B
O
C
I
i
, ∀ a, x. (196)
Definition 22
(General UTT-assemblage)
.
A general UTT-assemblage is a set of operators
{w
a|x
}
,
w
a|x
∈ L(H
B
I
B
O
⊗ H
C
I
), that satisfies
w
a|x
≥ 0 ∀ a, x (197)
Tr
X
a
w
a|x
= d
B
O
∀ x (198)
C
I
X
a
w
a|x
=
B
O
C
I
X
a
w
a|x
∀ x. (199)
Intuitively, behaviours are extracted from UTT-assemblages according to
p(abc|xyz) = Tr
h
w
a|x
(B
b|y
⊗ M
c|z
)
i
∀ a, b, c, x, y, z. (200)
Definition 23
(Causal UTT-assemblage)
.
A UTT-assemblage is causally ordered from Alice to Bob
to Charlie if it satisfies
C
I
w
A≺B≺C
a|x
=
B
O
C
I
w
A≺B≺C
a|x
∀ a, x, (201)
and from Bob to Alice to Charlie if it satisfies
C
I
X
a
w
B≺A≺C
a|x
=
C
I
X
a
w
B≺A≺C
a|x
0
∀ x, x
0
. (202)
A UTT-assemblage
{w
causal
a|x
}
is causal if it can be expressed as a convex combination of causally
ordered UTT-assemblages, i.e.,
w
causal
a|x
:
= qw
A≺B≺C
a|x
+ (1 − q)w
B≺A≺C
a|x
∀ a, x, (203)
where 0
≤ q ≤
1 is a real number. A UTT-assemblage that does not satisfy eq. (203) is called a
noncausal UTT-assemblage.
Notice that one can decide whether a UTT-assemblage is causal by means of an SDP.
It is not clear whether all general UTT-assemblages can be realized by tripartite process matrices nor
whether all causal UTT-assemblages can be realized by causally separable tripartite process matrices.
Nevertheless, the set of general UTT-assemblages is an outer approximation of the set of process UUT-
assemblages and the set of causal UUT-assemblages is an outer approximation of the set of process
UTT-assemblages that come from causally separable process matrices. What is not clear is whether
these approximations are tight.
E.6 Semi-device-independent – UUT
The final tripartite semi-device independent scenario studied in this work is the UUT scenario.
Definition 24
(Process UUT-assemblage)
.
A process UUT-assemblage is a set of operators
{w
ab|xy
}
,
w
ab|xy
∈ L
(
H
C
I
), for which there exist a tripartite process matrix
W
A
I
A
O
B
I
B
O
C
I
and sets of instruments
{A
A
I
A
O
a|x
}, {B
B
I
B
O
b|y
} such that
w
Q
ab|xy
= Tr
A
I
A
O
B
I
B
O
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
⊗ 1
C
I
)W
A
I
A
O
B
I
B
O
C
I
i
, ∀ a, b, x, y. (204)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 33
Definition 25
(General UUT-assemblage)
.
A general UUT-assemblage is a set of operators
{w
ab|xy
}
,
w
ab|xy
∈ L(H
C
I
), that satisfies
w
ab|xy
≥ 0 ∀ a, b, x, y (205)
Tr
X
a,b
w
ab|xy
= 1 ∀ x, y. (206)
Intuitively, behaviours are extracted from UUT-assemblages according to
p(abc|xyz) = Tr
w
ab|xy
M
c|z
∀ a, b, c, x, y, z. (207)
The set of general UUT-assemblages is an outer approximation of the set of process UUT-assemblages
but it is not clear to us whether this approximation is tight, i.e., it is not clear whether or not all
general UUT-assemblages can be obtained by tripartite process matrices.
Definition 26
(Causal UUT-assemblage)
.
A UUT-assemblage is causally ordered from Alice to Bob
to Charlie if it satisfies
Tr
X
b
w
A≺B≺C
ab|xy
= Tr
X
b
w
A≺B≺C
ab|xy
0
∀ a, x, y, y
0
, (208)
and from Bob to Alice to Charlie if it satisfies
Tr
X
a
w
B≺A≺C
ab|xy
= Tr
X
a
w
B≺A≺C
ab|x
0
y
∀ b, x, x
0
, y. (209)
A UUT-assemblage
{w
causal
ab|xy
}
is causal if it can be expressed as a convex combination of causally
ordered UUT-assemblages, i.e.,
w
causal
ab|xy
:
= qw
A≺B≺C
ab|xy
+ (1 − q)w
B≺A≺C
ab|xy
∀ a, b, x, y, (210)
where 0
≤ q ≤
1 is a real number. A UUT-assemblage that does not satisfy eq. (210) is called a
noncausal UUT-assemblage.
Notice that one can decide whether a UUT-assemblage is causal by means of an SDP.
For the case of causal UUT-assemblages, we prove that our approximation is indeed tight, i.e., that
all causal UUT-assemblages can be realized by causal tripartite process matrix, analogously to lemma 2
in the bipartite case and lemma 5 in the TTU tripartite case.
Lemma 6.
A UUT-assemblage is causal if and only if it is a process UUT-assemblage that can be
obtained from a causal tripartite process matrix.
Proof.
We begin by showing that all UUT-assemblages that come from a causal process matrix are
causal. Let {w
ab|xy
} be such that
w
ab|xy
= Tr
A
I
A
O
B
I
B
O
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
⊗ 1
C
I
)W
A≺B≺C
i
, ∀ a, b, x, y. (211)
Then, using eq. (152), which is
C
I
W
A≺B≺C
=
B
O
C
I
W
A≺B≺C
, it is possible to deduce that
Tr
P
b
w
ab|xy
is independent of
y
, and hence
{w
ab|xy
}
=
{w
A≺B≺C
ab|xy
}
is causally ordered. The equivalent is true for
the order B ≺ A ≺ C.
To prove the only if part we show that every causal UUT-assemblage can be reproduced by acting with
some instruments on a causal process matrix. Given a causal UUT-assemblage, it can be decomposed
into {w
A≺B≺C
ab|xy
} and {w
B≺A≺C
ab|xy
} with some convex weight q.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 34
From {w
A≺B≺C
ab|xy
}, we define
3
p
A≺B≺C
(ab|xy)
:
= Tr(w
A≺B≺C
ab|xy
) ∀ a, b, x, y, (212)
p
A≺B≺C
A
(a|x)
:
=
X
b
Tr(w
A≺B≺C
ab|xy
) ∀ a, x, (213)
ρ
ab|xy
:
=
w
A≺B≺C
ab|xy
Tr(w
A≺B≺C
ab|xy
)
∀ a, b, x, y, (214)
Using Bayes’ rule we also have
p
A≺B≺C
B
(
b|axy
) =
p
A≺B≺C
(
ab|xy
)
/p
A≺B≺C
A
(
a|x
). With this given
quantities, we can construct instruments {A
A≺B≺C
a|x
} and {B
A≺B≺C
b|y
},
A
A≺B≺C
a|x
= 1
A
I
⊗ p
A≺B≺C
A
(a|x)|axihax|
A
O
∀ a, x (215)
B
A≺B≺C
b|y
=
X
a,x
p
A≺B≺C
B
(b|axy)|axihax|
B
I
⊗ ρ
T B
O
ab|xy
∀ b, y, (216)
and also the process matrix
W
A≺B≺C
=
1
A
I
d
A
I
⊗ |Φ
+
ihΦ
+
|
A
O
B
I
⊗ |Φ
+
ihΦ
+
|
B
O
C
I
, (217)
and check that
Tr
A
I
A
O
B
I
B
O
h
(A
A≺B≺C
a|x
⊗ B
A≺B≺C
b|y
⊗ 1
C
I
)W
A≺B≺C
i
= (218)
= p
A≺B≺C
A
(a|x)p
A≺B≺C
B
(b|axy)ρ
ab|xy
(219)
= w
A≺B≺C
ab|xy
(220)
for every a, b, x, y. Analogously, the same holds for {w
B≺A≺C
ab|xy
}.
Finally, just like in the proof of lemma 1, by allowing the different causal orders to act in comple-
mentary subspaces, we can recover any convex combinations of causally ordered assemblages, i.e.,
causal assemblages.
Theorem 11.
A tripartite process matrix is certified to be causally nonseparable in a semi-device-
independent UUT way if and only if it can generate a noncausal UUT-assemblage for some choice of
instruments for Alice and Bob.
The proof is analogous to theorem 5.
F Proof that the quantum switch processes are causal in the UUT scenario
In this appendix we prove theorem 7, which we reestate for the convenience of the reader.
Theorem 7.
The quantum switch processes cannot be certified to be causally nonseparable on a
semi-device-independent scenario where Alice and Bob are untrusted and Charlie is trusted (UUT).
Moreover, any tripartite process matrix
W ∈ L
(
H
A
I
A
O
B
I
B
O
C
I
), with Charlie in the future of Alice
and Bob, that satisfies
Tr[(A
A
I
A
O
a|x
⊗B
B
I
B
O
b|y
⊗ 1
C
I
)W
A
I
A
O
B
I
B
O
C
I
] =
qp
A≺B
(ab|xy) + (1 − q)p
B≺A
(ab|xy),
(221)
for all
a, b, x, y
, where 0
≤ q ≤
1 is a real number, cannot be certified to be noncausal in a UUT
scenario.
3
If Tr(w
A≺B≺C
ab|xy
) = 0, we define ρ
ab|xy
as the null operator.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 35
Proof. Let
w
switch
ab|xy
:= Tr
A
I
A
O
B
I
B
O
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
⊗ 1
C
I
)W
switch
i
(222)
be a UUT-assemblage generated by the quantum switch. We define
p(ab|xy) := Tr(w
switch
ab|xy
) (223)
and
4
ρ
ab|xy
:=
w
switch
ab|xy
p(ab|xy)
. (224)
Given that
p(ab|xy) = Tr(w
switch
ab|xy
) (225)
= Tr
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
⊗ 1
C
I
)W
switch
i
(226)
= qp
A≺B
(ab|xy) + (1 − q)p
B≺A
(ab|xy), (227)
since the quantum switch is device-independent causal, one can write
w
switch
ab|xy
= p(ab|xy)ρ
ab|xy
(228)
= qp
A≺B
(ab|xy)ρ
ab|xy
+ (1 − q)p
B≺A
(ab|xy)ρ
ab|xy
. (229)
One can verify that the first term satisfies
Tr
X
b
p
A≺B
(ab|xy)ρ
ab|xy
=
X
b
p
A≺B
(ab|xy)Tr(ρ
ab|xy
) = p(a|x) (230)
and
Tr
X
a
p
B≺A
(ab|xy)ρ
ab|xy
=
X
a
p
B≺A
(ab|xy)Tr(ρ
ab|xy
) = p(b|y). (231)
Hence,
p
A≺B
(ab|xy)ρ
ab|xy
= w
A≺B
ab|xy
(232)
is an assemblage causally ordered from Alice to Bob and
p
B≺A
(ab|xy)ρ
ab|xy
= w
B≺A
ab|xy
(233)
is an assemblage causally ordered from Bob to Alice. Consequently,
w
switch
ab|xy
= qw
A≺B
ab|xy
+ (1 − q)w
B≺A
ab|xy
(234)
is UUT causal for all instruments of Alice and Bob.
This proof holds for all process matrices W for which
p(ab|xy) = Tr
h
(A
A
I
A
O
a|x
⊗ B
B
I
B
O
b|y
⊗ 1
C
I
)W
i
∈ CAUSAL. (235)
4
If Tr(w
switch
ab|xy
) = 0, we define ρ
ab|xy
as the null operator.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 36
G Analysis of experimental implementations of the quantum switch
In this appendix we present more details of our analysis of the experimental results involving the
quantum switch reported in refs. [29, 30]. Among other assumptions, both papers consider a device-
dependent scenario, where the analysis is made by assuming complete knowledge of all instruments in-
volved (although no assumption is made about the process matrix). Here, we show that the experiment
reported in both papers could have also certified indefinite causal order in a semi-device-independent
scenario, where no assumptions are made about the instruments of one of the parties.
We remark that (device-dependent) causal witnesses, including the ones of refs. [29, 30], are derived
assuming that all instruments are implemented perfectly. However, due to experimental imperfections,
this assumption is seldom true. One might then obtain a noncausal – even nonprocess – behaviour
simply because the implemented instruments are not the ideal ones, rather than the data being genu-
inely noncausal (or nonprocess). Indeed, we have analysed the experimental data
5
collected in ref. [29]
and verified that if the instruments performed by Alice, Bob, and Charlie are trusted, the experimental
behaviour is not a process behaviour. That is, there does not exist any process matrix W , causally
separable or not, that can generate the experimental behaviour consistently with the assumed instru-
ments
6
. To properly anaylse experimental data we would need to allow some leeway in the instruments
and probabilities, but developing the methods for that is beyond the scope of this paper.
Therefore, in order to avoid potential false positive results due to experimental error, we have
considered the statistics one would have obtained in an ideal version of the experiment instead of
considering the data collected on the actual experiments.
We start by analysing the device-dependent experiment described in ref. [30] by Goswami et al., which
considers an optical setup where three parties, Alice, Bob, and Charlie, have access to the reduced
quantum switch process W
red
(see eq. (42)). In this experiment, Alice and Bob can choose between
the 8 different unitary operations
7
and Charlie performs a measurement in the σ
X
basis on the control
qubit state. Theoretically, the statistics of an ideal device-dependent TTT (trusted-trusted-trusted)
experiment would be given by the behaviour
p
ideal1
(c|x, y) := Tr
h
U
x
⊗ U
y
⊗ M
c
W
red
i
, (236)
where the instruments {U
x
} and {M
c
} are the ones described on the supplemental material of ref. [30].
As pointed in ref. [30], the behaviour {p
ideal1
(
c|x, y
)
} certifies indefinite causal order in a device-
dependent way. Using the SDP formulation of the problem described in appendix E, we show that the
noisy behaviour
p
ideal1
η
(c|x, y) := (1 − η) p
ideal1
(c|x, y) + η
1
2
(237)
where p
I
(
c|x, y
) =
1
2
is a uniform probability distribution, cannot be described by a causally separable
tripartite process matrix, i.e.,
p
ideal1
η
(c|x, y) 6= Tr
h
U
x
⊗ U
y
⊗ M
c
W
sep
i
, (238)
in the range η ∈
[0
,
0
.
1989)
. Hence, in this range of η, the behaviour {p
ideal1
η
(
c|x, y
)
} certifies indefinite
causal order in a device-dependent (TTT) way.
In order to analyse the behaviour {p
ideal1
(
c|x, y
)
} in a semi-device-independent scenario, we drop the
hypothesis that the measurement {M
c
} performed by Charlie is trusted, working in a TTU (trusted-
trusted-untrusted) scenario. Using the SDP formulation of the problem described in appendix E, we
show that the noisy behaviour {p
ideal1
η
(c|x, y)} cannot be described by a causal TTU-assemblage, i.e.,
p
ideal1
η
(c|x, y) 6= Tr
h
U
x
⊗ U
y
w
causal
c
i
, (239)
5
We have analysed the data points of this experiment without taking the error bars into consideration.
6
Moreover, even if we drop the assumption about the knowledge of the instrument performed by Charlie, there is no
process matrix which is consistent with the experimental data collected.
7
Note that an unitary operation can be seen as an instrument with deterministic classical output.
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 37
in the same range of η ∈
[0
,
0
.
1989)
. Hence, in this range of η, the behaviour {p
ideal1
η
(
c|x, y
)
} certifies
indefinite causal order in a semi-device-independent (TTU) way.
Therefore, the experimental setup described in ref. [30] allows for certification of indefinite causal
order with weaker hypotheses – in a semi-device-independent scenario. Using the machinery developed
in this work, we could not show that the behaviour {p
ideal1
(
c|x, y
)
} can certify indefinite causal order
in the UTT or TUU scenarios. However, since some of our SDP methods for the tripartite case may
provide only an outer approximation of the sets of causal assemblages (see appendix E), we cannot
discard the possibility of such certification either. We remark, nevertheless, that we have proven that
it is possible to certify that the switch process is causally nonseparable in these scenarios for the right
choice of instruments (see section 3).
The experiment performed in ref. [29] by Rubino et al. considers a four-partite version of the switch
process which is slightly more general than the tripartite one discussed in this paper. While we consider
a tripartite switch process where the target state is embedded in the process, ref. [29] considers a four-
partite switch process in which the target state can be chosen by a fourth part in the common past of
all other parties. If we label the fourth party D
O
(David output), the process matrix of the four-partite
quantum switch is given by W
4
switch
:= |w
4
switch
ihw
4
switch
|, where
|w
4
switch
i :=
1
√
2
|Φ
+
i
D
O
A
I
|Φ
+
i
A
O
B
I
|Φ
+
i
B
O
C
t
I
|1i
C
c
I
+ |Φ
+
i
D
O
B
I
|Φ
+
i
B
O
A
I
|Φ
+
i
A
O
C
t
I
|0i
C
c
I
(240)
and |Φ
+
i := |00i + |11i is an unnormalized maximally entangled two-qubit state.
Reference [29] considers an optical setup where Alice, Bob, Charlie, and David have access to a
reduced four-partite quantum switch process given by W
4
red
:=
Tr
C
t
I
W
4
switch
. In this experiment,
David can choose between 3 different states ρ
d
, Alice can choose between 12 different dichotomic
instruments {A
a|x
}, and Bob can choose between 10 unitary operations represented by the instruments
{U
y
}. Theoretically, in an ideal device-dependent TTTT experiment, the statistics would be given by
the behaviour
p
ideal2
(a, c|x, y, d) := Tr
h
ρ
d
⊗ A
a|x
⊗ U
y
⊗ M
c
W
4
red
i
, (241)
where the instruments {ρ
d
}, {A
a|x
}, {U
y
}, and {M
c
} are the ones described on the supplemental
material of ref. [29].
In order to use the SDP formulations described in appendix E, we restrict our analysis to the
particular case where David outputs the state ρ
1
= |0ih0|. That is, we analyse the behaviour
p
ideal2
(a, c|x, y, 1) := Tr
h
|0ih0| ⊗ A
a|x
⊗ U
y
⊗ M
c
W
4
red
i
. (242)
Notice that when David is restricted to the choice of the state |
0
ih
0
|, the four-partite reduced switch
process W
4
red
relates to the tripartite reduced switch process W
red
(eq. (42)) via the identity
Tr
D
O
h
|0ih0|
D
O
⊗ 1
A
I
A
O
⊗ 1
B
I
B
O
⊗ 1
C
c
I
W
4
red
i
= W
red
. (243)
Moreover, if the behaviour {p
ideal2
(
a, c|x, y,
1)
} allows for certification of indefinite causal order, the
full behaviour {p
ideal2
(
a, c|x, y, d
)
} also allows for certification of indefinite causal order. We can, hence,
use our tripartite machinery to certify indefinite causal order in this particular four-partite case.
Using the SDP formulation of the problem described in appendix E, we show that the noisy behaviour
p
ideal2
η
(a, c|x, y, 1) := (1 − η) p
ideal2
(a, c|x, y, 1) + η
1
4
(244)
where p
I
(
a, c|x, y
) =
1
4
is a uniform probability distribution, cannot be described by a causally separable
tripartite process matrix, i.e.,
p
ideal2
η
(a, c|x, y, 1) 6= Tr
h
A
a|x
⊗ U
y
⊗ M
c
W
sep
i
, (245)
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 38
in the range η ∈
[0
,
0
.
2300)
. Hence, in this range of η, the behaviour {p
ideal2
η
(
a, c|x, y,
1)
} certifies
indefinite causal order in a device-dependent (TTT) way.
We now analyse the behaviour p
ideal2
in a semi-device-independent scenario, where we drop the
hypothesis that the measurement {M
c
} performed by Charlie is trusted, working in a TTU (trusted-
trusted-untrusted) scenario. Using the SDP formulation of the problem described in appendix E, we
show that the noisy behaviour {p
ideal1
η
(c|x, y)} cannot be described by a causal TTU-assemblage, i.e.,
p
ideal2
η
(a, c|x, y, 1) 6= Tr
h
A
a|x
⊗ U
y
w
causal
c
i
, (246)
in the same range of η ∈
[0
,
0
.
2300)
. Hence, in this range of η, the behaviour {p
ideal2
η
(
a, c|x, y,
1)
}
certifies indefinite causal order in a semi-device-independent (TTU) way.
Therefore, the experimental setup described in ref. [29] allows for certification of indefinite causal
order with weaker hypotheses – in a semi-device-independent scenario. Using the machinery developed
in this work, we could not show that the behaviour {p
ideal2
(
a, c|x, y, d
)
} can certify indefinite causal
order in other semi-device-independent scenarios. However, since we have only considered the case
where the state ρ
z
chosen by David is fixed and some of our SDP methods for the tripartite case may
provide only an outer approximation of the sets of causal assemblages (see appendix E), we cannot
discard the possibility of such certification either. We remark, nevertheless, that we have proven that it
is possible to certify that the switch process is causally nonseparable in other semi-device-independent
scenarios for the right choice of instruments (see section 3).
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 39