# Genuine network quantum nonlocality and self-testing

Ivan Šupić<EMAIL_ADDRESS>Département de Physique Appliquée, Université
de Genève, 1211 Genève, Switzerland  CNRS, LIP6, Sorbonne Université, 4 place
Jussieu, 75005 Paris, France  Jean-Daniel Bancal  Département de Physique
Appliquée, Université de Genève, 1211 Genève, Switzerland  Université Paris-
Saclay, CEA, CNRS, Institut de physique théorique, 91191, Gif-sur-Yvette,
France  Yu Cai  Département de Physique Appliquée, Université de Genève, 1211
Genève, Switzerland  Nicolas Brunner  Département de Physique Appliquée,
Université de Genève, 1211 Genève, Switzerland

(August 28, 2024)

###### Abstract

The network structure offers in principle the possibility for novel forms of
quantum nonlocal correlations, that are proper to networks and cannot be
traced back to standard quantum Bell nonlocality. Here we define a notion of
genuine network quantum nonlocality. Our approach is operational and views
standard quantum nonlocality as a resource for producing correlations in
networks. We show several examples of correlations that are genuine network
nonlocal, considering the so-called bilocality network of entanglement
swapping. In particular, we present an example of quantum self-testing which
relies on the network structure; the considered correlations are non-bilocal,
but are local according to the usual definition of Bell locality.

##  I Introduction

Discovered by Bell in the 1960s, the phenomenon of quantum nonlocality has far
reaching consequences Bell ; review . Arguably among the most counterintuitive
aspects of quantum theory, it also represents a powerful resource for
information processing, notably in the context of device-independent protocols
Acin2007 ; Pironio2010 ; Colbeck .

Quantum nonlocality manifests as strong correlations observed by distant
parties performing local measurements on a shared quantum system. This
phenomenon relies on the shared system to be in an entangled state, and the
local measurements to be incompatible (for each party). In this sense
nonlocality is based on the judicious combination of several inherently non-
classical properties of quantum mechanics, namely entanglement of quantum
states and incompatibility of quantum measurements.

Recently, growing interest has been devoted to the exploration of quantum
nonlocality in networks; see e.g. Armin  for a recent review. A general
framework has been developed for this problem Branciard_2010 ; Fritz_2012 ;
branciard2012bilocal . Consider a network with several parties, and $N$
sources distributing physical (classical or quantum) systems to various
subsets of parties. The main idea consists in assuming that all the sources of
the network are independent from each other. This allows one to define the
notion of $N$-locality, which can be seen as a natural extension of the
concept of Bell locality to networks with independent sources. Correlations
that do not admit an $N$-local model are termed “network nonlocal”.
Importantly, whether a given distribution is network nonlocal or not depends
on the topology of the considered network. In particular, for certain networks
there are correlations that are provably network nonlocal, while the same
correlation would be local when considering the usual Bell scenario, see e.g.
branciard2012bilocal .

Quantum theory allows for correlations in networks that are not $N$-local. The
first examples of “quantum network nonlocality” have been derived for the
scenario of entanglement swapping, now also referred to as “bilocality”
Branciard_2010 . In turn, a more astonishing effect was discovered, namely
that quantum network nonlocality can be demonstrated without the need for
measurement inputs, i.e. each party performs a single fixed measurement
Fritz_2012 ; branciard2012bilocal . This would of course be impossible in the
standard Bell scenario, and hence appears at first sight as a novel form of
quantum nonlocality proper to networks.

A first example of such “quantum nonlocality without inputs” was proposed by
Fritz Fritz_2012 , for the so-called triangle network. The latter features
three parties, each pair of them connected by a different source. After
inspection, it turns out, however, that Fritz’s example can be viewed as a
clever mapping of the standard Bell CHSH scenario into the triangle network
configuration Fraser ; Supic .

A second example was more recently proposed by Renou and co-authors renou_2019
, also for the triangle network. The authors argued, however, that their
example is fundamentally different from that of Fritz, and could represent a
form of quantum nonlocality genuine to networks, i.e. that cannot be mapped
back to any standard Bell scenario. This conjecture relies on the observation
that their quantum distribution was based on all three sources producing an
entangled state (contrary to Fritz’s example, where only one source needed to
produce entanglement), and all parties performing an entangled measurement on
their incoming subsystems (again, Fritz’s example uses only separable
measurements).

At this point, the main question is thus whether there exists indeed a form of
quantum nonlocality genuine to networks, and how to formalize this concept.
This is precisely the goal of the present paper. We first propose a formal
definition of genuine network nonlocality. Our definition is operational, and
considers standard Bell nonlocality as a resource. Correlations that can
provably not be obtained by these resources are therefore termed genuine
network nonlocal. In turn, we highlight several examples of this phenomenon.
In particular, we present a self-testing result for the bilocality network,
where the observation of specific quantum correlation certifies (up to local
isometries) the form of local measurements (including a joint entangled
measurement), and the underlying entangled states. This example shows that
self-testing can be enforced by the network structure, in the sense that the
quantum distribution is non-bilocal but can provably not violate any standard
Bell inequality. We conclude with a perspective on future possible
developments of these ideas.

##  II Setting

Figure 1: Bilocality (or entanglement swapping) network.

We focus our discussion on a simple tripartite quantum network, but our
analysis can be straightforwardly extended to more general networks. The
network we consider consists of three parties, Alice, Bob and Charlie, and two
independent sources (see Fig. 1). This network corresponds to the well-known
entanglement swapping experiment Zukowski . Alice and Bob share a bipartite
quantum state $\rho_{AB_{1}}$, distributed by a common source $S_{AB}$.
Similarly, Bob and Charlie share a state $\rho_{B_{2}C}$, distributed by a
second source $S_{BC}$.

Each party receives an input (i.e. a choice of measurement setting) and
outputs the corresponding measurement outcome. We denote $x$, $y$ and $z$ the
inputs of Alice, Bob and Charlie respectively, and $a$, $b$ and $c$ their
outputs. The resulting statistics are given by:

$p(abc|xyz)=\textrm{Tr}\left[\left(M^{A}_{a|x}\otimes
M^{B_{1}B_{2}}_{b|y}\otimes
M^{C}_{c|z}\right)\left(\rho_{AB_{1}}\otimes\rho_{B_{2}C}\right)\right]$ (1)

where $M^{A}_{a|x}$ denotes the operators of Alice’s measurements (i.e. the
POVM elements), satisfying $M^{A}_{a|x}\geq 0$ and
$\sum_{a}M^{A}_{a|x}=\openone$ for all $x$, and similarly for Bob’s and
Charlie’s measurements. Note that Bob’s measurement acts on both subsystems
$B_{1}$ and $B_{2}$. Also when computing (1), one should carefully associate
the Hilbert space of the various subsystems.

Given specific quantum correlations, one may now ask whether they could be
reproduced by a classical model. The standard approach in nonlocality consists
in asking whether the distribution admits a decomposition of the form:

$p(abc|xyz)=\int d\lambda p(\lambda)p(a|x\lambda)p(b|y\lambda)p(c|z\lambda)$
(2)

where $\lambda$ denotes the shared classical variable distributed to all
parties (Alice, Bob and Charlie) with distributions $p(\lambda)$. This
represents the usual definition of Bell locality for a tripartite system, see
e.g. review .

At this point it is important to realize that comparing quantum distributions
of the form (1) to classical correlations of the form (2) is arguably not
appropriate. The key point is that the underlying network structure is not the
same. Indeed, the decomposition in Eq. (2) assumes a common shared variable
distributed to all parties; hence requiring a common source. However, this
structure does not match the quantum network in Fig. 1.

This observation motivated the authors of Ref. Branciard_2010  to propose an
alternative definition of locality for this problem, taking into account the
network structure as in Fig. 1. Indeed, one should now consider a local model
involving two separate sources and enforce their independence. Just like the
quantum state has a product structure, i.e.
$\rho_{AB_{1}}\otimes\rho_{B_{2}C}$, the classical model should feature the
same structure. This motivates the notion of “bilocal” correlations, which
take the form:

$p(abc|xyz)=\int d\lambda p(\lambda)\int d\mu
q(\mu)p(a|x\lambda)p(b|y\lambda\mu)p(c|z\mu)$ (3)

where $\lambda$ and $\mu$ denote the two shared variables. Alice and Bob have
access to $\lambda$ (distributed by the source $S_{AB}$), while Bob and
Charlie have access to $\mu$ (via source $S_{BC}$). Importantly, $\lambda$ and
$\mu$ are distributed according to independent probability distributions
$p(\lambda)$ and $q(\mu)$.

There exist quantum correlations of the form (1) that are non bilocal, i.e. do
not admit a decomposition of the form of Eq. (3). A simple (perhaps almost
trivial) example consists in setting the state $\rho_{AB_{1}}$ to be a
maximally entangled two-qubit Bell state, and choosing the measurement of
Alice to be anti-commuting Pauli operators. Choosing Bob’s measurement to be
some appropriate anticommuting Pauli operators on the $B_{1}$ subsystem, and
trivial on $B_{2}$, the resulting distribution $p(abc|xyz)$ (as given in (1))
will be non-bilocal, no matter what state $\rho_{B_{2}C}$ is produced by the
second source and what measurements Charlie performs. This is because, the
marginal distribution $p(ab|xy)=\sum_{c}p(abc|xyz)$ violates the standard CHSH
Bell inequality. This deceptively simple example can be turned into a more
elegant form following the construction of Fritz Fritz_2012  by adding a third
independent source between Alice and Charlie. This results in a scenario with
two remarkable features: all inputs can be cleverly removed and replaced by
outputs, hence resulting in the effect of “quantum nonlocality without
inputs”, and the obtained construction allows to easily demonstrate the
presence of nonlocality in a network containing a loop. Nevertheless, one can
still argue that this instance of network nonlocality has its origins in a
standard Bell inequality test; see Fraser ; Supic  and below.

Intuitively, however, quantum theory should allow for conceptually a very
different (and possibly stronger) type of correlations in networks. This is
because quantum theory allows not only for entangled states, but also for
entangled measurements. Standard Bell tests exploit the entanglement of
states, but usually do not involve entangled measurements. In contrast, the
use of entangled measurements in networks is very natural. The combination of
entangled states and entangled measurements offers the possibility to create
strong correlations across the whole network.

An interesting, and well-known, example is that of entanglement swapping. Take
$\rho_{AB_{1}}$ and $\rho_{B_{2}C}$ to be two-qubit Bell states. Bob performs
a joint measurement on both of his quantum subsystems, the so-called Bell-
state measurement, projecting the two subsystems $B_{1}$ and $B_{2}$ in the
basis of four maximally entangled Bell states. This has the remarkable effect
of remotely preparing the systems of Alice and Charlie in an entangled state,
even though these systems where initially fully independent from each other.
By performing judicious local measurements, Alice and Charlie can then observe
a Bell inequality violation (conditioned on each given output of Bob’s
measurement). This is known as “event-ready nonlocality” Zukowski .

The question of characterizing this stronger form of quantum correlations
proper to networks is our starting point in this work. Clearly one needs to go
beyond the standard definition of bilocality (and its generalisation to
arbitrary networks), as the above simple examples demonstrate. In a sense the
question is to identify quantum correlations that truly take advantage of the
network structure, and therefore do not reduce to standard Bell inequality
tests. In the following, we take a first step in this direction and define a
notion of genuine network nonlocality for quantum systems, which we illustrate
with examples.

##  III Defining genuine network nonlocality

Figure 2: This figure explains the notion of quantum-wirable correlations.
Quantum resources (states and measurements) are shown in blue, forming the
quantum boxes, while classical information and processing is shown in black.
(a) A bipartite quantum box is a device featuring two terminals, that receive
inputs and produce outputs. The resulting correlations can be nonlocal, hence
the quantum box represents the basic resource for standard quantum Bell
nonlocality. (b) Quantum boxes distributed in a network: each party may have
access to one or more terminals, performing classical processing on their
inputs and outputs via a wiring. Correlations that can be achieved as such are
termed “quantum-wirable”. (c) An example of a quantum-wirable correlation in
the bilocality network: given the shared variables ($\lambda$ and/or $\mu$)
and their respective input, each party chooses a wiring of their terminals;
this includes classical pre- and post-processing of the inputs and outputs,
and the output of one terminal may be used as input to another. Notably, the
order in which the terminals are used may be different for each party and may
depend on the outputs of previous boxes.

To characterize genuine network nonlocality, we proceed by defining the
opposite notion, namely correlations that are not genuine to a network, in the
sense that they can be traced back to standard quantum Bell nonlocality. To do
so, we ask ourselves how to use the resources of standard quantum Bell
nonlocality in a network configuration. We take a rather resource-theoretic
approach to this question, and view standard quantum nonlocality as “quantum
boxes” that can be distributed by each source in the network to various
subsets of parties (see Fig. 2).

Let us first formalize the notion of a quantum box, which naturally follows
from the standard scenario of Bell nonlocality. Consider here two distant
parties (say Alice and Bob) sharing a quantum state $\sigma$. By performing
local quantum measurements on their subsystems (represented by sets of POVMs
$\\{N_{\alpha|X}\\}$ and $\\{N_{\beta|Y}\\}$, Alice and Bob can establish
correlations of the form:

$p(\alpha,\beta|X,Y)=\textrm{Tr}[(N_{\alpha|X}\otimes N_{\beta|Y})\sigma]$ (4)

where $X$ and $Y$ denote the inputs, and $\alpha$ and $\beta$ the outputs. We
can view this probability distribution as a quantum box; it represents a
(possibly nonlocal) box shared by Alice and Bob, which is realizable in
quantum theory. Each party has access locally to one end of the box, i.e. a
terminal, where they can give an input and then receive an output, see Fig.
2a. Note these quantum boxes can be viewed as resources, and recent works have
developed a resource-theoretic framework based on these ideas to characterize
and quantify Bell nonlocality Wolfe2020quantifyingbell ; see also BarrettPR ;
Allcock  for previous works along these lines.

Coming back to the network configuration, we can now consider that all sources
in the network distribute quantum boxes to the parties, as shown in Fig. 2b.
Each party therefore receives a number of terminals which can be combined. As
each terminal takes a classical input and delivers a classical output, the
most general operation can be represented as a “wiring” of the terminals. This
represents a classical measurement.

We now aim to characterize those correlations that can be established by using
as resources quantum boxes distributed across the network, which are then
locally combined via classical measurement, i.e. wirings. In the bilocality
network (our focus here), this amounts to assuming that the first source
distributes one (or more) quantum box(es) to Alice and Bob, and similarly for
the second source connecting Bob and Charlie. Hence, each party now receives a
number of terminals (each one featuring an input and an output), and can use
all of them in order to produce an output given the received input, as
depicted in Fig. 2c. Upon receiving her input $x$, Alice chooses a wiring for
processing her available terminals, in order to produce the final output $a$.
For instance, in the figure, Alice receives two terminals and starts by
inputting $x$ in the first one; then she uses the output of the first terminal
as an input for the second one; the final output $a$ is then be chosen as the
output of the second terminal.

Notably, these wiring operation can have non-trivial effects on nonlocality,
in particular when applied to boxes that are nonlocal, allowing for instance
to increase (or distill) nonlocality forster_2009 ; Brunner2009 . More
generally, wiring operations feature a rich structure, in particular when
three or more terminals are available; see e.g. ShortNJP ; Gallego2017 . For
instance, the output of the first terminal may then be used to decide on the
order in which the next two terminals will be used. Even more possibilities
arise, when considering a party that receives terminals from more than one
source. This is the case for the middle party, Bob, in the bilocality network.
At this point, it is also worth noting that it is in general not possible to
contract all quantum boxes shared by two parties into a single box. This is
because the order in which the boxes are used on one side might be different
than the order used on the other side.

We are now ready to state our main definition. We introduce the class of
“quantum wirable” correlations in a network as those that can be obtained when
all the sources distribute quantum boxes to the parties, and the parties
perform arbitrary local wiring operations. Formally, such distributions admit
a decomposition of the form (here stated for the bilocality network):

$\displaystyle p(a,b,c|x,y,z)$ $\displaystyle=$ (5)
$\displaystyle\sum_{\begin{subarray}{c}\lambda,\mu,\vec{k},\\\
\vec{l},\vec{l}^{\prime},\vec{m},\\\ \vec{X},\vec{Y},\vec{Z},\\\
\vec{\alpha},\vec{\beta},\vec{\gamma}\end{subarray}}\ $
$\displaystyle\overline{W}_{A}(a,\vec{k},\vec{X}|x,\lambda)\,\overline{W}_{B}(b,\vec{l},\vec{l}^{\prime},\vec{Y}|y,\lambda,\mu)$
$\displaystyle\
\times\overline{W}_{C}(c,\vec{m},\vec{Z}|z,\mu)\,p(\lambda)\,p(\mu)$
$\displaystyle\ \times
p_{Q}^{\vec{k},\vec{l}}(\vec{\alpha},\vec{\beta}|\vec{X},\vec{Y},\lambda)\,p_{Q}^{\vec{l}^{\prime},\vec{m}}(\vec{\beta}^{\prime},\vec{\gamma}|\vec{Y},\vec{Z},\mu)$

where

$\overline{W}_{A}(a,\vec{k},\vec{X}|x,\lambda)=\prod_{i=1}^{n+1}W_{A}^{i}(k_{i},X_{i}|[X]_{i-1},[\alpha]_{i-1})$
(6)

with the notation $[k]_{i}=\\{k_{0},k_{1},\ldots,k_{i}\\}$, $X_{0}=x$,
$\alpha_{0}=\lambda$, $k_{n+1}=0$ and $a=X_{n+1}$ describes the successive
wirings of Alice’s $n$ boxes (similarly for $\overline{W}_{B}$ and
$\overline{W}_{C}$), and
$p_{Q}^{\vec{k},\vec{l}}(\vec{\alpha},\vec{\beta}|\vec{X},\vec{Y},\lambda)$
(and similarly
$p_{Q}^{\vec{l}^{\prime},\vec{m}}(\vec{\beta}^{\prime},\vec{\gamma}|\vec{Y},\vec{Z},\mu)$)
are distributions of multiple quantum boxes with interactions ordered by
$\vec{k}$ and $\vec{l}$:

$p_{Q}^{\vec{k},\vec{l}}(\vec{\alpha},\vec{\beta}|\vec{X},\vec{Y},\lambda)=\prod_{i=1}^{n}p_{Q}(\alpha_{k^{-1}_{i}},\beta_{l^{-1}_{i}}|X_{k^{-1}_{i}},Y_{l^{-1}_{i}}).$
(7)

Here, $k$ and $k^{-1}$ (similarly for $l$, $l^{\prime}$ and $m$) satisfy
$k^{-1}_{i}\in\\{0,\ldots,n\\}$, $k_{k^{-1}_{i}}=i$, and
$l_{i}l_{i}^{\prime}=0$ for all $i=1,\ldots,n$.

As argued above, such distribution use as main resource the quantum boxes
(which are possibly nonlocal). The local wiring operations are basically
classical. One may thus argue that such distributions do not leverage the
structure of the network to extend the nonlocality, and thus that they can be
traced back to standard Bell nonlocality.

In our discussion above, we have already encountered an example of quantum
non-bilocal correlations that are however quantum-wirable. The Fritz example
is also clearly quantum-wirable; see Appendix A.

We have also already seen examples of quantum correlations that are provably
not quantum-wirable, i.e. do not admit a decomposition of the form (5). This
is the case for the entanglement swapping experiment discussed above. Indeed,
the observation of event-ready nonlocality rules out any quantum-wirable
model. This is because for any distribution of the form of (5), the
conditioned marginal distribution $p(ac|xyzb)$ is necessarily local, i.e.
cannot violate any Bell inequality. More generally, the use of a wiring at Bob
implies that, for every measurement $y$ and outcome $b$, the resulting quantum
state held by Alice and Charlie must be separable, even though all the quantum
boxes in the network may feature entangled states. Below we present a more
interesting example of genuine network quantum nonlocality, based on the
concept of self-testing.

##  IV Network self-testing

Self-testing refers to the characterization of quantum systems in a black-box
scenario MayersYao ; SupicBowles . In its standard form, self-testing involves
a quantum Bell experiment. The observation of specific quantum nonlocal
correlations (or maximal violation of some Bell inequality) can then be used
to precisely certify the underlying entangled state and local measurements, up
to local isometries.

Here we present an instance of self-testing in the bilocality network. The
interesting feature of this example is that the involved correlations are non-
bilocal, but are local following the usual definition of tripartite Bell
locality (see below). Hence, this represents an example of genuine network
quantum nonlocality which does not rely on event-ready nonlocality.

Consider both sources producing a Bell pair, i.e.
$\rho_{AB_{1}}={\lvert\phi_{+}\rangle\\!\langle\phi_{+}\rvert}$ and
$\rho_{B_{2}C}={\lvert\phi_{+}\rangle\\!\langle\phi_{+}\rvert}$. Alice and
Charlie perform the same local measurements, represented by the Pauli
observable
$A_{0}=C_{0}=\frac{\sigma_{\mathsf{x}}+\sigma_{\mathsf{z}}}{\sqrt{2}}$ and
$A_{1}=C_{1}=\frac{\sigma_{\mathsf{x}}-\sigma_{\mathsf{z}}}{\sqrt{2}}$. Bob
applies the Bell state measurement. With these resources the parties observe
correlations $p(abc|xz)$ with the following properties:

$\displaystyle\sum_{ac}p(abc|xz)=\textrm{Tr}\left[\left(\mathds{1}^{A}\otimes
M_{b}^{B_{1}B_{2}}\otimes\mathds{1}^{C}\right)\left(\rho_{AB_{1}}\otimes\rho_{B_{2}C}\right)\right]=\frac{1}{4},$
(8)

for all $b$, where $b$ can have four different values, and thus can be
represented as two bits $b=(b_{1},b_{2})$. Let us define the state of Alice
and Charlie remotely prepared by Bob’s measurement:

$\varrho_{b}=\frac{\textrm{Tr}_{B_{1}B_{2}}\left[\left(\mathds{1}^{A}\otimes
M_{b}^{B_{1}B_{2}}\otimes\mathds{1}^{C}\right)\left(\rho_{AB_{1}}\otimes\rho_{B_{2}C}\right)\right]}{\textrm{Tr}\left[\left(\mathds{1}^{A}\otimes
M_{b}^{B_{1}B_{2}}\otimes\mathds{1}^{C}\right)\left(\rho_{AB_{1}}\otimes\rho_{B_{2}C}\right)\right]}$
(9)

The statistics of Alice and Charlie, conditioned on Bob’s output, are given
by:

$\displaystyle\textrm{Tr}\left[(A_{0}\otimes\mathds{1})\varrho_{b}\right]$
$\displaystyle=\textrm{Tr}\left[(A_{1}\otimes\mathds{1})\varrho_{b}\right]=0,$
(10) $\displaystyle\textrm{Tr}\left[(\mathds{1}\otimes
C_{0})\varrho_{b}\right]$ $\displaystyle=\textrm{Tr}\left[(\mathds{1}\otimes
C_{1})\varrho_{b}\right]=0,$ (11) $\displaystyle\textrm{Tr}\left[(A_{0}\otimes
C_{0})\varrho_{b}\right]$ $\displaystyle=\textrm{Tr}\left[(A_{1}\otimes
C_{1})\varrho_{b}\right]=(-1)^{b_{1}}\delta_{b_{1}b_{2}},$ (12)
$\displaystyle\textrm{Tr}\left[(A_{0}\otimes C_{1})\varrho_{b}\right]$
$\displaystyle=\textrm{Tr}\left[(A_{1}\otimes
C_{0})\varrho_{b}\right]=(-1)^{b_{1}}\delta_{b_{1}1-b_{2}},$ (13)

where $\delta_{jk}$ is the Kronecker delta. We are now in position to state
our self-testing result:

######  Theorem 1.

The correlations in Eqs (8) and (10)-(13) imply that (i) the two local
measurements of both Alice and Charlie are anticommuting, (ii) the measurement
performed by Bob is the Bell-state measurement, and (iii), all states
$\varrho_{b}$ (as in Eq. (9)) are Bell states, i.e. Bob’s measurement remotely
steers maximally entangled state for Alice and Charlie.

We now sketch the proof; the full proof is given in Appendix B. First, from
Jordan’s lemma, we can bring Alice’s and Charlie’s measurement observables to
a block-diagonal form. A block diagonal unitary transformation can rotate
Alice’s observable to a form where $A_{0}$ has in every block Pauli’s
$\sigma_{\mathsf{z}}$, while all blocks of $A_{1}$ are rotated to the real
plane of the Bloch sphere. There exists a unitary making the equivalent
rotation of Charlie’s measurement observables. This allows to reduce the
problem to an ensemble of qubit states. The correlations given in eqs.
(10)-(13) imply that one of the two following statements must hold: either the
pairs of measurements of Alice and Bob commute, or they anticommute. The first
possibility is ruled out by using the fact that the measurement correlations
$\\{p(abc|xz)\\}$ are non-bilocal. Hence, the observables of Alice and Charlie
are unitarily equivalent to $\sigma_{\mathsf{z}}\otimes\mathds{1}$ and
$\sigma_{\mathsf{x}}\otimes\mathds{1}$. With this insight (10)-(13) imply that
for every output $b$ the state $\varrho_{b}$ corresponds to a Bell pair in
tensor product with some junk state (on which the measurements of Alice and
Charlie act trivially). By tracing out the uncorrelated degrees of freedom we
get a quantum channel mapping the steered states $\varrho_{b}$ to qubit Bell
pairs. Finally, this insight together with the argument presented in
renou2018self  concludes the self-testing of Bob’s measurement.

As mentioned above, an interesting feature of this self-testing result is that
it relies on the fact that the distribution is not bilocal. Indeed, one can
readily check that the quantum distribution is Bell local in the usual sense,
i.e. it admits a decomposition of the form (2). This in stark contrast to
previous self-testing results in networks renou2018self ; JD_BSM  (and
possibly with all known self-testing results) which rely on standard Bell
inequality violation.

Moreover, as the measurement performed by Bob must be the Bell-state
measurement, remotely preparing maximally entangled states $\varrho_{b}$ for
Alice and Charlie, it is clear that the quantum distribution is genuine
network nonlocal. But since the distribution is Bell local it does not feature
event-ready nonlocality.

##  V Discussion and outlook

We have defined a notion of genuine network quantum nonlocality. This
represents a stronger form of quantum nonlocality in networks (compared to the
previous notion of non-bilocality), which can only be obtained from the
judicious combination of entangled states and non-classical measurements.
Therefore, the nonlocality of such correlations cannot be traced back to
standard Bell inequality violations, and represents a phenomenon that is
proper to the network configuration. We presented an instance of this effect
based on self-testing in the bilocality network. Notably, this can be seen as
a novel form of self-testing, proper to networks, as it relies entirely on
non-bilocality, and does not involve the violation of a standard Bell
inequality.

Our approach (in particular the definition in Eq. (5)) can be readily
generalized to more complex networks. A case of particular interest is the
triangle network. We conjecture that the distribution recently proposed by
Renou et al. renou_2019  is genuine network nonlocal in the sense that we
define, contrary to the distribution of Fritz Fritz_2012  which is quantum-
wirable.

An interesting question is to identify the necessary quantum resources for
demonstrating genuine network nonlocality. From our examples, one may think
that the presence of an entangled measurement is necessary. However, this may
not be the case in general. Indeed, from the definition of quantum wirable
correlations in Eq. (5), we see that genuine network nonlocality requires the
use of a non-classical measurement, i.e. not describable as a wiring.
Interestingly there exist quantum measurement that feature only separable
eigenstates, but yet are non-classical Bennett ; see also VertesiMiguel . This
effect is known as “quanutm nonlocality without entanglement”, It would be
interesting to see if the use of such a measurement could lead to genuine
network nonlocality.

It is worth pointing out that our approach is completely different from the
one taken in a series of recent works MiguelElie ; Chao ; Bierhorst ; Coiteux
. The latter address a different question, namely they aim at characterizing
multipartite quantum states (such as GHZ states) and their correlations which
cannot be obtained from unbounded bipartite resources (quantum or even post-
quantum) distributed in a triangle network. In contrast, our work focuses on
quantum correlations that are possible in a given network structure, but which
cannot be decomposed using quantum boxes and local wirings.

Finally, note that one could consider a natural variation of our definition of
quantum-wirable correlations to “no-signaling-wirable” correlations. That is,
one would demand a decomposition as in Eq. (5), but where the shared nonlocal
boxes would no longer need to admit a quantum realization, but only satisfy
the no-signaling constraints (so-called non-signalling or Popescu-Rohrlich
boxes PR ; BarrettPR ). Note that any instance of event-ready quantum
nonlocality is then already an example of a quantum distribution that is not
no-signaling-wirable 111Similarly to the quantum case discussed in the main
text, the use of a wiring by Bob implies that the remotely prepared
correlations for Alice and Charlie must be local; see e.g. ShortNJP . It would
be interesting to see if this is also the case for our self-testing example.

_Note added._ In a recent pre-print Pozas-Kerstjens et al. pozas  propose a
notion of “full network nonlocality” which is different and complementary to
ours. Full network nonlocality, which certifies that all sources in the
network produce an entangled state, does not imply genuine network nonlocality
(as we define); indeed the presence of entanglement on all links can be
enforced via standard Bell tests (e.g. CHSH), which results in correlations
that are quantum-wirable.

###### Acknowledgements.

This work was supported by the Swiss National Science Foundation (project
2000021 192244/1 and NCCR SwissMap).

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##  Appendix A The Fritz distribution is quantum-wirable

In Fritz_2012 , Theorem 2.16, Fritz introduced a nonlocal distribution in the
triangle network based on CHSH violation. In the triangle network, as shown in
Fig. 3, two sources $S_{AB}$ and $S_{AC}$ send fully correlated classical
bits, while $S_{BC}$ sends the maximally entangled states. Alice outputs the
two bits that she receives, denoted as $(y,z)\equiv A$. Bob (Charlie) performs
the optimal CHSH measurements, treating $y$ from $S_{AB}$ ($z$ from $S_{AC}$)
as the input setting, and outputs two bits $(y,b)\equiv B$ ($(z,c)\equiv C$
respectively). The Bell nonlocality of $p(bc|yz)$ implies the network
nonlocality of $p(ABC)$.

However, we can clearly see that this distribution is quantum-wirable, since
all measurements are separable and the output are classical processing of
output of quantum boxes.

Figure 3: The underlying quantum network of Fritz’s distribution. The
distribution is network nonlocal but quantum-wirable.

##  Appendix B Proof of Theorem 1

Jordan’s lemma states that for any two dichotomic observables there exist a
unitary which transforms both of them to a block diagonal form in the same
basis, where the blocks are of size $2\times 2$ or $1\times 1$. In the same
way like in Jed , we embed every $1\times 1$ into a Hilbert space of larger
dimension. This operation does not affect the correlation probabilities, and
it simplifies our analysis, as we work with a Jordan decomposition in which
all blocks are of the size $2\times 2$. Henceforth, through a simple basis
change, we can bring Alice’s and Charlie’s measurement observables to a block-
diagonal form

$\displaystyle V_{A}A_{0}V_{A}^{\dagger}$
$\displaystyle=\bigoplus_{\alpha}\sigma_{\mathsf{z}}^{\alpha},$ (14)
$\displaystyle V_{A}A_{1}V_{A}^{\dagger}$
$\displaystyle=\bigoplus_{\alpha}\left(\cos\theta_{\alpha}\sigma_{\mathsf{z}}^{\alpha}+\sin\theta_{\alpha}\sigma_{\mathsf{x}}^{\alpha}\right),$
(15) $\displaystyle V_{C}C_{0}V_{C}^{\dagger}$
$\displaystyle=\bigoplus_{\gamma}\sigma_{\mathsf{z}}^{\gamma},$ (16)
$\displaystyle V_{C}C_{1}V_{C}^{\dagger}$
$\displaystyle=\bigoplus_{\gamma}\left(\cos\phi_{\gamma}\sigma_{\mathsf{z}}^{\gamma}+\sin\phi_{\gamma}\sigma_{\mathsf{x}}^{\gamma}\right),$
(17)

where $\theta_{\alpha},\phi_{\gamma}\in[0,2\pi)$ for all $\alpha$ and
$\gamma$. In turn, this implies that we can assume that

$V\varrho_{b_{1}b_{2}}V^{\dagger}=\bigoplus_{\alpha,\gamma}p_{b_{1}b_{2}}^{\alpha,\gamma}\varrho_{b_{1}b_{2}}^{\alpha,\gamma},$
(18)

where $V=V_{A}\otimes V_{C}$, without loss of generality.

The correlations between $A_{0}$ and $C_{0}$ imply

$\displaystyle\textrm{Tr}\left[\left(A_{0}\otimes
C_{0}\right)\varrho_{00}\right]$
$\displaystyle=\sum_{\alpha,\gamma}p_{00}^{\alpha,\gamma}\textrm{Tr}\left[\left(\sigma_{\mathsf{z}}^{\alpha}\otimes\sigma_{\mathsf{z}}^{\gamma}\right)\varrho_{00}^{\alpha,\gamma}\right]$
(19) $\displaystyle=1$ (20)

Given that $\sum_{\alpha,\gamma}p_{00}^{\alpha,\gamma}=1$ the last equation
implies
$\textrm{Tr}\left[\left(\sigma_{\mathsf{z}}^{\alpha}\otimes\sigma_{\mathsf{z}}^{\gamma}\right)\varrho_{0}^{\alpha,\gamma}\right]=1$
for all $\alpha$ and $\gamma$ such that $p_{00}\neq 0$ (components of
$\varrho_{00}$ with $p_{00}=0$ are not relevant because they do not contribute
to the observed statistics). This condition imposes the following structure of
blocks on the diagonal of $V\varrho_{00}V^{\dagger}$:

$\varrho_{00}^{\alpha,\gamma}=\begin{bmatrix}q_{00}^{\alpha,\gamma}&0&0&r_{00}^{\alpha,\gamma}\\\
0&0&0&0\\\ 0&0&0&0\\\
{r_{00}^{\alpha,\gamma}}^{*}&0&0&1-q_{00}^{\alpha,\gamma}\end{bmatrix}.$ (21)

Given this form of $\varrho_{00}$ and the fact that
$\textrm{Tr}[\sigma_{\mathsf{x}}^{\alpha}\otimes\sigma_{\mathsf{z}}^{\gamma}\rho_{00}^{\alpha,\gamma}]=\textrm{Tr}[\sigma_{\mathsf{z}}^{\alpha}\otimes\sigma_{\mathsf{x}}^{\gamma}\rho_{00}^{\alpha,\gamma}]=0$
for all $\alpha$ and $\gamma$ with $p_{00}\neq 0$, the correlations between
$A_{1}$ and $C_{1}$ imply

$\displaystyle\textrm{Tr}\left[\left(A_{1}\otimes
C_{1}\right)\varrho_{00}\right]=$
$\displaystyle=\sum_{\alpha,\gamma}p_{00}^{\alpha,\gamma}\textrm{Tr}\big{[}\big{(}\left(\cos\theta_{\alpha}\sigma_{\mathsf{z}}^{\alpha}+\sin\theta_{\alpha}\sigma_{\mathsf{x}}^{\alpha}\right)\otimes$
$\displaystyle\qquad\qquad\otimes\left(\cos\phi_{\gamma}\sigma_{\mathsf{z}}^{\gamma}+\sin\phi_{\gamma}\sigma_{\mathsf{x}}^{\gamma}\right)\big{)}\varrho_{00}^{\alpha,\gamma}\big{]}$
(22)
$\displaystyle=\sum_{\alpha,\gamma}p_{00}^{\alpha,\gamma}\left[\cos\theta_{\alpha}\cos\phi_{\gamma}+\sin\theta_{\alpha}\sin\phi_{\gamma}2\text{Re}(r_{00}^{\alpha,\gamma})\right]$
$\displaystyle=1.$ (23)

The positivity of the blocks imposes $|r_{00}^{\alpha,\gamma}|^{2}\leq
q_{00}^{\alpha,\gamma}(1-q_{00}^{\alpha,\gamma})$, which further implies
$|r_{00}^{\alpha,\gamma}|\leq 1/2$ and therefore
$2|\text{Re}(r_{00}^{\alpha,\gamma})|\leq 1$. Hence, the maximal value of the
expression
$\cos\theta_{\alpha}\cos\phi_{\gamma}+\sin\theta_{\alpha}\sin\phi_{\gamma}2\text{Re}(r_{00}^{\alpha,\gamma})$
for all $\alpha$ and $\gamma$ such that $p_{00}\neq 0$ is equal to $1$. The
last equation thus implies that for every $\alpha$ and $\gamma$ with
$p_{00}\neq 0$ this expression must take value $1$ since
$\sum_{\alpha,\gamma}p_{00}^{\alpha,\gamma}=1$. For every $\alpha$ and
$\gamma$ such that $p_{00}^{\alpha,\gamma}\neq 0$, the equation

$\cos\theta_{\alpha}\cos\phi_{\gamma}+\sin\theta_{\alpha}\sin\phi_{\gamma}2\text{Re}(r_{00}^{\alpha,\gamma})=1$
(24)

admits two groups of solutions :

  1. 1.

$2\text{Re}(r_{00}^{\alpha,\gamma})=1$ and
$\theta_{\alpha}=\phi_{\gamma}\neq\pi$ or
$2\text{Re}(r_{00}^{\alpha,\gamma})=-1$ and
$\theta_{\alpha}=2\pi-\phi_{\gamma}\neq\pi$;

  2. 2.

$\theta_{\alpha}=\phi_{\gamma}=\pi$ and unrestricted value of
$r_{00}^{\alpha,\gamma}$.

Observing the correlations between $A_{i}$ and $C_{j}$ for Bob’s outcome
$b={11}$ we come to similar conclusions, i.e, two groups of solutions exist
which satisfy all reference correlations for all $\alpha$ and $\gamma$ such
that $p_{11}^{\alpha,\gamma}\neq 0$:

  1. 1.

$2\text{Re}(r_{11}^{\alpha,\gamma})=-1$ and $\theta_{\alpha}=\phi_{\gamma}\neq
l\pi$ or $2\text{Re}(r_{11}^{\alpha,\gamma})=1$ and
$\theta_{\alpha}=2\pi-\phi_{\gamma}\neq\pi$ ;

  2. 2.

$\theta_{\alpha}=\phi_{\gamma}=\pi$ and unrestricted value of
$r_{11}^{\alpha,\gamma}$.

Observe now that the unitary $U=\oplus_{\gamma}U_{\gamma}$, where $U_{\gamma}$
is

$U_{\gamma}=\begin{bmatrix}\cos(\frac{\phi_{\gamma}}{2})&\sin(\frac{\phi_{\gamma}}{2})\\\
-\sin(\frac{\phi_{\gamma}}{2})&\cos(\frac{\phi_{\gamma}}{2})\end{bmatrix},$
(25)

induces the following map
$\tilde{C}_{0}=UV_{C}C_{1}V_{C}^{\dagger}U^{\dagger}=\bigoplus_{\gamma}\sigma_{\mathsf{z}}^{\gamma}$
and
$\tilde{C}_{1}=UV_{C}C_{0}V_{C}^{\dagger}U^{\dagger}=\bigoplus_{\gamma}(\cos(\phi_{\gamma})\sigma_{\mathsf{z}}^{\gamma}-\sin(\phi_{\gamma})\sigma_{\mathsf{x}}^{\gamma})$.
With this local change of Charlie’s basis the correlations for Bob’s outcome
$b={01}$ are

$\displaystyle\textrm{Tr}[A_{0}\otimes\tilde{C}_{0}\tilde{\varrho}_{01}]$
$\displaystyle=1$
$\displaystyle\textrm{Tr}[A_{1}\otimes\tilde{C}_{1}\tilde{\varrho}_{01}]$
$\displaystyle=1$
$\displaystyle\textrm{Tr}[A_{0}\otimes\tilde{C}_{1}\tilde{\varrho}_{01}]$
$\displaystyle=0$
$\displaystyle\textrm{Tr}[A_{1}\otimes\tilde{C}_{0}\tilde{\varrho}_{01}]$
$\displaystyle=0$ (26)

where $\tilde{\varrho}_{01}=(\mathds{1}\otimes
U)V\varrho_{01}V^{\dagger}(\mathds{1}\otimes U^{\dagger})$. The first equation
(26) implies

$\tilde{\varrho}_{01}^{\alpha,\gamma}=\begin{bmatrix}{\tilde{q}}_{01}^{\alpha,\gamma}&0&0&{\tilde{r}}_{01}^{\alpha,\gamma}\\\
0&0&0&0\\\ 0&0&0&0\\\
{\tilde{r}}_{01}^{{\alpha,\gamma}^{*}}&0&0&1-{\tilde{q}}_{01}^{\alpha,\gamma}\end{bmatrix},$
(27)

where $\tilde{\varrho}_{01}^{\alpha,\gamma}=(\mathds{1}\otimes
U)V\varrho_{01}^{\alpha,\gamma}V^{\dagger}(\mathds{1}\otimes U^{\dagger})$.
Writing explicitly all the correlations, analogously to the previous cases, we
again obtain two groups of solutions for all $\alpha$ and $\gamma$ such that
$p_{01}^{\alpha,\gamma}\neq 0$:

  1. 1.

$2\text{Re}({\tilde{r}}_{01}^{\alpha,\gamma})=1$ and
$\theta_{\alpha}=2\pi-\phi_{\gamma}\neq\pi$ or
$2\text{Re}({\tilde{r}}_{01}^{\alpha,\gamma})=-1$ and
$\theta_{\alpha}=\phi_{\gamma}\neq\pi$;

  2. 2.

$\theta_{\alpha}=\phi_{\gamma}=\pi$ and unrestricted value of
${\tilde{r}}_{01}^{\alpha,\gamma}$.

Equivalently, for Bob’s output $b={10}$ we define
$\tilde{\varrho}_{10}=(\mathds{1}\otimes
U)V\varrho_{10}V^{\dagger}(\mathds{1}\otimes U^{\dagger})$ and obtain two
groups of solutions for all $\alpha$ and $\gamma$ such that
$p_{10}^{\alpha,\gamma}\neq 0$:

  1. 1.

$2\text{Re}({\tilde{r}}_{10}^{\alpha,\gamma})=1$ and
$\theta_{\alpha}=\phi_{\gamma}\neq\pi$ or
$2\text{Re}({\tilde{r}}_{10}^{\alpha,\gamma})=-1$ and
$\theta_{\alpha}=2\pi-\phi_{\gamma}\neq\pi$;

  2. 2.

$\theta_{\alpha}=\phi_{\gamma}=\pi$ and unrestricted value of
${\tilde{r}}_{10}^{\alpha,\gamma}$.

Let us assume the second case solution holds for all outputs $b$, _i.e._ for
all $\alpha$, $\gamma$ and $b$ such that $p_{b}^{\alpha,\gamma}\neq 0$. This
implies $\cos\theta_{\alpha}=\cos\phi_{\gamma}=\pm 1$ for all $\alpha$ and
$\gamma$ such that $p_{b}^{\alpha,\gamma}\neq 0$ for either of
$b=00,11,01,10$. In this case measurement observables $A_{0}$ and $A_{1}$ are
commuting on the support of all states $\varrho_{b}$, as well as $C_{0}$ and
$C_{1}$.

Suppose Alice and Charlie both perform commuting measurements. The probability
distribution $p(a,b,c|x,z)$ is then such that one can write a parent
distribution $p(a_{0},a_{1},b,c_{0},c_{1})$ without inputs, where $a_{x}$
denotes the output $a$ for the setting $x$, and similarly for $c_{z}$
fine1982hidden . The distribution $p(a,b,c|x,z)$ can then be obtained as a
marginal of the parent distribution, e.g. by selecting the desired outcome
$a_{x}$ as a function of the input $x$.

It has been proven by Branciard _et al._ branciard2012bilocal , that one
cannot generate a non-bilocal distribution in this scenario if Bob has no
inputs. We present the argument here. Take an arbitrary distribution
$p(a,b,c)$. Due to the bilocality assumption, it can be decomposed as
$p(a,b,c)=p(a)p(c)p(b|a,c)$, where we used that $p(c|a)=p(c)$. Now consider
that the source between Alice and Bob samples $\lambda$ from Alice’s marginal
distribution $p(a)$ and Alice outputs $a=\lambda$. Similarly, the source
between Bob and Charlie samples $\mu$ from Charlie’s marginal distribution
$p(c)$ and Charlie outputs $c=\mu$. Then upon receiving $\lambda$ and $\mu$,
Bob can produce an output according to $p(b|a=\lambda,c=\mu)$, hence
reproducing the arbitrary distribution $p(a,b,c)$. Therefore, any local
distribution admits a bilocal model in absence of inputs for Bob.

Thus we conclude that if Alice and Charlie’s measurements commute, then the
resulting statistics admits a bilocal model, and can therefore not violate any
bilocality inequality. Since the correlations we consider maximally violate
the bilocality inequality from Branciard_2010  the possibility that both Alice
and Charlie perform compatible measurements must be discarded.

Therefore, there must be $\alpha$ and $\gamma$ such that
$r_{b}^{\alpha,\gamma}=\pm 1$ (or $\tilde{r}_{b}^{\alpha,\gamma}=\pm 1$). In
that case the block $\varrho_{b}^{\alpha,\gamma}$ is maximally entangled, and
thus the full state $\varrho_{b}$ is entangled.

We denote the set of values of $\alpha$ such that there is $\gamma$ such that
$\cos(\theta_{\alpha})\cos(\phi_{\gamma})=1$ with $\mathscr{A}^{\prime}$.
Similarly define the set of values of $\gamma$ such that there is $\alpha$
such that $\cos(\theta_{\alpha})\cos(\phi_{\gamma})=1$ with
$\mathscr{G}^{\prime}$. The sets of all other values of $\alpha$ and $\gamma$
is denoted with $\mathscr{A}^{\prime\prime}$ and $\mathscr{G}^{\prime\prime}$,
respectively. Without loss of generality assume there is
$p_{00}^{\alpha,\gamma}\neq 0$ for $\alpha\in\mathscr{A}^{\prime\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$ such that
$\cos(\theta_{\alpha})\cos(\phi_{\gamma})\neq 1$. There are then two
possibilities, namely $\theta_{\alpha}=\phi_{\gamma}\neq\pi$ implying
$2r_{00}^{\alpha,\gamma}=1$ and $\theta_{\alpha}=2\pi-\phi_{\gamma}\neq\pi$
implying $2r_{00}^{\alpha,\gamma}=-1$. For the time being, we chose the first
one, and remember that it might as well have been the other one, implying that
$\varrho_{00}^{\alpha\gamma}=\Phi_{-}$. We deal with this other possibility
later.

We thus have $2\text{Re}(r_{00}^{\alpha,\gamma})=1$, which implies also
$q_{00}^{\alpha,\gamma}=1/2$. Hence,

$\varrho_{00}^{\alpha\gamma}=\Phi_{+},$ (28)

for $\alpha\in\mathscr{A}^{\prime\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$. At this stage, we also notice that eq.
(23) can only hold if $\alpha\in\mathscr{A}^{\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$ or $\alpha\in\mathscr{A}^{\prime\prime}$
and $\gamma\in\mathscr{G}^{\prime}$ implies $p_{00}^{\alpha,\gamma}=0$.
Furthermore when $\alpha\in\mathscr{A}^{\prime}$ and
$\gamma\in\mathscr{G}^{\prime}$, $\theta_{\alpha}\neq\phi_{\gamma}$ only if
$p_{00}^{\alpha,\gamma}=0$.

Similarly we see that the correlations between $A_{0}$ and $C_{0}$, and
$A_{1}$ and $C_{1}$ allow to conclude that the state $\varrho_{11}$ has the
following form $\varrho_{11}=\oplus
p_{11}^{\alpha\gamma}\varrho_{11}^{\alpha\gamma}$ where for
$\alpha\in\mathscr{A}^{\prime\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$

$\varrho_{11}^{\alpha\gamma}=\Psi_{-}.$ (29)

For all the cases when $\theta_{\alpha}=2\pi-\phi_{\gamma}$ we have
$\varrho_{11}^{\alpha\gamma}=\Psi_{+}$. The analysis of the correlations of
the states $\varrho_{01}$ and $\varrho_{10}$ can be done in an equivalent way.
For $\alpha\in\mathscr{A}^{\prime\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$ we have

$\tilde{\varrho}_{01}^{\alpha,\gamma}=\Phi_{-}.$ (30)

and if $\theta_{\alpha}=2\pi-\phi_{\gamma}$ then
$\tilde{\varrho}_{01}^{\alpha,\gamma}=\Phi_{+}$. For
$\alpha\in\mathscr{A}^{\prime}$ and $\gamma\in\mathscr{G}^{\prime\prime}$ or
$\alpha\in\mathscr{A}^{\prime}$ and $\gamma\in\mathscr{G}^{\prime\prime}$ it
holds $p_{01}^{\alpha,\gamma}=0$. By repeating the same procedure for the
correlations on $\varrho_{10}$ we obtain that for
$\alpha\in\mathscr{A}^{\prime\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$ we have

$\tilde{\varrho}_{10}^{\alpha,\gamma}=\Psi_{+}.$ (31)

or if $\theta_{\alpha}=2\pi-\phi_{\gamma}$ then
$\tilde{\varrho}_{10}^{\alpha,\gamma}=\Psi_{-}$, while for
$\alpha\in\mathscr{A}^{\prime}$ and $\gamma\in\mathscr{G}^{\prime\prime}$ or
$\alpha\in\mathscr{A}^{\prime}$ and $\gamma\in\mathscr{G}^{\prime\prime}$ it
must be $p_{10}^{\alpha,\gamma}=0$. Let us now observe that

$\textrm{Tr}_{\text{B}_{1}\text{B}_{2}}[\varrho^{\text{A}\text{B}_{1}}\otimes\varrho^{\text{B}_{2}\text{C}}]=\sum_{b}p(b)\varrho_{b}.$
(32)

On the left hands side we have the tensor product state
$\varrho^{\text{A}}\otimes\varrho^{\text{C}}$, where
$\varrho_{\text{A}}=\textrm{Tr}_{\text{B}_{1}}\left[\varrho^{\text{A}\text{B}_{1}}\right]$
and
$\varrho_{\text{C}}=\textrm{Tr}_{\text{B}_{2}}\left[\varrho^{\text{B}_{2}\text{C}}\right]$.
On the right hand side we have the state of the form

$\bigoplus_{\alpha\in\mathscr{A^{\prime}},\gamma\in\mathscr{G}^{\prime}}p_{\alpha,\gamma}^{\prime}\varrho_{\alpha,\gamma}^{\prime}\bigoplus_{\alpha\in\mathscr{A^{\prime\prime}},\gamma^{\prime}\in\mathscr{G}^{\prime\prime}}p_{\alpha,\gamma}^{\prime\prime}\varrho_{\alpha,\gamma}^{\prime\prime}$
(33)

which has a tensor product structure only if $p^{\prime}_{\alpha,\gamma}=0$
for all $\alpha\in\mathscr{A}^{\prime}$ and $\gamma\in\mathscr{G}^{\prime}$ or
$p^{\prime\prime}_{\alpha,\gamma}=0$ for all
$\alpha\in\mathscr{A}^{\prime\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$. Given that the second case implies that
Alice’s and Charlie’s measurements are compatible on the support of
$\varrho_{\text{A}}$ and $\varrho_{\text{C}}$, respectively, for the reasons
already explained this cannot be the case. A similar argument impedes having
$\theta_{\alpha}\neq\phi_{\gamma}$ when $\alpha\in\mathscr{A}^{\prime\prime}$
and $\gamma\in\mathscr{G}^{\prime\prime}$, implying that for all
$\alpha\in\mathscr{A}^{\prime\prime}$ and
$\gamma\in\mathscr{G}^{\prime\prime}$ it must hold
$\theta_{\alpha}=\phi_{\gamma}\equiv\theta$. This excludes the possibility
$\theta_{\alpha}=2\pi-\phi_{\gamma}$, so we have to abandone this possibility.
Hence, we conclude that Alice’s and Charlie’s measurements on the support of
$\varrho_{\text{A}}$ and $\varrho_{\text{C}}$ respectively have the following
form:

$\displaystyle V_{A}A_{1}V_{A}^{\dagger}$
$\displaystyle=\bigoplus_{\alpha}(\cos(\theta)\sigma_{\mathsf{z}}^{\alpha}+\sin(\theta)\sigma_{\mathsf{x}}^{\alpha})$
(34) $\displaystyle V_{C}C_{1}V_{C}^{\dagger}$
$\displaystyle=\bigoplus_{\gamma}(\cos(\theta)\sigma_{\mathsf{z}}^{\gamma}+\sin(\theta)\sigma_{\mathsf{x}}^{\gamma})$
(35)

Now we can return to the measurement statistics $\textrm{Tr}\left[A_{0}\otimes
C_{1}\varrho_{0}\right]=0$. This implies

$0=\cos\theta\sum_{\alpha,\gamma}p^{\prime\prime}_{\alpha,\gamma}=\cos\theta,$
(36)

implying

$\displaystyle V_{A}A_{1}V_{A}^{\dagger}$
$\displaystyle=\bigoplus_{\alpha}\sigma_{\mathsf{x}}^{\alpha}$ (37)
$\displaystyle=\sigma_{\mathsf{x}}\otimes\mathds{1}$ $\displaystyle
V_{C}C_{1}V_{C}^{\dagger}$
$\displaystyle=\bigoplus_{\gamma}\sigma_{\mathsf{x}}^{\gamma}$ (38)
$\displaystyle=\sigma_{\mathsf{x}}\otimes\mathds{1}$

This implies that on the support of $\varrho_{\text{A}}$ and
$\varrho_{\text{C}}$ Alice’s and Charlie’s measurements, respectively,
anticommute. The corresponding Hilbert spaces $\mathscr{H}^{A}$ and
$\mathscr{H}^{C}$ decompose:
$\mathscr{H}^{A}=\mathscr{H}^{A^{\prime}}\otimes\mathscr{H}^{A^{\prime\prime}}$
and
$\mathscr{H}^{C}=\mathscr{H}^{C^{\prime}}\otimes\mathscr{H}^{C^{\prime\prime}}$,
where $\mathscr{H}^{A^{\prime}}$ and $\mathscr{H}^{C^{\prime}}$ are qubit
Hilbert spaces. Finally we notice that since $\phi_{\gamma}=\pi$ for all
$\gamma$ the unitary $U$ simply decomposes as $U=\tilde{U}\otimes\mathds{1}$,
where

$\tilde{U}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ -1&1\end{bmatrix}.$ (39)

Furthermore the states have the following form:

$\displaystyle V\varrho_{00}V^{\dagger}$
$\displaystyle=\bigoplus_{\alpha,\gamma}\Phi_{+}^{\alpha,\gamma}=\Phi_{+}\otimes\sigma_{00},$
(40) $\displaystyle V\varrho_{11}V^{\dagger}$
$\displaystyle=\bigoplus_{\alpha,\gamma}\Psi_{-}^{\alpha,\gamma}=\Psi_{-}\otimes\sigma_{11},$
(41) $\displaystyle V{\varrho}_{01}V^{\dagger}$
$\displaystyle=(\mathds{1}\otimes
U^{\dagger})\bigoplus_{\alpha,\gamma}\Phi_{-}^{\alpha,\gamma}(\mathds{1}\otimes
U)$ (42) $\displaystyle=(\mathds{1}\otimes
U^{\dagger})\Phi_{-}\otimes\sigma_{01}(\mathds{1}\otimes U),$ (43)
$\displaystyle V{\varrho}_{10}V^{\dagger}$ $\displaystyle=(\mathds{1}\otimes
U^{\dagger})\bigoplus_{\alpha,\gamma}\Psi_{+}^{\alpha,\gamma}(\mathds{1}\otimes
U)$ (44) $\displaystyle=(\mathds{1}\otimes
U^{\dagger})\Psi_{+}\otimes\sigma_{10}(\mathds{1}\otimes U),$ (45)

where $\sigma_{jk}$ are junk states of additional degrees of freedom. Let us
now define the CPTP map
$\Lambda_{A}:\mathscr{H}^{A}\longrightarrow\mathscr{H}^{A^{\prime}}$ such that
$\Lambda_{A}(\rho)=\textrm{Tr}_{A^{\prime\prime}}\left[V_{A}\rho
V_{A}^{\dagger}\right]$. Analogously we can define
$\Lambda_{A}:\mathscr{H}^{C}\longrightarrow\mathscr{H}^{C^{\prime}}$ such that
$\Lambda_{C}(\rho)=\textrm{Tr}_{C^{\prime\prime}}\left[V_{C}\rho
V_{C}^{\dagger}\right]$. With these CPTP maps we have:

$\displaystyle\Lambda_{A}\otimes\Lambda_{C}(\varrho_{00})$
$\displaystyle=\Phi_{+},$ (46)
$\displaystyle\Lambda_{A}\otimes\Lambda_{C}(\varrho_{11})$
$\displaystyle=\Psi_{-},$ (47)
$\displaystyle\Lambda_{A}\otimes\Lambda_{C}(\varrho_{01})$
$\displaystyle=(\mathds{1}\otimes\tilde{U}^{\dagger})\Phi_{-}(\mathds{1}\otimes\tilde{U}),$
(48) $\displaystyle\Lambda_{A}\otimes\Lambda_{C}(\varrho_{10})$
$\displaystyle=(\mathds{1}\otimes\tilde{U}^{\dagger})\Psi_{+}(\mathds{1}\otimes\tilde{U}),$
(49)

These equations can be phrased as Step 1 in the proof of self-testing theorem
of the Bell state measurement in renou2018self . Since Step 2 which completes
the proof depends completely on Step 1, we can use exactly the same methods to
show that Bob’s measurement can be self-tested to be exactly the Bell state
measurement.