# Single-copy activation of Bell nonlocality via broadcasting of quantum
states

Joseph Bowles  ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute
of Science and Technology, 08860 Castelldefels (Barcelona), Spain  Flavien
Hirsch  Institute for Quantum Optics and Quantum Information (IQOQI), Austrian
Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria  Daniel Cavalcanti
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels (Barcelona), Spain

###### Abstract

Activation of Bell nonlocality refers to the phenomenon that some entangled
mixed states that admit a local hidden variable model in the standard Bell
scenario nevertheless reveal their nonlocal nature in more exotic measurement
scenarios. We present such a scenario that involves broadcasting the local
subsystems of a single-copy of a bipartite quantum state to multiple parties,
and use the scenario to study the nonlocal properties of the two-qubit
isotropic state:

$\displaystyle\rho_{\alpha}=\alpha\,|\Phi^{+}\rangle\langle\Phi^{+}|+(1-\alpha)\frac{\mathbb{1}}{4}.$

We present two main results, considering that Nature allows for (i) the most
general no-signalling correlations, and (ii) the most general quantum
correlations at the level of any hidden variable theory. We show that the
state does not admit a local hidden variable description for $\alpha>0.559$
and $\alpha>\frac{1}{2}$, in cases (i) and (ii) respectively, which in both
cases provides a device-independent certification of the entanglement of the
state. These bounds are significantly lower than the previously best-known
bound of $0.697$ for both Bell nonlocality and device-independent entanglement
certification using a single copy of the state. Our results show that strong
examples of non-classicality are possible with a small number of resources.

###### Contents

  1. 1 Overview
  2. 2 Preliminaries
    1. 2.1 Bell nonlocality
      1. 2.1.1 Bell’s theorem
      2. 2.1.2 The CHSH inequality
      3. 2.1.3 Bell-local quantum states
      4. 2.1.4 Activation of Bell nonlocality
    2. 2.2 Device-independent entanglement certification
  3. 3 The broadcasting scenario
  4. 4 Activation of two-qubit states
    1. 4.1 Simple inequality for three parties
      1. 4.1.1 Quantum violation
    2. 4.2 Symmetric four-partite inequality
      1. 4.2.1 Quantum violation
  5. 5 Device-independent entanglement certification in the broadcasting scenario
    1. 5.1 Entanglement certification of isotropic states
  6. 6 Numerical methods
    1. 6.1 Linear programming methods
    2. 6.2 Semi-definite programming hierarchy for entanglement certification
  7. 7 Discussion
  8. 8 Conclusion
  9. 9 Acknowledgements
  10. A Linear optimisation of visibilities
  11. B Heuristic optimisation of visibility
    1. B.1 Seesaw procedure for maximising inequality violation
    2. B.2 Global heuristic procedure

##  1 Overview

The discovery by John Bell that quantum theory cannot be reproduced by local
hidden variables (LHVs) is one of the most fascinating outcomes of modern
physics and is made possible by the strange nature of quantum correlations.
More precisely, measurement outcomes arising from local measurements made on
entangled quantum systems exhibit correlations which provably forbid any
explanation in terms of LHVs [1, 2]. This phenomenon—termed _Bell nonlocality_
—can be experimentally witnessed via the violation of Bell inequalities:
functions of the measurement statistics that are bounded for LHV theories.
Since Bell’s original theorem, many experiments have observed Bell inequality
violations, proving that Nature—quantum or otherwise—evades any LHV
description [2, 3, 4, 5, 6, 7].

In order to observe Bell nonlocality one performs a Bell test. Here, a source
produces a bipartite quantum state that is sent to two separated devices, and
a number of possible measurements are performed locally on the state. To show
Bell nonlocality one violates a Bell inequality, which proves that the
resulting statistics cannot be reproduced by a source of classical randomness,
i.e. by LHVs. This scenario, that we call the _standard scenario_ , is shown
in figure 1, left. In the standard scenario, all pure entangled states violate
a Bell inequality [8], however for mixed states the situation is more subtle.
Notably, there exist a large class of mixed entangled states—called Bell-local
states—that do not violate any Bell inequality, due to the explicit
construction of LHV models that reproduce the outcome statistics of all
possible local measurements made on them [9, 10, 11, 12, 13, 14, 15].

For some time it was thus presumed that Bell-local states could not exhibit
Bell nonlocal behaviour. This intuition has since been proven wrong due to the
phenomenon that is generally known as the _activation of Bell nonlocality_
[16, 17, 18, 19, 20, 21, 22, 23, 24]. Here, one subjects the quantum state to
a more complex measurement scenario, which is able to prove a lack of LHV
description for some (although still not all) states that are local in the
standard scenario. Such scenarios typically come in two flavours (see figure
1):

  1. 1.

single-copy sequential scenarios [24, 17, 21, 23], where one makes a sequence
of time-ordered local measurements on a single copy of the state;

  2. 2.

multiple-copy scenarios [16, 18, 19, 20, 22], where one prepares multiple
copies of the state in some network structure and makes joint measurements on
the local subsystems.

Both scenarios are known to exhibit Bell nonlocality activation, although for
generally different classes of states. One state that is commonly studied is
the two-qubit isotropic state:

$\displaystyle\rho_{\alpha}=\alpha\,|\Phi^{+}\rangle\langle\Phi^{+}|+(1-\alpha)\frac{\mathbb{1}}{4},$
(1)

which can be seen as the result of passing one half of the two-qubit maximally
entangled state
$\left|\Phi^{+}\right\rangle=\frac{1}{\sqrt{2}}[\left|00\right\rangle+\left|11\right\rangle]$
through a depolarising channel with depolarising probability $\alpha$. The
state is entangled for $\alpha>1/3$, and has an LHV model111This LHV model
holds only for projective measurements, and so these examples of activation
are relative to projective measurements in the standard scenario. It is
generally much harder to construct LHV models for POVM measurements, although
the state is known to have an LHV model for POVM measurements for $\alpha\leq
0.455$ [25]. No Bell inequality violation using POVMs is known below
$\alpha=1/K_{3}$ however, and it is generally believed that the true value is
significantly higher—if not equal to $1/K_{3}$. in the standard scenario if
and only if $\alpha\leq 1/K_{3}$, where $K_{3}$ is Grothendieck’s constant of
order 3 [26, 27] and the best known bounds are $0.683<1/K_{3}<0.697$ [25, 28];
see figure 2, right. Several works have shown Bell nonlocality activation of
this state. In the multiple-copy scenario, it was first shown that activation
is possible for $\alpha>0.64$ [19, 20] by placing copies of the state in a
star network topology, and later in the entire range of entanglement [18]
(extending [16]) by processing many copies in a parallel scheme. In both cases
one requires a large number of copies to reach a value of $\alpha$ smaller
than $0.683$ (for example at least 21 states in [19]). In the single-copy
sequential scenario, no example of activation is known for the two-qubit
isotropic state.

Figure 1: Scenarios for showing nonlocality of quantum states. Arrows in and
out of boxes denote measurement inputs and outputs. Left: the standard Bell
scenario. centre: The single-copy sequential scenario used to show ‘hidden
nonlocality’. Here, at least one of the local subsystems undergoes a time-
ordered sequence of measurements. Right: the multiple-copy scenario. Many
copies of the state are prepared and joint measurements are made between the
local subsystems.

In this work, we introduce a new single-copy scenario—that we call the
_broadcasting_ scenario—that leads to a form of Bell nonlocality activation.
The scenario is inspired from the work [29], where it was shown that such
scenarios imply Bell inequalities that can outperform the CHSH Bell inequality
for some classes of states. The causal structure of this scenario is shown in
figure 2, left. The main idea is to map the bipartite state in question to a
multipartite state via the application of local transformations that broadcast
the local systems to a number of additional parties. The correlations arising
from measurements made on the multipartite state then rule out the possibility
of an LHV description for the original bipartite state. Using this scenario,
we show that one can show a lack of LHV description of the state (1) for
values $\alpha>0.559$. For $\alpha>\frac{1}{\sqrt{3}}\approx 0.577$, we are
able to prove this via the construction of a simple Bell inequality tailored
to the broadcasting scenario. We also study the class of two qubit states

$\displaystyle\rho_{\alpha,\theta}=\alpha|\psi_{\theta}\rangle\langle\psi_{\theta}|+(1-\alpha)\rho_{A}\otimes\mathbb{1}/2,$
(2)

where
$\left|\psi_{\theta}\right\rangle=\frac{1}{\sqrt{2}}[\cos\theta\left|00\right\rangle+\sin\theta\left|11\right\rangle]$
and
$\rho_{A}=\operatorname{Tr}_{B}[|\psi_{\theta}\rangle\langle\psi_{\theta}|]$.
By numerically optimising the violation of our inequality, we obtain values
$\alpha,\theta$ for which $\rho_{\alpha,\theta}$ is Bell nonlocal in the
broadcast scenario; see figure 4.

Our results hold under the assumption that any hidden variables used to
describe the full experiment obey the no-signalling principle, which is a
common assumption in definitions of multipartite nonlocality [30]. This is
required because one needs to consider the possibility that the transformation
device broadcasts additional hidden variables, which may not be LHVs, but may
exhibit nonlocal correlations (see also figure 3). Phrased more precisely, our
examples of Bell nonlocality prove that in any no-signalling hidden variable
theory reproducing quantum theory, those variables that describe the bipartite
state in question cannot be LHVs. This is an interesting subtlety that does
not arise in the other scenarios of activation that we elaborate on in section
3.

Figure 2: Left: the broadcasting scenario. One (or more) of the local systems
is broadcast via the application of a quantum channel, resulting in a
multipartite state. Local measurements are then performed on this state, and
the resulting statistics are used to rule out a local hidden variable
description for the original bipartite state. Right: Bounds for for the two-
qubit isotropic state (1). In this work we give bounds for which the state
does not have a local hidden variable description in any no-signalling hidden
variable theory (broadcast-nonlocal) and for which the entanglement of the
state can be certified device-independently (D.I. entanglement cert.).

Bell nonlocality is also closely linked to protocols of entanglement
certification. In particular, the observation of a Bell inequality violation
immediately implies that the underlying quantum state is entangled.
Furthermore, since all that is required to observe a Bell inequality violation
is the output statistics of the experiment, this implies that entanglement can
be certified even while treating the local measurement devices as black boxes;
a desirable property from both a practical and cryptographic perspective. Such
protocols are consequently called _device-independent_ (DI), and a proof of
Bell nonlocality is a DI certification of the entanglement of the state. This
property carries over to the broadcasting scenario as well, and our results
can equivalently be viewed as new protocols for DI entanglement certification.
Moreover, by assuming that the devices are limited to preparing quantum
resources—a standard assumption in DI entanglement certification—we are able
to device-independently certify the entanglement of the state (1) for all
$\alpha>\frac{1}{2}$. This is significantly lower than the best known bounds
of $1/\sqrt{2}$ (using CHSH) or $0.697$ [28] (using 90 local measurements for
each party) in the single copy regime, and we thus expect our results to be
generally useful for the design of DI protocols that can tolerate high levels
of noise in the state.

##  2 Preliminaries

We give here a brief recap of Bell’s theorem, Bell-nonlocal and Bell-local
entangled states, Bell nonlocality activation scenarios and DI certification.
Readers that are familiar with these concepts may want to skip to section 3.

###  2.1 Bell nonlocality

Consider two distant observers, called Alice and Bob, sharing a physical
system described by variable $\lambda$ (which can be any mathematical object).
Alice can interact with her part of the system by performing different
measurements, which we label by an integer: $x=0,1,...$. We also label the
possible outcomes with an integer $a=0,1,...$ 222We consider a scenario with
finitely many measurements and results for convenience, note however that the
analysis applies to continuous scenarios too.. Similarly, Bob can perform
measurements labelled by $y=0,1,...$ and get outcomes $b=0,1,...$. Upon
repeating the experiment many times, Alice and Bob gain access to the joint
statistics $p(ab|xy)$, that is, the probability of obtaining _outcomes_ $a$
and $b$, given _inputs_ $x$ and $y$. We call the joint statistics $p(ab|xy)$ a
_behaviour_.

We say that a behaviour $p(ab|xy)$ is _Bell-local_ if it admits the following
decomposition

$\displaystyle
p(ab|xy)=\int\Pi(\lambda)\;p_{A}(a|x,\lambda)\;p_{B}(b|y,\lambda)\;d\lambda,$
(3)

where $\Pi(\lambda)$ is a probability density function, and
$p_{A}(a|x,\lambda)$, $p_{B}(b|y,\lambda)$ are probability distributions (for
all $x,y$ and $\lambda$). That is, the statistics can be explained by assuming
the shared variable $\lambda$—called a local hidden variable (LHV)—distributed
with density $\Pi(\lambda)$, and local probabilities of obtaining outcomes
$a$, respectively $b$, that are independent of distant choice of input $y$,
respectively $x$. A decomposition (3) is called a _local hidden variable
model_ (LHV model) of the behaviour $p(a,b|x,y)$, and captures those
behaviours that can be reproduced using classical shared randomness alone. If
a behaviour $p(ab|xy)$ cannot be written in form (3), we say that it is _Bell-
nonlocal_.

####  2.1.1 Bell’s theorem

Bell’s theorem states that quantum theory cannot be described by an LHV model;
i.e. there exist behaviours in quantum theory that are Bell-nonlocal. More
precisely, let Alice and Bob share an entangled quantum state $\rho$. Alice
performs measurements from a set $\\{A_{a|x}\\}$ ($A_{a|x}\geq 0$ and
$\sum_{a}A_{a|x}=\mathbb{1}$), and Bob performs measurements from a set
$\\{B_{b|y}\\}$ (with similar conditions). The resulting behaviour is given
through the Born rule by

$\displaystyle p(ab|xy)=\operatorname{Tr}(A_{a|x}\otimes B_{b|y}\;\rho).$ (4)

Bell’s theorem states that there exist entangled states $\rho$ and sets of
local measurements $A_{a|x}$, $B_{b|y}$ such that the behaviour $p(ab|xy)$ is
Bell-nonlocal. This is typically shown with the use of a _Bell inequality_ ; a
function on behaviours that is bounded for all LHV theories, but can be
violated by a quantum behaviour. We give an example below.

####  2.1.2 The CHSH inequality

The simplest Bell inequality is the CHSH inequality

$\displaystyle\langle A_{0}B_{0}\rangle+\langle A_{0}B_{1}\rangle+\langle
A_{1}B_{0}\rangle-\langle A_{1}B_{1}\rangle\leq 2$ (5)

where correlators $\langle A_{x}B_{y}\rangle$ are defined as

$\displaystyle\langle A_{x}B_{y}\rangle=p(00|xy)-p(01|xy)-p(10|xy)+p(11|xy),$
(6)

that is, the expected value of the product $A\cdot B$, where $A=+1$ for $a=0$
, $A=-1$ for $a=1$, and similarly for $B$.

In order to prove the bound of 2, note that since (5) is linear in the
probabilities, it is enough to consider only the extremal behaviours of the
set (3). It is well known (see e.g. [2] section II.B.1) that these behaviours
take the form of deterministic distributions

$\displaystyle p(ab|xy)=D_{A}(a|x)D_{B}(b|y)$ (7)

where $D_{A}(a|x)$ is a conditional probability distribution such that
$D(a|x)\in\\{0,1\\}$ $\forall a,x$, and similarly for $D_{B}$.

For these behaviours it follows that $\langle A_{x}B_{y}\rangle=\langle
A_{x}\rangle\langle B_{y}\rangle$, where

$\displaystyle\langle A_{x}\rangle=D_{A}(0|x)-D_{A}(1|x),\quad\langle
B_{y}\rangle=D_{B}(0|y)-D_{B}(1|y)$ (8)

and thus $\langle A_{x}\rangle,\langle B_{y}\rangle\in\\{-1,1\\}$. One may
therefore rewrite the left-hand side of (5) as

$\displaystyle\langle A_{0}\rangle\Big{(}\langle B_{0}\rangle+\langle
B_{1}\rangle\Big{)}+\langle A_{1}\rangle\Big{(}\langle B_{0}\rangle-\langle
B_{1}\rangle\Big{)}.$ (9)

Since we have $\langle B_{y}\rangle\in\\{-1,1\\}$, it follows that one of
$\langle B_{0}\rangle\pm\langle B_{1}\rangle$ is zero, and the bound of 2
follows.

Now, taking the state $\rho=\left|\Phi^{+}\right\rangle\langle\Phi^{+}|\ $,
where
$\left|\Phi^{+}\right\rangle=1/\sqrt{2}(\left|00\right\rangle+\left|11\right\rangle)$,
and measurements in direction $\sigma_{z}$, $\sigma_{x}$ for Alice,
$\sigma_{z}+\sigma_{x}$, $\sigma_{z}-\sigma_{x}$ for Bob, a textbook
calculation finds

$\displaystyle\langle A_{0}B_{0}\rangle+\langle A_{0}B_{1}\rangle+\langle
A_{1}B_{0}\rangle-\langle A_{1}B_{1}\rangle=4\cdot 1/\sqrt{2}=2\sqrt{2}.$ (10)

Ensuring that the underlying behaviour cannot be reproduced by any LHV model.

####  2.1.3 Bell-local quantum states

A quantum state that is capable of producing Bell-nonlocal behaviours is said
to be a _Bell-nonlocal state_. It is easy to see that such states must be
entangled. Consider a generic separable state:

$\displaystyle\rho=\int\Pi(\lambda)\sigma^{A}_{\lambda}\otimes\sigma^{B}_{\lambda}\text{d}\lambda.$
(11)

Performing local measurements on this state leads to

$\displaystyle
p(ab|xy)=\int\Pi(\lambda)\operatorname{Tr}(A_{a|x}\sigma^{A}_{\lambda})\operatorname{Tr}(B_{b|y}\sigma^{B}_{\lambda})\text{d}\lambda$
(12)

which is of the form (3) and therefore admits an LHV model. A natural question
is thus: is every entangled state Bell nonlocal?

This question was first posed by Werner and answered in the negative [9].
Werner presented a class of entangled mixed states—now known as Werner
states—and proved that all behaviours arising from local projective
measurements on the states admit an LHV model. This was later extended to a
model for all POVM measurement by Barrett [10], and later extended to other
classes of entangled states [31, 12, 13, 14, 11, 15]. Such states are known as
_Bell-local_ states.

The archetypal example of a Bell-local state is the two-qubit isotropic state

$\displaystyle\rho_{\alpha}=\alpha\,|\Phi^{+}\rangle\langle\Phi^{+}|+(1-\alpha)\frac{\mathbb{1}}{4}.$
(13)

which is local unitary equivalent to a two-qubit Werner state. This state is
entangled for $\alpha>\frac{1}{3}$ and known to be Bell-local for projective
measurements for $\alpha\leq 0.683$. For POVM measurements, the state is known
to be Bell-local for $\alpha\leq 0.455$ [25], although no example of Bell
nonlocality for $\alpha<0.683$ using POVM measurements is known and this bound
is thus not expected to be tight. The state is known to be Bell nonlocal for
$\alpha>0.697$ [28] (see also figure 2).

####  2.1.4 Activation of Bell nonlocality

Some years after the result of Werner it was noticed that by subjecting some
Bell-local quantum states to more complex measurement procedures, one could
nevertheless prove a lack of LHV description of the state. This phenomenon is
generally known as Bell nonlocality _activation_. The known examples of Bell
nonlocality activation can be grouped into two scenarios.

The first group we call the _single-copy sequential_ scenario. Here, one
performs sequences of measurements on the local subsystems, which obey a time-
ordered causal structure (see [24] for a detailed discussion of these
scenarios). The first example of Bell nonlocality activation, shown by Popescu
[32], belongs to this scenario and is known as hidden nonlocality. Here, the
parties apply local filters to $\rho$, that is, a local stochastic quantum
channel given by the Kraus operators $K_{A},K_{B}$, such that the state

$\displaystyle\frac{K_{A}\otimes K_{B}\;\rho\;K_{A}^{\dagger}\otimes
K_{B}^{\dagger}}{\operatorname{Tr}[K_{A}\otimes
K_{B}\;\rho\;K_{A}^{\dagger}\otimes K_{B}^{\dagger}]}$ (14)

violates a Bell inequality. This stochastic map can be realised by performing
local measurements with measurement operators
$\\{K_{A}^{\dagger}K_{A},\mathbb{1}-K_{A}^{\dagger}K_{A}\\}$ and
$\\{K_{B}^{\dagger}K_{B},\mathbb{1}-K_{B}^{\dagger}K_{B}\\}$, where one
obtains the state (14) upon obtaining the first outcomes of each measurement.
One can then perform the measurements that violate the Bell inequality in a
second measurement round, where the violation will be observed given the
correct outputs in the first measurement round. In this way, hidden
nonlocality can be seen as a subset of the possible strategies in the
sequential scenario, where the Bell inequality violation from the latter
measurements in the sequence precludes the existence of an LHV model for the
full sequence. Popescu’s method was able to activate a subset of Werner states
of local dimension greater than 5 that are Bell-local for all projective
measurements. This result was later strengthened by Hirsch et. al. [21] to
apply to POVM-local states, and then by Bowles et. al. [23] to the general
multipartite scenario. Other notable examples of hidden nonlocality can be
found in [33, 34, 35, 36, 37]. To date, all known examples of activation in
the single-copy regime can be understood as hidden nonlocality, and it is an
open question whether stronger results are possible by exploiting the full
sequential structure. It is known however that there exist entangled states
that do not have hidden nonlocality [38]. In particular, the isotropic state
(13) was shown to display no hidden nonlocality for $\alpha\leq 0.36$. In
fact, no example of Bell nonlocality activation is known for this state in the
single-copy regime, even considering arbitrary sequences of measurements.

The second scenario for Bell nonlocality activation we call the _multiple-
copy_ scenario. Here, one proves a lack of LHV description for a state by
performing measurements on more than one copy of the state and exploiting the
possibility to make joint measurements between the local subsystems of each
copy. The usefulness of using multiple copies of a state in this context was
first pointed out by Peres [39], who showed that by using entanglement
distillation one can generate Bell nonlocal statistics from many copies of a
local (and distillable) state. With respect to nonlocality, it was shown that
processing many copies of a state can increase CHSH violation [40]. It was
later proved that there exist states $\rho$ not violating CHSH, such that
$\rho\otimes\rho$ does violate it [41].

The first examples of multiple-copy activation were given by Aditi et. al.
[19] and Cavalcanti et. al. [20], where one arranges the multiple copies of
the state in a star-shaped network. Using this method, one can show activation
of the state (13) for $\alpha>0.64$. A later breakthrough in the activation of
nonlocality in the multiple-copy scenario was made by Palazuelos [16], who
showed examples of Bell nonlocality activation by processing many copies of a
state in parallel (as in figure 1). Building on Palazuelos’ result, a
criterion for activation of nonlocality in the same scenario was exhibited in
[18]. Namely, all states useful for teleportation (or equivalently states with
an entanglement fraction $>1/d$, where $d$ is the local Hilbert space
dimension) can be activated in the multiple-copy regime. Applied to the state
(13), these examples prove that the state can be activated in the entire range
of entanglement.

We note that by combining the sequential and multiple-copy scenarios, one can
show Bell nonlocality activation of any state that is distillable. This
follows from two facts: (i) The existence of an entanglement distillation
protocol implies a local stochastic map that maps multiple copies of the state
arbitrarily close to the maximally entangled state of two qubits; (ii) The
maximally entangled state violates the CHSH Bell inequality and so the
multiple-copy state exhibits hidden nonlocality. In particular, since all
entangled two-qubit states are distillable, this implies all such states
exhibit Bell nonlocality in the multiple-copy sequential scenario.

###  2.2 Device-independent entanglement certification

Bell nonlocality is closely linked to protocols of DI entanglement
certification [42, 43, 44, 45, 46]. From (12), it follows that separable
states always lead to Bell-local correlations. The observation of a behaviour
$p(ab|xy)$ that is Bell-nonlocal therefore implies that the underlying state
is entangled. Bell nonlocality can thus be used as a means for entanglement
certification which furthermore is device-independent since the measurement
devices can be treated as black boxes. Bell nonlocality activation also
directly implies a protocol for DI entanglement certification of the
underlying state, since separable states cannot exhibit activation. As a
result the best known bounds for DI entanglement certification of bipartite
entangled states coincide with the best bounds for Bell nonlocality
activation. In the single-copy regime, this means that DI entanglement
certification of the isotropic state (13) is possible for $\alpha>0.697$ [28].
In section 5 we show how one can in fact go beyond these bounds exploiting the
quantum structure of the experiment.

Figure 3: The causal model corresponding to the simplest broadcasting
scenario. A source $S$ produces a bipartite system, one half of which is sent
through a transformation $T$ that broadcasts it to two separated parties.
Measurements labelled $x,y,z$ with outcomes $a,b,c$ are preformed on the
resulting tripartite system. The resulting behaviour $p(a,b,c|x,y,z)$ is then
used to rule out a local hidden variable description of the source.

##  3 The broadcasting scenario

We now present the broadcasting scenario, focusing on the specific scenario of
figure 3 for clarity. A source, denoted $S$, produces a bipartite system. One
of these systems undergoes a transformation, denoted $T$, that broadcasts this
system to a number of additional parties (in this case, two). Classical inputs
$x,y,z$ are given to the parties (called Alice, Bob and Charlie), that return
outcomes $a,b,c$, resulting in the behaviour $p(abc|xyz)$. In the quantum
mechanical setting, the source corresponds to some bipartite quantum state
$\rho_{AB_{0}}$, and the transformation to a completely positive trace
preserving (CPTP) quantum channel $\Omega_{B_{0}\rightarrow BC}$ that enlarges
the $B_{0}$ space to a tensor product space of $B$, $C$. The resulting
tripartite state is thus

$\displaystyle\rho_{ABC}=\mathbb{1}\otimes\Omega_{B_{0}\rightarrow
BC}[\rho_{AB_{0}}].$ (15)

The final behaviour obtained by making local measurements on this state is

$\displaystyle p(abc|xyz)=\operatorname{Tr}(A_{a|x}\otimes B_{b|y}\otimes
C_{c|z}\;\rho_{ABC}).$ (16)

The question we pose is the following. _Can the behaviour ( 16) be explained
by replacing the state $\rho_{AB_{0}}$ with local hidden variables?_ A
negative reply to this question would imply a type of Bell nonlocality of the
state $\rho_{AB_{0}}$.

To this end we consider again the causal structure of figure 3, where the
source produced LHVs $\lambda$ distributed with density $\Pi(\lambda)$.
Alice’s outcome is governed by a local response function $p(a|x\lambda)$. The
transformation device receives $\lambda$ and sends some system
$\tau_{\lambda}$ to Bob and Charlie that they use to produce outcomes
according to the joint probability distribution
$p_{BC}(bc|yz,\tau_{\lambda})$. The behaviour therefore admits a decomposition

$\displaystyle
p(abc|xyz)=\int\Pi(\lambda)\;p_{A}(a|x,\lambda)\;p_{BC}(bc|yz,\tau_{\lambda})\;d\lambda.$
(17)

At this point, one arrives at a subtlety. Namely, what constraints does one
put on the distribution $p_{BC}(bc|yz,\tau_{\lambda})$? Equivalently, what
systems $\tau_{\lambda}$ can the transformation device prepare?

One natural option is to demand that, like the source, the transformation
device produce another LHV $\lambda^{\prime}$ that depends on $\lambda$ to
send to Bob and Charlie. That is, $\tau_{\lambda}$ is an LHV
$\lambda^{\prime}(\lambda)$ and $p_{BC}(bc|yz,\tau_{\lambda})$ therefore
factorises as
$p_{B}(b|y,\lambda^{\prime}(\lambda))p_{C}(c|z,\lambda(\lambda^{\prime}))$,
leading to333Note that we have assumed that $\lambda^{\prime}$ is a
deterministic function of $\lambda$. In principle, the transformation device
could act stochastically and produce a new LHV $\lambda^{\prime}$ conditioned
on $\lambda$ described by some conditional probability distribution
$q(\lambda^{\prime}|\lambda)$. By defining a new LHV
$\mu=(\lambda,\lambda^{\prime})$ distributed with density
$\Pi(\lambda)q(\lambda^{\prime}|\lambda)$, this in-determinism can be absorbed
by the source, so that the distribution admits the decomposition (19) with
$\lambda$ replaced by $\mu$. Thus, the form (19) is general.

$\displaystyle p(abc|xyz)$
$\displaystyle=\int\Pi(\lambda)\;p_{A}(a|x,\lambda)\;p_{B}(b|y,\lambda^{\prime}(\lambda))p_{C}(c|z,\lambda^{\prime}(\lambda))\;d\lambda$
(18)
$\displaystyle=\int\Pi(\lambda)\;p_{A}(a|x,\lambda)\;\tilde{p_{B}}(b|y,\lambda)\tilde{p_{C}}(c|z,\lambda))\;d\lambda.$
(19)

Note that this is simply the standard definition of Bell nonlocality for a
three-party scenario. Suppose now that one sees a violation of
$\eqref{model1b}$. This implies Bell nonlocality somewhere due to a lack of
LHV model for the three parties. The problem, however, is that one cannot
conclude that this nonlocality originated from the original state
$\rho_{AB_{0}}$. Consider for instance the following experiment. The source
prepares a state $\rho_{AB_{0}}$, however the transformation device ignores
the $B_{0}$ subsystem and instead creates a fresh maximally entangled state
which it sends to Bob and Charlie (i.e. if $B_{0}$ is a qubit space, the Kraus
operators for the channel are $|\Phi^{+}\rangle\langle 0|$ and
$|\Phi^{+}\rangle\langle 1|$). Bob and Charlie now hold a maximally entangled
state and can violate the CHSH inequality. The resulting behaviour will not
admit a decomposition (19) regardless of what Alice measures. This will be the
case for any choice of $\rho_{AB_{0}}$, including separable $\rho_{AB_{0}}$.
Thus, although a violation of (19) implies some Bell nonlocality, one cannot
rule out a LHV description of $\rho_{AB_{0}}$, since the nonlocality may have
originated from the transformation device alone.

To make a statement about $\rho_{AB_{0}}$, we thus need to consider models
that rule out the above possibility. To do this, we allow the systems
$\tau_{\lambda}$ to be arbitrary non-signalling resources, so that the
distribution $p_{BC}(bc|yz,\tau_{\lambda})$ is any normalised joint
conditional probability distribution that obeys the no-signalling constraints:

$\displaystyle\sum_{b}p_{BC}(bc|yz,\tau_{\lambda})$
$\displaystyle=\sum_{b}p_{BC}(bc|y^{\prime}z,\tau_{\lambda})\;\forall
y,y^{\prime},c,z,\tau_{\lambda},$ (20)
$\displaystyle\sum_{c}p_{BC}(bc|yz,\tau_{\lambda})$
$\displaystyle=\sum_{c}p_{BC}(bc|yz^{\prime},\tau_{\lambda})\;\forall
z,z^{\prime},b,y,\tau_{\lambda}.$ (21)

Thus, the full distribution can be decomposed

$\displaystyle
p(abc|xyz)=\int\Pi(\lambda)\;p_{A}(a|x,\lambda)\;p^{NS}_{BC}(bc|yz,\lambda)\;d\lambda.$
(22)

where the $p^{NS}_{BC}(b,c|y,z,\lambda)$ are arbitrary non-signalling
distributions satisfying (20), (21) between Bob and Charlie, for each value
$\lambda$. Note that for a finite number of inputs and outcomes, the sets of
possible distributions $p_{A}(a|x,\lambda)$ and $p^{NS}_{BC}(bc|yz,\lambda)$
are both convex polytopes, which follows from the fact that they are defined
by a finite number of linear inequalities. There are therefore a finite number
of extremal distributions in each of these sets. In particular the extremal
$p_{A}(a|x,\lambda)$ correspond to deterministic distributions, and the
extremal $p^{NS}_{BC}(bc|yz,\lambda)$ to the (possibly non-deterministic and
Bell non-local) extremal points of the no-signalling set (see e.g. [2]). The
set (22) can thus be equivalently defined

$\displaystyle
p(abc|xyz)=\sum_{\lambda}p(\lambda_{1},\lambda_{2})\;p_{A}(a|x,\lambda_{1})\;p^{NS}_{BC}(bc|yz,\lambda_{2}),$
(23)

where $p_{A}(a|x,\lambda_{1})$ and $p^{NS}_{BC}(bc|yz,\lambda_{2})$ are these
extremal distributions, and is therefore a convex polytope.

Assuming the no-signalling condition, any violation of (22) cannot originate
from the transformation device alone, so long as it produces non-signalling
resources (as is the case for quantum theory and the majority of GPT
theories). One can thus make the following statement.

_In any no-signalling hidden variable theory reproducing quantum theory, a
violation of ( 22) implies that those variables describing the state
$\rho_{AB_{0}}$ cannot be LHVs._

We thus take the set of behaviours admitting a decomposition (22) as our
analogue of the local set in the broadcasting scenario. If a state leads to
behaviours that violate (22), we say it is _broadcast-nonlocal_.

##  4 Activation of two-qubit states

We now study Bell nonlocality activation of the isotropic state (13) in the
broadcasting scenario. As a main result, we show analytically that the state
is broadcast-nonlocal for $\alpha>1/\sqrt{3}$. To find this result we employed
the convex optimisation methods in section 6 and the heuristic algorithm
described in appendix B. A slightly lower value of $\alpha=0.559$ was found
with the same methods, although we failed to convert this strategy to a nice
analytic form (see auxiliary files for the numerical files needed to verify
this result).

###  4.1 Simple inequality for three parties

To prove our results, we construct a correlation inequality that is satisfied
by all behaviours admitting a decomposition (22). Consider a scenario in which
Alice has 3 inputs $x=0,1,2$, and Bob and Charlie both have 2 inputs $y=0,1$,
$z=0,1$. All outcomes are dichotomic with $a,b,c=\pm 1$. Define the three body
correlators

$\displaystyle\langle A_{x}B_{y}C_{z}\rangle=\sum_{a,b,c=\pm
1}abc\;p(abc|xyz),$ (24)

the two body correlators for Alice and Bob

$\displaystyle\langle A_{x}B_{y}\rangle=\sum_{a,b=\pm 1}ab\;p(ab|xy)$ (25)

(and similarly for other pairs of parties), and the single body correlators
for Alice

$\displaystyle\langle A_{x}\rangle=\sum_{a=\pm 1}a\;p(a|x)$ (26)

(and similarly for Bob and Charlie). The inequality reads

$\mathcal{I}=\langle A_{0}B_{0}C_{0}\rangle+\langle
A_{0}B_{1}C_{1}\rangle+\langle A_{1}B_{1}C_{1}\rangle-\langle
A_{1}B_{0}C_{0}\rangle\\\ +\langle A_{0}B_{0}C_{1}\rangle+\langle
A_{0}B_{1}C_{0}\rangle+\langle A_{1}B_{0}C_{1}\rangle-\langle
A_{1}B_{1}C_{0}\rangle\\\ -2\langle A_{2}B_{0}\rangle+2\langle
A_{2}B_{1}\rangle\leq 4,$ (27)

which can be shown to be a facet of the broadcast-local polytope. Note that
with a slight abuse of notation the above can be written

$\displaystyle\mathcal{I}=\langle\text{CHSH}[A_{0},A_{1},C_{0},C_{1}](B_{0}+B_{1})\rangle+\langle
A_{2}(B_{1}-B_{0})\rangle\leq 4,$ (28)

where
$\text{CHSH}[A_{0},A_{1},C_{0},C_{1}]=A_{0}C_{0}+A_{0}C_{1}-A_{1}C_{0}+A_{1}C_{1}$
is a relabelling of the well known CHSH Bell inequality [47]. We first prove
the bound of $4$ for behaviours (22).

###### Proof.

Since (27) is linear in the probabilities, we can restrict ourselves to
extremal points of the set (22). These points take the form

$\displaystyle p(abc|xyz)=D(a|x)P(bc|yz)$ (29)

where $D$ is a deterministic behaviour such that $D(a|x)\in\\{0,1\\}\;\forall
a,x$, and $P$ is an extremal behaviour of the 2 parties, 2 inputs, 2 outputs
scenario. For the points (29), it follows that the three body correlators
factorise

$\displaystyle\langle A_{x}B_{y}C_{z}\rangle=\langle A_{x}\rangle\langle
B_{y}C_{z}\rangle.$ (30)

We thus have

$\mathcal{I}=\Big{(}\langle A_{0}\rangle-\langle
A_{1}\rangle\Big{)}\Big{(}\langle B_{0}C_{0}\rangle+\langle
B_{1}C_{0}\rangle\Big{)}\\\ +\Big{(}\langle A_{0}\rangle+\langle
A_{1}\rangle\Big{)}\Big{(}\langle B_{0}C_{1}\rangle+\langle
B_{1}C_{1}\rangle\Big{)}\\\ +2\langle A_{2}\rangle\Big{(}\langle
B_{1}\rangle-\langle B_{0}\rangle\Big{)}.$ (31)

Since Alice’s strategy is deterministic, one has $\langle A_{x}\rangle=\pm 1$
and thus either the first or the second term of the above must be equal to
zero. Due to the symmetry of the inequality, we can take this to be the second
term without loss of generality. Using $|\langle A_{x}\rangle|\leq 1$ we thus
have

$\displaystyle\mathcal{I}$ $\displaystyle\leq 2|\langle
B_{0}C_{0}\rangle+\langle B_{1}C_{0}\rangle|+2|\langle B_{1}\rangle-\langle
B_{0}\rangle|.$ (32)

We now consider the extremal behaviours $P(bc|yz)$ for Bob and Charlie. These
are known to be of two types [48]: (i) local deterministic behaviours and (ii)
local relabellings of the PR box. In case (i), the correlators $\langle
B_{y}C_{z}\rangle$ factorise as $\langle B_{y}\rangle\langle C_{z}\rangle$ and
one has

$\displaystyle\mathcal{I}\leq 2|\big{(}\langle B_{0}\rangle+\langle
B_{1}\rangle\big{)}\langle C_{0}\rangle|+2|\langle B_{1}\rangle-\langle
B_{0}\rangle|\leq 4$ (33)

since either $\langle B_{0}\rangle+\langle B_{1}\rangle$ or $\langle
B_{0}\rangle-\langle B_{1}\rangle$ is zero. In case (ii), since the PR box is
such that $\langle B_{0}\rangle=\langle B_{1}\rangle=0$, from (32) one also
obtains the bound 4. ∎

####  4.1.1 Quantum violation

The inequality (27) can be violated by the isotropic state (13). Let Alice
measure the triple of anti-commuting qubit observables

$\displaystyle A_{0}=\sigma_{z},\;A_{1}=\sigma_{x},\;A_{2}=\sigma_{y}.$ (34)

Charlie and Bob measure the observables

$\displaystyle C_{0}$ $\displaystyle=\sigma_{z},\;C_{1}=\sigma_{x},$ (35)
$\displaystyle B_{0}$
$\displaystyle=\cos(\phi)\sigma_{x}+\sin(\phi)\sigma_{y},\;B_{1}=\cos(\phi)\sigma_{x}-\sin(\phi)\sigma_{y},$

with $\tan(\phi)=1/\sqrt{2}$. The channel $\Omega_{B_{0}\rightarrow BC}$ is an
isometry given by

$\displaystyle U=\left|\psi_{0}\right\rangle\langle
0|+\left|\psi_{1}\right\rangle\langle 1|$ (36)

with

$\displaystyle\left|\psi_{0}\right\rangle$
$\displaystyle=\sin\beta\left|\Phi^{-}\right\rangle+\cos\beta\left|\Psi^{+}\right\rangle$
(37) $\displaystyle\left|\psi_{1}\right\rangle$
$\displaystyle=-\cos\beta\left|\Phi^{-}\right\rangle+\sin\beta\left|\Psi^{+}\right\rangle,$
(38)

and where $\left|\Phi^{-}\right\rangle$, $\left|\Psi^{+}\right\rangle$ are the
usual Bell states and $\beta=\pi/8$. Taking the isotropic state, one finds via
direct calculation that

$\displaystyle\mathcal{I}=\alpha\;4\sqrt{3}.$ (39)

Thus, for $\alpha>1/\sqrt{3}$ one obtains a violation.

We have also investigated the violation of inequality (27) for the larger
class of two-qubit states

$\displaystyle\rho_{\alpha,\theta}=\alpha|\psi_{\theta}\rangle\langle\psi_{\theta}|+(1-\alpha)\rho_{A}\otimes\mathbb{1}/2,$
(40)

where
$\left|\psi_{\theta}\right\rangle=\frac{1}{\sqrt{2}}[\cos\theta\left|00\right\rangle+\sin\theta\left|11\right\rangle]$
and
$\rho_{A}=\operatorname{Tr}_{B}[|\psi_{\theta}\rangle\langle\psi_{\theta}|]$.
This state is entangled for $\alpha>1/3$ for all $\theta>0$. We plot the
critical visibility at which the Bell inequality (27) is no longer violated,
and compare this to the equivalent value for the widely used CHSH Bell
inequality [47] in the standard scenario. As with the isotropic state, the
inequality (27) allows for lower visibilities than the CHSH inequality for all
values of $\theta$, see figure 4. Unlike the isotropic state however, it is
not known if there exists an LHV model for the state (40) in the region above
the broadcast-nonlocal bound for $\theta\neq\pi/4$, and so it is not known if
these values of the visibility correspond to activation or not. Given that
this is the case for $\theta=\pi/4$ however, we expect there to be some
activation, at least for a range of $\theta$ close to $\pi/4$.

$0$$0.2$$0.4$$0.6$$0.6$$0.7$$0.8$$0.9$$1$$\theta$visibility $\alpha$Broadcast-
nonlocalCHSH-nonlocal Figure 4: Critical value of the visibility $\alpha$
above which the state (40) is broadcast-nonlocal (blue line), and Bell
nonlocal using the CHSH inequality in the standard scenario (red line). The
blue curve was obtained by numerically optimising the violation of inequality
(27).

###  4.2 Symmetric four-partite inequality

One can also obtain activation of the isotropic state in the same range as
above in a symmetric scenario where one applies a channel to both local
subsystems resulting in a 4-partite state $\rho_{ABCD}$ (see figure 5), where
each party now has two inputs. Following the same logic as before, the
relevant behaviours are now of the form

$\displaystyle
p(abcd|xyzw)=\int\Pi(\lambda)\;p^{NS}_{AB}(a,b|x,y,\lambda)\;p^{NS}_{CD}(c,d|z,w,\lambda)\;d\lambda.$
(41)

Where $p^{NS}_{AB}$ and $p^{NS}_{CD}$ are non-signalling distributions. The
inequality we use to show activation in this scenario then reads

$\displaystyle\mathcal{I}=\langle A_{0}B_{0}C_{0}D_{0}\rangle+\langle
A_{0}B_{0}C_{1}D_{0}\rangle+\langle A_{0}B_{1}C_{0}D_{0}\rangle+\langle
A_{0}B_{1}C_{1}D_{0}\rangle+$ $\displaystyle-\langle
A_{1}B_{0}C_{0}D_{0}\rangle+\langle A_{1}B_{0}C_{1}D_{0}\rangle-\langle
A_{1}B_{1}C_{0}D_{0}\rangle+\langle A_{1}B_{1}C_{1}D_{0}\rangle-$
$\displaystyle 2\langle B_{0}D_{1}\rangle+2\langle B_{1}D_{1}\rangle\leq 4.$
(42)

We first derive the bound of $4$.

###### Proof.

We again may focus on extremal behaviours of the set (4.2). Since the extremal
points of non-signalling behaviours for 2 inputs and 2 outputs are either
local deterministic distributions, or symmetries of the PR box behaviour [48],
the extremal behaviours of the set (4.2) can be grouped into three groups: (i)
All parties have local deterministic outcomes, (ii) A and B have local
deterministic outcomes, and C and D share a PR box (or vice-versa) and (iii)
Both A and B, and C and D share a PR box. In all cases, the full body
correlators factorise as

$\displaystyle\langle A_{x}B_{y}C_{z}D_{w}\rangle=\langle
A_{x}B_{y}\rangle\langle C_{z}D_{w}\rangle.$ (43)

We thus write (4.2) as

$\mathcal{I}=\Big{(}\langle A_{0}B_{0}\rangle+\langle
A_{0}B_{1}\rangle\Big{)}\Big{(}\langle C_{1}D_{0}\rangle+\langle
C_{0}D_{0}\rangle\Big{)}\\\ +\Big{(}\langle A_{1}B_{0}\rangle+\langle
A_{1}B_{1}\rangle\Big{)}\Big{(}\langle C_{1}D_{0}\rangle-\langle
C_{0}D_{0}\rangle\Big{)}\\\ -2\Big{(}[\langle B_{1}\rangle-\langle
B_{0}\rangle]\langle D_{1}\rangle\Big{)}.$ (44)

Consider the final term of the inequality. This term is non-zero if and only
if neither pair uses a PR box and $\langle B_{0}\rangle=-\langle
B_{1}\rangle$. In this case, since A and B use a deterministic strategy, one
has

$\displaystyle\langle A_{0}B_{0}\rangle+\langle A_{0}B_{1}\rangle=\langle
A_{0}\rangle(\langle B_{0}\rangle+\langle B_{1}\rangle)=0$ (45)

and so the first term of (44) is zero (and similarly for the second term).
Thus, one obtains the bound $4$. For all other cases, the final term of (44)
is zero and we can focus on the first two terms. Here, it follows that for all
strategies of C and D either $\langle C_{1}D_{0}\rangle+\langle
C_{0}D_{0}\rangle$ or $\langle C_{1}D_{0}\rangle-\langle C_{0}D_{0}\rangle$ is
zero, and again one obtains $\mathcal{I}\leq 4$.

∎

Figure 5: A four-partite broadcast scenario. Transformations are applied to
both subsystems of the source.

####  4.2.1 Quantum violation

Again we consider the isotropic state (13). The two local channels applied to
the state are identical and are again given by (36). Alice and Bob’s local
observables are

$A_{0}=\sigma_{x},\;A_{1}=\sigma_{z},\quad
B_{0}=(\sigma_{x}+\sigma_{y}+\sigma_{z})/\sqrt{3},\;B_{1}=(\sigma_{x}-\sigma_{y}+\sigma_{z})/\sqrt{3}.$
(46)

Charlie’s and Bob’s local observables are

$C_{0}=\sigma_{x},\;C_{1}=\sigma_{z},\quad
D_{0}=\sigma_{x},\;D_{1}=\sigma_{y}.$ (47)

Using this strategy $\mathcal{I}=12\alpha/\sqrt{3}=4\alpha\cdot
3/\sqrt{3}=4\alpha\sqrt{3}$, thus leading to activation for
$\alpha>1/\sqrt{3}$.

##  5 Device-independent entanglement certification in the broadcasting
scenario

In DI entanglement certification, one aims at certifying the entanglement of
the state in an experiment based solely on the observation of the experimental
statistics. Here, we show how the broadcasting scenario can be used to design
protocols for DI entanglement certification. To do this, we consider the four-
partite scenario of figure 5. To achieve a DI certification of entanglement,
one needs to show that the correlations are incompatible with the source
producing any separable state, that is, one must show that the correlations
are incompatible with the causal network of figure 5, where the source is
replaced by a separable state.

Consider a state $\rho_{A_{0}C_{0}}$ produced by the source. This state is
separable iff it admits a decomposition

$\displaystyle\rho_{SEP}=\int\text{d}\lambda\Pi(\lambda)\sigma_{\lambda}^{A_{0}}\otimes\sigma_{\lambda}^{C_{0}}.$
(48)

Following the local transformations, the most general four-partite state than
one can obtain from this state is of the form

$\displaystyle\rho_{ABCD}=\int\text{d}\lambda\Pi(\lambda)\sigma_{\lambda}^{AB}\otimes\sigma_{\lambda}^{CD}$
(49)

where $\sigma^{AB}_{\lambda}=\Omega_{A_{0}\rightarrow
AB}[\sigma_{\lambda}^{B_{0}}]$ and
$\sigma^{CD}_{\lambda}=\Omega_{C_{0}\rightarrow
CD}[\sigma_{\lambda}^{C_{0}}]$. Performing local measurements on this state
thus leads to behaviours of the form

$\displaystyle p(abcd|xyzw)$
$\displaystyle=\operatorname{Tr}\left[\rho_{ABCD}A_{a|x}\otimes B_{b|y}\otimes
C_{c|z}\otimes D_{d|w}\right]$ (50)
$\displaystyle=\int\Pi(\lambda)\operatorname{Tr}\left[\sigma^{AB}_{\lambda}A_{a|x}\otimes
B_{b|y}\right]\operatorname{Tr}\left[\sigma^{CD}_{\lambda}C_{c|z}\otimes
D_{d|w}\right]$ (51)
$\displaystyle=\int\Pi(\lambda)\;p^{Q}_{AB}(ab|xy\lambda)\;p^{Q}_{CD}(cd|zw\lambda)\;d\lambda,$
(52)

where $p^{Q}_{AB}$, $p^{Q}_{CD}$ are arbitrary quantum behaviours. Thus, the
possible correlations correspond to those that can be obtained by a general
four-partite state $\rho_{AB|CD}$ (of arbitrary dimension), that is separable
with respect to the bipartition $AB$ vs $CD$. Thus, if we consider an
expression $\mathcal{I}[p(abcd|xyzw)]$ in this scenario and define $\beta$ the
maximum value obtained by biseperable states (49), a violation of the
inequality $\mathcal{I}[p(abcd|xyzw)]\leq\beta$ certifies that the state
$\rho_{A_{0}C_{0}}$ is entangled in a DI manner.

Before proceeding to a specific example, we clarify the assumptions of the
certification. As is standard in DI certification we assume that all
measurement and transformation devices act on separate locations in space and
thus correspond to disjoint Hilbert spaces. Strictly speaking, the space
$A_{0}$ should be understood to encapsulate all the degrees of freedom ‘to the
left’ of the source in figure 5; i.e. the space on which Alice’s
transformation device acts as well as the spaces $A$ and $B$ (and analogously
for $C_{0}$). No assumption is needed on the causal ordering of the
transformation and measurements (although it would clearly be useless to apply
a transformation after the measurement process).

###  5.1 Entanglement certification of isotropic states

We now apply this reasoning to the isotropic state, where we show that DI
entanglement certification is possible for $\alpha>1/2$. The expression we
will use is the four-partite Klyshko-Belinskii expression [49, 50]:

$\displaystyle\mathcal{I}=2^{-\frac{3}{2}}\sum_{\vec{x}\in\\{0,1\\}^{4}}\cos\left[\frac{\pi}{4}(2|\vec{x}|-3)\right]\langle
A_{x_{1}}B_{x_{2}}C_{x_{3}}D_{x_{4}}\rangle$ (53)

where $|\vec{x}|=\sum_{i=1}^{4}x_{i}$. The bound $\mathcal{I}\leq 1$ for LHV
theories is a well known Bell inequality in the standard four-party Bell
scenario. The maximum quantum violation of the expression is $2^{3/2}$ and is
achieved by the four-partite GHZ state

$\displaystyle\left|\text{GHZ}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0000\right\rangle+\left|1111\right\rangle\right)$
(54)

and measurements observables that lie in the $x-y$ plane of the Bloch sphere
for each of the parties. In [51] (using [52]), it was proven that the
$k$-producible bound of the inequality—the maximum possible value obtainable
by states that contain entanglement between at most $k$ parties—is given by
$\mathcal{I}\leq 2^{(k-1)/2}$. Note that the state (49) is $2$-producible,
since there are at most two parties entangled. Thus, a violation of the
inequality $\mathcal{I}\leq 2^{1/2}$ is a DI entanglement certification of
$\rho_{A_{0}C_{0}}$ and (53) can be used as a DI entanglement witness in the
broadcasting scenario.

Now, take the local channels in figure 5 to be isometries given by

$\displaystyle U=|00\rangle\langle 0|+|11\rangle\langle 1|.$ (55)

With this choice, and taking $\rho_{A_{0}C_{0}}$ to be the isotropic state we
obtain

$\displaystyle\rho_{ABCD}=\alpha|\text{GHZ}\rangle\langle\text{GHZ}|+\frac{1}{4}(1-\alpha)\left[|0000\rangle\langle
0000|+|0011\rangle\langle 0011|+|1100\rangle\langle 1100|+|1111\rangle\langle
1111|\right].$ (56)

We now perform the measurements that give the maximum violation of
$\mathcal{I}$ for the GHZ state. Note that the second term in the state will
contribute zero to $\mathcal{I}$ since it is diagonal in the $z$ basis, and
the measurements are chosen in the $x-y$ plane. One therefore obtains the
value $\alpha\cdot 2^{3/2}$, and has a DI certification of entanglement for

$\displaystyle\alpha\cdot 2^{3/2}>2^{1/2},$ (57)

that is, for $\alpha>1/2$. We suspect that further investigation into the
biseparable bounds of known Bell inequalities could lead to even stronger
examples.

It may also be possible that stronger examples are already possible in the
three partite scenario of figure (3), and in the next section we give a
numerical method that can be useful to find such examples. We note that the
Bell inequality (27) has the same bound of $4$ when considering quantum
resources, since this bound can be achieved by a local deterministic (and thus
quantum) strategy. This means that the same critical visibility of
$1/\sqrt{3}$ is obtained for DI entanglement certification.

##  6 Numerical methods

Here we outline two numerical methods that allow one to tackle the problem of
testing compatibility with the causal networks considered in the previous
sections. In the case of section 4, this can be achieved via a linear program,
and the methods we present (combined with the algorithm in appendix B) were
used to find the results of section 4. We present two approaches to this end;
the second uses a possibly unknown trick to reduce memory usage and may be of
independent interest. For DI entanglement certification, we present a method
based on semi-definite programming to define a relaxation to the problem that
in practice can give good results. This method works for scenarios in which
the broadcasting channel is applied to only one party. It would be interesting
to use this method to look for stronger examples that that presented in
section 5.

###  6.1 Linear programming methods

Let us consider again the set of behaviours defined in (22):

$\displaystyle
p(abc|xyz)=\int\Pi(\lambda)\;p_{A}(a|x,\lambda)\;p^{NS}_{BC}(bc|yz,\lambda)\;d\lambda,$
(58)

and denote by $\mathcal{L}_{NS}$ the set of behaviours admitting the above
decomposition. It is well known (see e.g. [2]) that the sets of distributions
$p_{A}(a|x,\lambda)$ and $p^{NS}_{BC}(bc|yz,\lambda)$ are both convex and have
a finite number of extremal elements, that is, they are convex polytopes.
Denote these extremal elements by $\\{D_{A}(a|x,i)\\}_{i}$ and
$\\{E_{BC}(bc|yz,j)\\}_{j}$ where $i$ and $j$ index the elements. In this
scenario, these elements correspond to deterministic behaviours for Alice, and
the extremal behaviours of the no-signalling polytope for Bob and Charlie.
Thus, there always exist non-negative weights $q(i|\lambda)$, $w(j|\lambda)$
such that $\sum_{i}q(i|\lambda)=\sum_{j}w(j|\lambda)=1$ such that

$\displaystyle p_{A}(a|x,\lambda)=\sum_{i}q(i|\lambda)D_{A}(a|x,i),\quad\quad
p^{NS}_{BC}(bc|yz,\lambda)=\sum_{j}w(j|\lambda)E_{BC}(bc|yz,j).$ (59)

Substituting this in (58) we find

$\displaystyle p(abc|xyz)=\sum_{i,j}p_{ij}\;D_{A}(a|x,i)\;E_{BC}(bc|yz,j),$
(60)

where $p_{ij}=\int\Pi(\lambda)q(i|\lambda)w(j|\lambda)d\lambda$. Hence, since
the distributions $D_{A}$ and $E_{BC}$ are fixed, the problem of deciding
membership in $\mathcal{L}_{NS}$ is equivalent to finding positive numbers
$p_{ij}$ such that the above is satisfied. Moreover, since the above
expression is linear in $p_{ij}$ this can be cast as a linear programming
instance, i.e., one solves the linear program

$\displaystyle\text{find }p_{ij}\geq 0\;\;\text{such that}$ (61)
$\displaystyle
p(abc|xyz)=\sum_{i,j}p_{ij}\;D_{A}(a|x,i)\;E_{BC}(bc|yz,j)\;\;\forall
a,b,c,x,y,z.$

If the behaviour is found to be outside of $\mathcal{L}_{NS}$, the dual of the
optimisation returns a corresponding separating hyperplane that one can
interpret as a generalised Bell inequality for the broadcasting scenario.

In practice, one is limited to scenarios in which the number of optimisation
variables $p_{ij}$ can be stored in memory. Since the number of extremal
distributions increases exponentially in the number of each input, this means
that (61) can become intractable for relatively simple scenarios. One method
to partly overcome this, is to use the fact the distributions
$p^{NS}_{BC}(bc|yz,\lambda)\in\mathcal{P}_{NS}$ have a linear
characterisation, given in (20), (21). That is

$\displaystyle p^{NS}_{BC}(bc|yz,\lambda)\in\mathcal{P}_{NS}\iff\quad$
$\displaystyle\sum_{b}p^{NS}_{BC}(bc|yz,\lambda)=\sum_{b}p^{NS}_{BC}(bc|y^{\prime}z,\lambda)\;\forall
y,y^{\prime},c,z,\lambda,$ (62)
$\displaystyle\sum_{c}p^{NS}_{BC}(bc|yz,\lambda)=\sum_{c}p^{NS}_{BC}(bc|yz^{\prime},\lambda)\;\forall
z,z^{\prime},b,y,\lambda.$ (63)

If in (58) we replace $p_{A}(a|x,\lambda)$ only (through (59)) then we find

$\displaystyle p(abc|xyz)$
$\displaystyle=\sum_{i}D_{A}(a|x,i)\left[\int\Pi(\lambda)q(i|\lambda)\;p^{NS}_{BC}(bc|yz,\lambda)\;d\lambda\right]$
$\displaystyle=\sum_{i}D_{A}(a|x,i)\,\tilde{p}^{NS}_{BC}(bc|yz,i),$ (64)

where $\tilde{p}^{NS}_{BC}(bc|yz,i)$ is a sub-normalised non-signalling
distribution with normalisation $\int\Pi(\lambda)q(i|\lambda)d\lambda$ that is
the same for all $y,z$, and $\tilde{p}^{NS}_{BC}(bc|yz,i)\in\mathcal{P}_{NS}$
since it still satisfies the constraints (20), (21). One can therefore also
test membership in (58) with an equivalent LP by treating the distributions
$\tilde{p}^{NS}_{BC}(bc|yz,i)$ as optimisation variables:

$\displaystyle\text{find }\tilde{p}^{NS}_{BC}(bc|yz,i)\geq 0\;\;\text{such
that}$ $\displaystyle
p(abc|xyz)=\sum_{i}D_{A}(a|x,i)\tilde{p}^{NS}_{BC}(bc|yz,i),\;\;\forall
a,b,c,x,y,z;$
$\displaystyle\tilde{p}^{NS}_{BC}(bc|yz,i)\in\mathcal{P}_{NS}\;\;\forall
i,\quad\sum_{bc}\tilde{p}^{NS}_{BC}(bc|yz,i)=\sum_{bc}\tilde{p}^{NS}_{BC}(bc|y^{\prime}z^{\prime},i)\;\;\forall
y,z,y^{\prime},z^{\prime},i.$ (65)

The number of optimisation variables now scales with the number of extremal
elements of Alice’s distribution, at the expense of introducing a similar
number of constraints to the optimisation. In practice, this requires less
memory than (61) and allows one to tackle significantly larger scenarios. Both
linear programs can be applied to the causal structure of figure 5 (or
scenarios with any number of parties) by decomposing the relevant
distributions as convex mixtures of their extremal elements in a similar
fashion. One can also define closely related linear programs to calculate the
robustness to noise to the set $\mathcal{L}_{NS}$ given a state with a linear
noise parameter, and fixed measurements and channel(s); see Appendix A.

###  6.2 Semi-definite programming hierarchy for entanglement certification

Here we wish to tackle membership in scenarios where the transformation device
prepares quantum systems. The approach we present is applicable to the
scenario of figure 3. Denote by $\mathcal{L}_{Q}$ the set of relevant
behaviours obtainable via a separable state:

$\displaystyle
p(abc|xyz)=\int\Pi(\lambda)\;p_{A}(a|x,\lambda)\;p^{Q}_{BC}(bc|yz,\lambda)\;d\lambda,$
(66)

where we have written $p_{A}(a|x,\lambda)$ rather than
$\operatorname{Tr}[A_{a|x}\sigma_{\lambda}^{A}]$ since any distribution can be
obtained with a suitable choice of $\sigma_{\lambda}^{A}$ and $A_{a|x}$.
Denote by $\mathcal{Q}$ the set of quantum distributions
$p^{Q}_{BC}(bc|yz,\lambda)$. Unlike $\mathcal{L}_{NS}$, the set $\mathcal{Q}$
has an infinite number of extremal points, and the same approach as above is
not possible. We may however do the following. Similarly to (6.1), replace
$p_{A}(a|x,\lambda)$ via (59) to obtain

$\displaystyle p(abc|xyz)$
$\displaystyle=\sum_{i}D_{A}(a|x,i)[\int\Pi(\lambda)q(i|\lambda)\;p^{Q}_{BC}(bc|yz,\lambda)d\lambda]$
$\displaystyle=\sum_{i}D_{A}(a|x,i)\,\tilde{p}^{Q}_{BC}(bc|yz,i),$ (67)

where since $\mathcal{Q}$ is convex, $\tilde{p}^{Q}_{BC}(bc|yz,i)$ is a sub-
normalised distribution of $\mathcal{Q}$, with normalisation
$\int\Pi(\lambda)q(i|\lambda)d\lambda$ independent of $y,z$. Denote the set of
un-normalised distributions of $\mathcal{Q}$ by $\tilde{\mathcal{Q}}$. A
solution to the membership problem is thus given by

$\displaystyle\text{find }\tilde{p}^{Q}_{BC}(bc|yz,i)\;\;\text{such that}$
$\displaystyle p(abc|xyz)=\sum_{i}D_{A}(a|x,i)\,\tilde{p}^{Q}_{BC}(bc|yz,i)$
$\displaystyle\tilde{p}^{Q}_{BC}(bc|yz,i)\in\tilde{\mathcal{Q}}\;\;\forall
i,\quad\sum_{bc}\tilde{p}^{Q}_{BC}(bc|yz,i)=\sum_{bc}\tilde{p}^{Q}_{BC}(bc|y^{\prime}z^{\prime},i)\;\forall
y,z,y^{\prime},z^{\prime},i.$ (68)

This optimisation is thought to be hard, since no efficient characterisation
of the set $\tilde{\mathcal{Q}}$ is known. However, it is known that the set
$\mathcal{Q}$ can be characterised from the outside by a sequence of semi-
definite programs, known as the NPA hierarchy [53]. Formally, one defines a
sequence of sets $\mathcal{Q}_{k}$ such that
$\mathcal{Q}_{1}\supseteq\mathcal{Q}_{2}\supseteq\cdots\supseteq\mathcal{Q}_{\infty}\supseteq\mathcal{Q}$
where membership in $\mathcal{Q}_{k}$ for any finite $k$ can be tested via an
(increasingly large) semi-definite program. This amounts to checking if a
matrix $\Gamma$—which is a linear function of the probabilities—is positive
semi-definite, and where the constraint $\Gamma_{00}=1$ defines the
normalisation of the distributions in the set. If one replaces this
normalisation constraint by a positivity constraint $\Gamma_{00}\geq 0$, one
obtains an analogous hierarchy for un-normalised distributions, whose sets we
denote $\tilde{\mathcal{Q}}_{k}$. One can thus relax the problem (6.2) by
relaxing the condition $\tilde{p}^{Q}_{BC}(bc|yz,i)\in\tilde{\mathcal{Q}}$ to
$\tilde{p}^{Q}_{BC}(bc|yz,i)\in\tilde{\mathcal{Q}}_{k}$ for some $k$:

$\displaystyle\text{find }\tilde{p}^{Q}_{BC}(bc|yz,i)\;\;\text{such that}$
$\displaystyle p(abc|xyz)=\sum_{i}D_{A}(a|x,i)\,\tilde{p}^{Q}_{BC}(bc|yz,i)$
$\displaystyle\tilde{p}^{Q}_{BC}(bc|yz,i)\in\tilde{\mathcal{Q}}_{k}\;\;\forall
i,\quad\sum_{bc}\tilde{p}^{Q}_{BC}(bc|yz,i)=\sum_{bc}\tilde{p}^{Q}_{BC}(bc|y^{\prime}z^{\prime},i)\;\forall
y,z,y^{\prime},z^{\prime},i.$ (69)

This problem now has only linear and positive semi-definite constraints and
can be solved via a semi-definite program. Since the problem is a relaxation,
if this semi-definite program is found to be infeasible, this implies
$p(abc|xyz)\notin\mathcal{L}_{Q}$, and so the state is entangled. As before,
in the case of infeasibility, the dual optimisation problem (also a semi-
definite program), provides a separating hyperplane that can be understood as
a DI entanglement witness.

##  7 Discussion

We have shown that the broadcasting scenario allows for a strong form of Bell
nonlocality and DI entanglement certification for a large class of two-qubit
states. It is desirable to know the limits of this method. For example, for
which values of $\alpha$ is the isotropic state broadcast nonlocal and is this
the entire range of entanglement? Note that in the simplest scenario of figure
3 where each party has two inputs and two outcomes, all facet inequalities
have been enumerated [29]. Numerical optimisation of these inequalities
suggests they do not lead to activation of the isotropic state. Stronger
examples than those presented here thus likely require more more inputs or
broadcasted parties. For the case of DI entanglement certification, we have
shown promising results that suggest DI entanglement certification may be
possible in the entire range of entanglement, and it would be interesting to
pursue this further. In both cases, the connection between multipartite Bell
inequalities and the broadcasting scenario may be useful, as was the case in
section 5 that makes use of the Klyshko-Belinskii Bell inequality. Another
line of investigation could involve generalising the broadcasting scenario so
that the transformation device has an input and output, which we have not
investigated in this work.

One could also consider other resources than non-signalling resources in the
definition of broadcast-nonlocality (equation (22)), as is done in the study
of multipartite non-locality [30, 54]. One possibility is to allow the
transformation device to prepare signalling resources as well as non-
signalling resources, although this is known to cause problems in multipartite
scenarios [30]. Note that this effectively means the local parties can be
considered as a single party (since they can communicate) and so this renders
the broadcasting scenario equivalent to the standard Bell scenario in which
the transformation and local parties’ measurements is equivalent to performing
a POVM measurement on the original bipartite state. A more interesting
possibility would be to consider the one-way signalling resources (see [54,
55]), which avoid the logical problems that arise with general signalling
resources. In this case, we have not established if our examples of activation
still hold. We note that for practical applications however, one should not
allow for signalling correlations. For example, if one is interested whether
or not the state can be replaced by a source of physical classical randomness
(that is to say, one in which the $\lambda$’s can be known by an experimenter,
such as a commercial random number generator) then any other physical system
produced that is conditioned on $\lambda$ must be non-signalling, since
otherwise it would allow the experimenter prepare signalling resources on
demand. Thus, it only makes sense to consider signalling resources if one
assumes that the $\lambda$’s are truly hidden variables, so that they cannot
be known even in principle. This is the case for example in the de
Broglie–Bohm theory.

The examples of Bell nonlocality activation of the isotropic state in section
4 are relative to projective measurements, since for $\alpha\leq 0.683$ the
isotropic state is known to have an LHV model for projective measurements.
Since it is unknown whether a model for general (POVM) measurements exists in
the same range, a natural question is whether an example of ‘genuine’
activation with respect to POVM measurements can be achieved in the
broadcasting scenario. This is in fact possible by leveraging a previously
known result of genuine hidden nonlocality [21], in which it is shown that
there exists a family of two qubit states that have an LHV model for general
measurements in the standard scenario, and which exhibit hidden nonlocality.
Following the connection between hidden nonlocality and the single-copy
sequential scenario discussed in section 4, one may do the following. Consider
a sequential scenario where Bob performs a sequence of two measurements. This
can be realised in the broadcasting scenario of figure 3 by performing the
first measurement at the transformation device, but sending the outcome to
Bob, who then outputs this value (he has no input). The post-measurement state
is then sent to Charlie. This process is conceptually equivalent to a
sequential measurement protocol in which the first measurement device performs
a single measurement, where Bob is simply used to output the value of this
measurement. Consequently, results in the hidden nonlocality scenario apply to
the broadcasting scenario. Thus, the example of hidden nonlocality relative to
POVM measurements also implies an example in the broadcasting scenario.
Nevertheless, it would be interesting to find stronger examples that rely
naturally on the structure of the broadcasting scenario.

Our results also suggest the possibility of strengthening existing DI
protocols. In particular, since we have shown that DI entanglement
certification of the isotropic state at significantly lower visibilities that
were previously known, one may hope that this could lead to higher levels of
noise tolerance of more complex protocols, such as DI quantum key
distribution, DI randomness extraction or self-testing. This warrants
additional investigation since—although the state can tolerate more noise—the
broadcasting protocol involves additional transformation and measurement
devices which themselves will compound further noise to the experimental
statistics, and the isometry transformations we find may be difficult to
implement in practice. Moving beyond simple entanglement certification, it may
also be possible to leverage the semi-definite hierarchy [45] to certify not
only entanglement, but quantify this via a lower bound on the state
negativity.

##  8 Conclusion

In this work we have shown that there exists a simple scenario that allows one
to reveal the non-classicality of a large class of two-qubit mixed states. The
non-classicality we study takes the form of (i) a type of Bell nonlocality
relative to no-signalling hidden variable models, that we call broadcast
nonlocality; and (ii) DI entanglement certification. In both cases we show
that the broadcasting scenario allows for examples of non-classicality that
surpass what is known for all previously studied scenarios that process a
single copy of the state. Our work raises many questions that we aim to answer
in a future publication, and suggests further applications to DI protocols.

##  9 Acknowledgements

We would like to thank Marco Túlio Quintino, Pei-Sheng Lin, Cristian Boghiu,
Marcus Huber, Yeong-Cherng Liang, Gláucia Murta, Charles Xu and Mateus Araújo
for valuable comments and discussions. FH acknowledges funding from the Swiss
National Fund (SNF) through the Early Postdoc Mobility fellowship
P2GEP2$\\_$181509\. JB and DC acknowledges funding from the Spanish MINECO
(Severo Ochoa SEV-2015-0522), Fundacio Cellex and Mir-Puig, Generalitat de
Catalunya (SGR 1381 and CERCA Programme). JB acknowledges funding from the AXA
Chair in Quantum Information Science. DC acknowledges the Ramon y Cajal
fellowship.

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##  Appendix A Linear optimisation of visibilities

Here we consider calculating the robustness to noise for behaviours generated
via a family of states with a linear parameter, with fixed local measurements
and channel(s). For example, consider the family of states

$\displaystyle\rho_{v}=v\rho_{ent}+(1-v)\rho_{noise}$ (70)

with $0\leq v\leq 1$, and where $\rho_{ent}$ is an entangled bipartite state,
while $\rho_{noise}$ is a separable state (for example, the isotropic state
(13) is already in that form). Fix a channel $\Omega_{B_{0}\rightarrow BC}$
and local measurements $A_{a|x},B_{b|y},C_{c|z}$. The behaviour is thus given
by

$\displaystyle p_{v}(abc|xyz)$ $\displaystyle=\operatorname{Tr}(A_{a|x}\otimes
B_{b|y}\otimes C_{c|z}\;\mathbb{1}\otimes\Omega_{B_{0}\rightarrow
BC}(\rho_{v}))$ $\displaystyle=v\;p_{ent}(abc|xyz)+(1-v)p_{noise}(abc|xyz),$
(71)

where

$\displaystyle p_{ent}(abc|xyz)=\operatorname{Tr}(A_{a|x}\otimes
B_{b|y}\otimes C_{c|z}\;\mathbb{1}\otimes\Omega_{B_{0}\rightarrow
BC}(\rho_{ent}))$ (72)

and similarly for $p_{noise}$. Since $p_{v}(abc|xyz)$ is linear with respect
to $v$, one can find the exact value $v^{*}$ for which the behaviour becomes
nonlocal using the following linear program (using (61)):

$\displaystyle\text{maximise}\;v\quad\text{such that}$ (73) $\displaystyle
v\;p_{ent}(abc|xyz)+(1-v)p_{noise}(abc|xyz)=\sum_{i,j}p_{ij}\;D_{A}(a|x,i)\;E_{BC}(bc|yz,j)\;\;\forall
a,b,c,x,y,z.$ $\displaystyle p_{i,j}\geq 0$

The optimal dual variables to the euqality constraints provide a Bell-type
inequality that is violated by $p_{v}(abc|xyz)$ for $v>v^{*}$.

##  Appendix B Heuristic optimisation of visibility

Here we detail the algorithm used for the heuristic minimisation of state
visibility. Consider again a one-parameter family of state (70). We want to
find the lowest visibility $v$ such that the state is nonlocal (in some
scenario), that is, the general task is

$\displaystyle\text{minimise}\;v$ (74)
$\displaystyle\text{s.t.}\;\rho_{v}=v\rho_{ent}+(1-v)\rho_{noise}\;\text{is
nonlocal}.$

In order to show that the state is nonlocal in the standard Bell scenario one
needs to violate a Bell inequality, i.e., one needs measurements
$\\{A_{a|x}\\}$ and $\\{B_{b|y}\\}$ such that behaviour
$p(ab|xy)=\operatorname{Tr}(A_{a|x}\otimes B_{b|y}\;\rho)$ violates a Bell
inequality. In the broadcasting scenario of e.g. figure 3 one needs to find a
map $\Omega_{B_{0}\rightarrow BC}$, and measurements $\\{A_{a|x}\\}$,
$\\{B_{b|y}\\}$ and $\\{C_{c|z}\\}$ such that behaviour

$\displaystyle p(abc|xyz)=\operatorname{Tr}(A_{a|x}\otimes B_{b|y}\otimes
C_{c|z}\;\mathbb{1}\otimes\Omega_{B_{0}\rightarrow BC}(\rho_{v}))$ (75)

violates an inequality tailored to the broadcasting scenario (for example
inequality (27)). Below, we show how to optimise the quantum value of such
inequalities.

###  B.1 Seesaw procedure for maximising inequality violation

Let us fix an inequality in the broadcasting scenario, for instance with three
parties as in figure 3 (where we have renamed the input/output labels):

$\displaystyle\sum_{a_{1}a_{2}a_{3}x_{1}x_{2}x_{3}}\mathcal{I}_{a_{1}a_{2}a_{3}}^{x_{1}x_{2}x_{3}}\;p(a_{1}a_{2}a_{3}|x_{1}x_{2}x_{3})\leq
L$ (76)

where the local bound $L$ depends on the underlying model one wants to rule
out (i.e. the scenarios of sections 4 or 5). For a fixed strategy, the quantum
value is given by

$\displaystyle\sum_{a_{1}a_{2}a_{3}x_{1}x_{2}x_{3}}\mathcal{I}_{a_{1}a_{2}a_{3}}^{x_{1}x_{2}x_{3}}\;\operatorname{Tr}(M^{(1)}_{a_{1}|x_{1}}\otimes
M^{(2)}_{a_{2}|x_{2}}\otimes
M^{(3)}_{a_{3}|x_{3}}\;\mathbb{1}\otimes\Omega_{B}(\rho_{AB}))$ (77)

Below we propose an algorithm to optimise the quantum violation over local
measurements and map(s), for a fixed inequality and state $\rho_{AB}$.

Seesaw for optimising inequality violation for fixed state

Assume one want to maximise the violation of a general inequality
$\mathcal{I}_{a_{1}a_{2}\cdots a_{n}}^{x_{1}x_{2}\cdots x_{n}}$ (in a general
broadcasting scenario) for a fixed state $\rho_{AB}$ and (for now) fixed
channels $\Omega_{A/B}$, and sets of measurements $M^{(1)}_{a_{1}|x_{1}}$,
$M^{(2)}_{a_{2}|x_{2}}$, … $M^{(n)}_{a_{n}|x_{n}}$. One can do the following.

  1. 1.

Fix randomly all sets of measurements

  2. 2.

Set $k$=1

  3. 3.

Optimise the inequality with respect to set of measurements
$M^{(k)}_{a_{k}|x_{k}}$, update variables $M^{(k)}_{a_{k}|x_{k}}$ accordingly

  4. 4.

Repeat point 3 for $k=2...n$

  5. 5.

Repeat point 2 - 4 until two successive values of the inequality are equal up
to some small precision $\epsilon$.

The key point is to notice that point 3 corresponds to optimising a linear
function, with SDP constraints (namely $M_{a_{k}|x_{k}}\geq 0$ and
$\sum_{a}M_{a_{k}|x_{k}}=\mathbb{1}\,\forall x_{k}$), that is, it can be
framed as a semi-definite program, for which there exist efficient algorithms
[56].

One also needs to optimise over the channel(s). This can be done via the Choi-
Jamiolkowski isomorphism [57]. Taking the case of figure 3 of section 4 as a
concrete example, for the map $\Omega_{B_{0}\rightarrow BC}$ one has

$\displaystyle\Omega_{B_{0}\rightarrow
BC}(\sigma_{B_{0}})=\operatorname{Tr}_{1}(\rho_{\Omega}(\sigma_{B_{0}}^{T}\otimes\mathbb{1}_{BC}))$
(78)

where $\sigma_{B_{0}}$ is some operator acting on $\mathcal{H}_{B_{0}}$, and
$\left|\Phi^{+}\right\rangle$ is the maximally entangled state of local
dimension $d=\text{dim}(\mathcal{H}_{B_{0}})$ so that
$\rho_{\Omega_{B_{0}\rightarrow
BC}}=d\cdot\mathbb{1}\otimes\Omega[|\Phi^{+}\rangle\langle\Phi^{+}|]$ is the
Choi state of map $\Omega_{B_{0}\rightarrow BC}$. The Choi state thus
satisfies $\rho_{\Omega}\geq 0$ and
$\operatorname{Tr}_{BC}(\rho_{\Omega})=\mathbb{1}_{B_{0}}$. That is, variable
$\rho_{\Omega}$ can be treated as an SDP variable. Furthermore, the reverse is
true: any operator satisfying the SDP conditions is the Choi state of some
channel. The probabilities can thus be written

$\displaystyle
p(abc|xyz)=\operatorname{Tr}((A_{a|x}\otimes\mathbb{1}_{B_{0}}\otimes
B_{b|y}\otimes
C_{c|z})\;(\mathbb{1}_{A}\otimes\rho_{\Omega})(\rho_{AB_{0}}\otimes\mathbb{1}_{BC}))$
(79)

This expression is linear in the Choi state, which means for fixed state and
measurements, the inequality can be maximised over all quantum channels of the
given dimension. A similar procedure allows the same to be done for Alice’s
channel (if she has one). Combining this with the above method to optimise the
measurements, one can ‘see-saw’ between the two until convergence has been
reached, which in practice performs well for simple scenarios.

###  B.2 Global heuristic procedure

Relying on the previous section and on a seesaw procedure described in
Appendix B.1, we describe here a global heuristic procedure to minimise the
critical visibility of a fixed family of states.

  1. 1.

Set random measurements $\\{M_{a_{i}|x_{i}}\\}$ and channels $\Omega_{A}$,
$\Omega_{B}$

  2. 2.

Compute $v^{*}$ (using the linear program (73))

  3. 3.

Repeat step 1-2 until $v^{*}<1$, and keep the corresponding inequality
$\mathcal{I}$

  4. 4.

Set k=0

  5. 5.

Optimise the violation of $\mathcal{I}_{0}$ for state $\rho_{v^{*}}$ (using
seesaw from B.1 )

  6. 6.

Compute new $v^{*}$ using optimal measurements and map found in 5 (using
linear program (73)), and keep corresponding inequality $I_{(k+1)}$

  7. 7.

Set $k=k+1$

  8. 8.

Repeat steps 5-7 until two successive $v^{*}$ are equal up to some small
precision.

Using this procedure in the symmetric case described in 4.2, with the two-
qubit isotropic state and using $3$ inputs for parties A and C, $2$ for B and
D, and $2$ outputs for each, we found a violation for $\alpha\simeq 0.559$,
showing that activation for values of $\alpha$ lower than $1/\sqrt{3}$ is
indeed possible in the broadcasting scenario.

In this example, the two local channels applied to the state are identical and
are again given by (36), while the local measurements as well as the optimal
inequality are given as auxiliary files. We suspect that even stronger
examples could be found with further investigation, and it is still unknown
whether activation in the entire range of entanglement is possible.