# No-broadcasting theorem for non-signalling boxes and assemblages

Carlos Vieira<EMAIL_ADDRESS>Departamento de Matemática
Aplicada, Instituto de Matemática, Estatística e Computação Cientıfica,
Universidade Estadual de Campinas, 13083-859, Campinas, São Paulo, Brazil
Adrian Solymos<EMAIL_ADDRESS>Wigner Research Centre for Physics,
H-1525 Budapest P.O. Box 49, Hungary  Eötvös Loránd University, Pázmány Péter
sétány 1/A 1117 Budapest, Hungary  Cristhiano Duarte<EMAIL_ADDRESS>School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom  International Institute of Physics, Universidade Federal do Rio
Grande do Norte, 59070-405 Natal, Brazil  Institute for Quantum Studies,
Chapman University, One University Drive, Orange, CA, 92866, USA  Zoltán
Zimborás<EMAIL_ADDRESS>Wigner Research Centre for Physics, H-1525
Budapest P.O. Box 49, Hungary  Eötvös Loránd University, Pázmány Péter sétány
1/A 1117 Budapest, Hungary  Algorithmiq Ltd, Kanavakatu 3C 00160 Helsinki,
Finland

###### Abstract

The no-broadcasting theorem is one of the most fundamental results in quantum
information theory; it guarantees that the simplest attacks on any quantum
protocol, based on eavesdropping and copying of quantum information, are
impossible. Due to its fundamental importance, it is natural to ask whether it
is an inherent quantum property or holds also for a broader class of non-
classical theories. A relevant generalization is to consider non-signalling
boxes. Subsequently Joshi, Grudka and Horodecki⊗4 conjectured that one cannot
locally broadcast nonlocal boxes. In this paper, we prove their conjecture
based on fundamental properties of the relative entropy of boxes. Following a
similar reasoning, we also obtain an analogous theorem for steerable
assemblages.

##  I Introduction

Among the many no-go theorems in quantum information theory, it is fair to
assume that the most famous are the no-cloning and the no-broadcasting
theorems. Broadly speaking, the no-cloning theorem states that there is no
universal quantum cloning machine that can always create two exact copies from
unknown states $\rho$ [1]. To a certain extent, the no-broadcasting result can
be thought of as a generalization of the no-cloning theorem. In general lines,
the impossibility of broadcasting quantum information can be seen as a
fundamental limit, a statement showing that there is no universal quantum
channel such that given an unknown quantum state $\rho$, it produces a
bipartite quantum state whose either marginal coincides with the original
state $\rho$ [2, 3]. An even stronger formulation of the no-broadcasting
theorem states that given two non-commuting quantum states $\rho_{1}$ and
$\rho_{2}$, there is no quantum channel that could broadcast both of them [2].

No-cloning and no-broadcasting play a vital role in fields such as quantum
cryptography, as they show that eavesdroppers are unable to perfectly copy
unknown quantum information [4, 5, 6] transmitted between two sites.
Additionally, these no-go theorems together with the Heisenberg uncertainty
principle can be used to obtain a limit to the amount of information an agent
can learn from a single copy of a quantum state [7].

Following the earlier results, a generalized no-broadcasting theorem was
demonstrated for any non-classical finite-dimensional probabilistic model
satisfying a no-signalling criterion [8]. Therefore, it can be said, that the
impossibility put forward by the standard versions of the no-broadcasting
theorem is not distinctive to quantum theory, as it is also present in broader
non-classical theories as well.

Alternative and initially counterintuitive variations of the no-broadcasting
theorem have also been discussed in the literature: local versions of the
theorem are perhaps the best example. In these variations, the focus is on the
potential broadcasting of a known bipartite quantum states using local
operations. It turns out that only separable states can be broadcast using
LOCC (local operations and classical communication) [9, 10, 11]. If one
considers only LO (local operations), then an even stronger restriction is
imposed: nothing other than classical-classical bipartite states can be
broadcast [12]. Later it was shown that this kind of broadcasting is
equivalent to the original one in the case of quantum theory [13, 14].

In [11], the authors considered yet another variant of the broadcasting
scenario. Their idea was to explore local broadcasting of bipartite non-
signalling boxes with binary inputs and outputs—put another way, boxes in the
usual $(2,2,2)$ Bell-scenario. They consider a class of operations that
transform local boxes into local ones (ones that admit a LHV model) and prove
that a nonlocal box cannot be broadcast in this scenario if we are restricted
to these transformations. The unfortunate drawback of ref. [11] is that their
proof heavily relies on some properties of the $(2,2,2)$ scenario. Our work
draws from that scholarship as we want to answer the question they raise: ‘is
there a no-broadcasting theorem in the general scenario?’, or in other words,
‘is there a proof for general boxes?’.

A similar question also arises regarding the case of assemblages [15, 16, 17].
We have discussed that the no-broadcasting theorem is not an exclusivity of
quantum theories and that alternative local versions exist both for quantum
states and for boxes. In a rough approximation, assemblages can be thought of
as an object halfway between bipartite quantum states and boxes—assemblages
combine probabilities and quantum states. Inspired by this crude
approximation, a natural line of inquiry is the question of whether or not it
is possible to broadcast with the appropriate transformations certain classes
of assemblages. In this work, we prove a general local no-broadcasting theorem
for boxes and also for assemblages using fundamental properties of the
relative entropy of nonlocality and the relative entropy of steering.

##  II No-broadcasting for boxes

In this section, we will state and prove a theorem about the impossibility of
broadcasting nonlocal boxes (Thm. 1). Before we state our main result, we
introduce what we mean by correlation scenarios, boxes, local boxes, and the
like. For a more extensive introduction, we suggest ref. [18].

###  II.1 Correlation scenarios

Fig. 1: Pictorial representation of a (box) correlation scenario. Measurements
are represented by the colored buttons on the top of each box. A single
outcome is returned after a measurement is performed, and schematically one
light bulb goes off (colors online).

Correlation scenarios are usually formulated in a device-independent language.
Think of it as a collection of $N$ boxes. Each box has $m$ inputs, and for
each input, there are $o$ possible associated outputs (see Fig. 1). The
framework is formulated to hide the inner physical mechanism of each box. As
we do not have access to the physical details producing an outcome given that
a certain button was pressed, the only description for this $(N,m,o)$-scenario
is via the aggregated joint statistics called the behavior:

$\textbf{P}=\\{P(ab...c|xy...z)\\}_{ab...c|xy...z}\in\mathbb{R}^{(om)^{N}}.$
(1)

Each $P(ab...c|xy...z)$ means the joint probability of obtaining the outcome
$a$ out of the first box when the $x$ button is pressed, and outcome $b$ out
of the second box when the $y$ button is pressed, …, and outcome $c$ out of
the $N$th-box when button $z$ is pressed. Fig. 1 shows a pictorial
representation of this situation.

###  II.2 Local Boxes

######  Definition 1.

In a given $(N,m,o)-$scenario, we say that a behavior
$\textbf{P}=\\{P(ab...c|xy...z)\\}_{ab...c|xy...z}\in\mathbb{R}^{(om)^{N}}$ is
_local_ whenever it satisfies

$\displaystyle
P(ab...c|xy...z)=\sum_{\lambda}P(a|x,\lambda)P(b|y,\lambda)...P(c|z,\lambda)q(\lambda),$
(2)

for all measurements and for all outcomes, with $q(\lambda)$ a probability
distribution over an exogenous variable.

In sum, locality for boxes says that there is a hidden variable we do not have
access to that fully explains the correlation across the boxes, or, more
succinctly, that there is a _classical_ explanation for such correlations.
That there are more than classical and more than quantum correlations is not
immediate, but they exist. The next sections address precisely these cases.

###  II.3 Quantum Boxes

######  Definition 2.

In a given $(N,m,o)-$scenario, we say that a behavior
$\textbf{P}=\\{P(ab...c|xy...z)\\}_{ab...c|xy...z}\in\mathbb{R}^{(om)^{N}}$ is
_quantum_ whenever it satisfies:

$\displaystyle
P(ab...c|xy...z)=\tr(\Pi_{a}^{x}\otimes\Pi_{b}^{y}\otimes...\otimes\Pi_{c}^{z}\rho_{A_{1}A_{2}...A_{N}}),$
(3)

where $\\{\Pi_{a}^{x}\\}$, $\\{\Pi_{b}^{y}\\}$, …, $\\{\Pi_{c}^{z}\\}$ are
POVMs and $\rho_{A_{1}A_{2}...A_{N}}$ is a density operator acting on
$\bigotimes_{i=1}^{N}\mathcal{H}_{A_{i}}$.

While the set of local correlations is a convex polytope, the set of quantum
correlations has a far more complex geometry. It is a convex set that properly
contains the local polytope, but its exact shape is only known in certain
regions and in certain scenarios. For an in-depth analysis of the problem, we
refer to [19]. Behaviors that can be obtained by post-quantum theories also
exist, and although they have never been realized in the laboratory, their
study still can enlighten the nature of physical theories and what they all
have in common. The best example is given by non-signalling correlations that
we go on to discuss now.

###  II.4 Non-Signalling Boxes

For sake of simplicity, we introduce the notion of non-signalling only for two
boxes. All of the reasoning here can be easily extended for larger scenarios
with $N>2$ boxes. We also refer to ref. [20], where non-signalling is
motivated and rigorously defined for correlation scenarios with more than two
boxes.

######  Definition 3.

In a given $(2,m,o)-$scenario, we say that a behavior
$\textbf{P}=\\{P(ab|xy)\\}_{ab|xy}\in\mathbb{R}^{(om)^{2}}$ is _non-
signalling_ whenever it satisfies

$\displaystyle\sum_{b}P(ab|xy)\eqqcolon
P(a|x)\coloneqq\sum_{b}P(ab|xy^{\prime});\,\,\forall\,\,a,x,y,y^{\prime}\text{,}$
(4) $\displaystyle\sum_{a}P(ab|xy)\eqqcolon
P(b|y)\coloneqq\sum_{a}P(ab|x^{\prime}y);\,\,\forall\,\,b,y,x,x^{\prime}.$ (5)

Remark. Simply saying, one may want to consider non-signalling behaviors as
those behaviors for which all marginal probabilities are well-defined. For the
$(2,m,o)$ case, it is equivalent of saying that
$P(a|x,y)=P(a|x)=P(a|x,y^{\prime})$ for all choices of measurements and
outcomes.

###  II.5 KL-divergence

We prove our main result for boxes using information measures. In general
lines, our argument is the following: on the one hand, we will argue that
these measures are monotones for a set of free transformation. On the other
hand, we will show that broadcasting-like transformations increase the
quantity of resource, so they are not allowed. To do so, we need, first and
foremost, to decide upon an information measure. In this paper, we work with
well-adapted extensions of the Kullback-Leibler divergence.

The Kullback-Leibler divergence (also called relative entropy) is an
information-based measure of disparity among probability distributions [21,
22]. Formally, given two probability distributions $p(a)$ and $q(a)$, the
Kullback-Leibler divergence (KL-divergence) of $p(a)$ from $q(a)$ is denoted
by $S(p(a)||q(a))$ and defined as:

$S(p(a)||q(a))\coloneqq\sum_{a}p(a)\log\frac{p(a)}{q(a)},$ (6)

where the index $a$ runs over the alphabet that $p$ and $q$ are defined over.
The KL-divergence is $(i)$ non-negative; $(ii)$ it is zero if, and only if,
the distributions match exactly; $(iii)$ it is not symmetric in $p(a)$ and
$q(a)$; and can potentially be equal infinity. The KL-divergence is, in other
words, a type of statistical distance: a measure of how one probability
distribution $p(a)$ is different from a second, reference probability
distribution $q(a)$. Especially, the KL-divergence measures the information
lost when $q(a)$ is used to approximate $p(a)$ [23]. It plays a central role
in the theory of statistical inference [24].

Given its extreme importance, it is natural to study extensions of KL-
divergence to other objects. For correlation scenarios, a possible
generalization was introduced in [25, 26]. The main idea is to associate a
single probability distribution to a box, once this is done, we use the KL-
divergence defined in eq. (6) to compare two boxes. More concretely, if we
consider that the probability distribution of Alice’s and Bob’s choices of
inputs is given by $\pi(x,y)$, then $P(a,b,x,y)\coloneqq\pi(x,y)P(a,b|x,y)$ is
the probability of the event where the input choices were $x,y$ and the
outputs were $a,b$. Therefore, we can compare two boxes $P(a,b|x,y)$,
$Q(a,b|x,y)$ through the KL-divergence between $\pi(x,y)P(a,b|x,y)$ and
$\pi(x,y)Q(a,b|x,y)$. More generally, if we want to measure how different two
boxes are, it is natural to grant Alice and Bob the freedom to choose their
inputs according to any probability distribution available to them, and then
single out the one that optimizes this distance. That process can be used to
define the KL-divergence for boxes between $P(a,b|x,y)$, and $Q(a,b|x,y)$ as:

$\displaystyle S_{b}(P(a,b|x,y)||Q(a,b|x,y))$
$\displaystyle\coloneqq\sup_{\pi(x,y)}S(\pi(x,y)P(a,b|x,y)||\pi(x,y)Q(a,b|x,y)),$
(7)

where the supremum is taken over all possible discrete probability
distributions over Alice and Bob’s inputs.

Remarkably, the expression in eq. (7) can be simplified. As shown in ref.
[26], the optimal probability $\pi^{\ast}(x,y)$ is such that the KL-divergence
for boxes can be expressed as:

$\displaystyle S_{b}(P(a,b|x,y)||Q(a,b|x,y))$
$\displaystyle=\max_{x,y}S(P(a,b|x,y)||Q(a,b|x,y)).$ (8)

###  II.6 Broadcasting of a box

We start by recalling the standard definition of broadcasting of quantum
states [2]. Given a quantum state $\rho\in\mathcal{L}(\mathcal{H})$, we say
that $\rho_{XY}\in\mathcal{L}(\mathcal{H}\otimes\mathcal{H})$ is a
broadcasting of $\rho$ when both reduced state of $\rho_{XY}$ is equal to
$\rho$. Put another way, $\tr_{X}(\rho_{XY})=\rho=\tr_{Y}(\rho_{XY})$. It is
worth emphasizing that broadcasting is a generalization of cloning, where we
now allow for the possibility of correlations between the two parts.

An analogous definition for boxes can be crafted in the following way:
consider a scenario with four agents, labelled $A_{0},A_{1},B_{0},B_{1}$,
where $x_{i}\in\mathcal{X}$ and $a_{i}\in\mathcal{A}$ are the inputs and
outputs for the agents $A_{i}$, respectively. Similarly, $y_{i}\in\mathcal{Y}$
and $b_{i}\in\mathcal{B}$ are the inputs and outputs for the agents $B_{i}$,
respectively. The aggregated statistics for the experiment is described by the
behavior $P(a_{0},a_{1},b_{0},b_{1}|x_{0},x_{1},y_{0},y_{1})$. We say that the
agents are realizing a broadcasting of a box $P^{\prime}(a,b|x,y)$ when the
marginal statistics for the pair of agents $A_{0},B_{0}$ is equal to the
marginal statistics for the pair $A_{1},B_{1}$ which must be, in turn, equal
to $P^{\prime}(a,b|x,y)$. We summarize this discussion in the definition
below.

######  Definition 4.

We say that $P(a_{0},a_{1},b_{0},b_{1}|x_{0},x_{1},y_{0},y_{1})$ is a
broadcasting of $P^{\prime}(a,b|x,y)$ if:

$\sum_{a_{1},b_{1}}P(a_{0},a_{1},b_{0},b_{1}|x_{0},x_{1},y_{0},y_{1})=P^{\prime}(a_{0},b_{0}|x_{0},y_{0}),$
(9a) and
$\sum_{a_{0},b_{0}}P(a_{0},a_{1},b_{0},b_{1}|x_{0},x_{1},y_{0},y_{1})=P^{\prime}(a_{1},b_{1}|x_{1},y_{1}),$
(9b)

where the equations above must be true for every choice of inputs and outputs.

Recalling the usual notion of non-signalling behaviors, we should notice that
eq. (9) implies that $P(a_{0},a_{1},b_{0},b_{1}|x_{0},x_{1},y_{0},y_{1})$ is
non-signalling with respect to the partition $A_{0}B_{0}|A_{1}B_{1}$.

Remark: In order to not overload the notation, we will usually denote
$(a_{0},a_{1})$ by $\bm{a}$ and similarly for $\bm{b},\bm{x}$ and $\bm{y}$.
Nonetheless, we will alternate between those two notations when we feel it is
convenient.

###  II.7 Nonlocality in the broadcasting scenario

Although our broadcasting scenario could be seen as a quadripartite situation,
the concept of locality we will use is definitively bipartite, for it is
related to the bipartition $A_{0}A_{1}|B_{0}B_{1}$. This partition is not
artificial, as the resource is spatially divided between $A_{0},A_{1}$ on one
side and $B_{0},B_{1}$ on the other. With that in mind, we define the set of
Local-Realistic Non-Signalling boxes [11]. This set consists of boxes that are
local on the partition $A_{0}A_{1}|B_{0}B_{1}$, and in addition, the local
decomposition has non-signalling components between $A_{0}$ and $A_{1}$ and
between $B_{0}$ and $B_{1}$. The second assumption is also natural since we
are interested in the broadcasting boxes, so the marginals $A_{0}B_{0}$ and
$A_{1}B_{1}$ must be well defined. More formally,

######  Definition 5.

A behavior $Q(\bm{a},\bm{b}|\bm{x},\bm{y})$ is local realistic non-signalling
if it can be decomposed as:

$\displaystyle Q(a_{0},a_{1},b_{0},b_{1}|x_{0},x_{1},y_{0},y_{1})$
$\displaystyle=\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})Q_{\lambda}(b_{0},b_{1}|y_{0},y_{1}),$
(10)

where $Q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})$ and
$Q_{\lambda}(b_{0},b_{1}|y_{0},y_{1})$ are non-signalling for all $\lambda$.
The set of all boxes with such properties will be denoted by
$\mathrm{LR_{ns}}$.

Now that we have introduced the set we will be interested in the most, we can
ask whether or not a box belongs to this set. In the negative case, when a
behavior does not belong to that set, it is fair to be still interested in how
“distant” this box is from the elements of that set. To answer, we can use the
KL-divergence for correlations scenarios, which, as discussed, compares how
different two boxes are. Using the KL-divergence for boxes, we introduce the
relative entropy of nonlocality in the following way [26]:

######  Definition 6.

The relative entropy of nonlocality of a behavior
$P(\bm{a},\bm{b}|\bm{x},\bm{y})$ is given by:

$E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))\coloneqq\inf_{Q\in\mathrm{LR_{ns}}}S_{\mathrm{b}}(P(\bm{a},\bm{b}|\bm{x},\bm{y})||Q(\bm{a},\bm{b}|\bm{x},\bm{y})).$
(11)

Remark: The KL-divergence for correlation scenarios is not a proper distance
measure, as it is not symmetric. Nevertheless, it is fair to say that the
relative entropy of nonlocality defined by eq. (11) measures how distinct
$P(\bm{a},\bm{b}|\bm{x},\bm{y})$ is from the probability distributions that
admit a local-realistic hidden model.

###  II.8 Transforming boxes into boxes

####  II.8.1 LOSR-Tranformations

A significant part of our work is dedicated to the (im)possibility of
broadcasting boxes—in the sense of def. 4. To do so, we need to specify which
set of box transformations we will be considering and, more importantly, why.
Naturally, we will work with transformations that map a bipartite system $AB$
into a quadripartite system $A_{0}A_{1}B_{0}B_{1}$. Nonetheless, we still want
to keep some sort of locality, so that we will deal with transformations that
can be implemented without any communication between Alice and Bob. Finally,
in our framework, any kind of classical correlation between the parts is
allowed. This class of operations is called LOSR (Local Operations and Shared-
Randomness) transformations [27]. A useful characterization of these
transformations is given by the following definition [26].

######  Definition 7.

A map $\mathcal{M}$ from $AB$ to $A_{0}A_{1}B_{0}B_{1}$ is a LOSR
transformation if there exist local boxes $I(x,y|\bm{x},\bm{y})$ and
$O(\bm{a},\bm{b}|\bm{x},x,a,\bm{y},y,b)$ such that:

$\displaystyle
P(\bm{a},\bm{b}|\bm{x},\bm{y})=\mathcal{M}(P^{\prime}(a,b|x,y))$
$\displaystyle=\sum_{x,y,a,b}I(x,y|\bm{x},\bm{y})P^{\prime}(a,b|x,y)O(\bm{a},\bm{b}|\bm{x},x,a,\bm{y},y,b).$
(12)

Fig. 2: Illustration of a LOSR transformation of a box. The box $P^{\prime}$
is in green in the middle, with two wavy lines between the two parts
representing possibly non-classical correlations, its inputs are $x$ and $y$,
while the outputs are $a$ and $b$. The red boxes represent the input box $I$,
while the blue ones the output box $O$. They have single wavy lines between
them to represent classical correlations. The final box has inputs
$x_{0},x_{1},y_{0},y_{1}$ and outputs $a_{0},a_{1},b_{0},b_{1}$ (colors
online).

An illustration of LOSR transformations is depicted in Fig. 2.

An important property of LOSR transformations is that it transforms local
boxes into local boxes, i.e, if $P(a,b|x,y)$ is local, then
$P_{f}(\bm{a},\bm{b}|\bm{x},\bm{y})=\mathcal{M}(P(a,b|x,y))$ will be also
local [27]. It should be highlighted that the set of transformations that
preserves the local boxes is a larger set than that of LOSR transformations.
The standard example is WPICC maps, these are transformations that preserve
locality, but their implementation requires communication between the parties
[25, 26]. Despite being a subset of the transformations that preserve
locality, as discussed in [27], the set of LOSR transformations is the most
natural set of “free” transformations in the context of nonlocality. This
comes from the fact that its implementation only uses resources that preserve
the initial causal structure of the scenario, namely, classical correlations
and local operations.

####  II.8.2 LOSR Transformations and KL-Divergence

An interesting property of the KL-divergence is that it is contractible over
the set of LOSR transformations [26], i.e, if $P(a,b|x,y),Q(a,b|x,y)$ are two
non-signalling boxes and $\mathcal{M}$ is a LOSR transformation, then

$\displaystyle
S_{b}\left(\mathcal{M}(P(a,b|x,y))||\mathcal{M}(Q(a,b|x,y))\right)$
$\displaystyle\leq S_{b}(P(a,b|x,y)||Q(a,b|x,y)).$ (13)

As discussed in the previous section, for the problem of broadcasting, we are
interested in the set of local boxes whose local decomposition is given by
non-signalling components. With this in mind, the transformations that we will
consider must also preserve the set of local realistic boxes in addition to
being LOSR. Informally, we can summarize this discussion in the definition
below.

######  Definition 8.

A box transformation is said to be an $LR_{ns}$-LOSR transformation when $(i)$
it belongs to the set of LOSR transformations, and in addition, $(ii)$ it
transforms boxes from $LR_{ns}$ into boxes in $LR_{ns}$.

###  II.9 No-broadcasting theorem for nonlocal boxes

This section deals with our main no-go theorem for boxes. We show that it is
impossible to broadcast a nonlocal box using only $\mathrm{LR_{ns}}$-LOSR
transformations. Our argument is by contradiction. In fact, we will establish
that if it were possible to broadcast a nonlocal box $P^{\prime}(a,b|x,y)$ by
$\mathrm{LR_{ns}}$-LOSR transformations, then we would have
$E_{\mathrm{LR}}(P^{\prime}(a,b|x,y))<E_{\mathrm{LR}}(P^{\prime}(a,b|x,y))$,
which is a contradiction.

The proof will be divided into two parts, which we will state as propositions.
First, we will show that the relative entropy of nonlocality is contractive
over the set of $\mathrm{LR_{ns}}$-LOSR transformations.

######  Proposition 1.

If $\mathcal{M}$ is a $LR_{ns}$-LOSR transformation, then

$E_{\mathrm{LR}}(\mathcal{M}(P(a,b|x,y))\leq
E_{\mathrm{LR}}(P(a,b|x,y))\text{.}$ (14)

The proof of the Proposition above is mainly based on two special properties.
One is the fact that $\mathrm{LR_{ns}}$-LOSR transformations preserve the set
of $\mathrm{LR_{ns}}$ boxes. The other is the contractivity of the boxes KL-
divergence over LOSR transformations (see eq. (II.8.2)). Details of the proof
are given in Appendix A.

The second proposition tells us that the amount of nonlocality that a
broadcasting from a nonlocal box has is greater than the initial box. In fact,

######  Proposition 2.

If $P(\bm{a},\bm{b}|\bm{x},\bm{y})$ is a broadcasting of a nonlocal box
$P^{\prime}(a,b|x,y)$, then

$E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))>E_{\mathrm{LR}}(P^{\prime}(a,b|x,y)).$
(15)

The proof of this proposition is more elaborate and the details are given in
Appendix B. To summarize, we adapt the well-known chain rule of probability
distributions for the box case. The chain rule is an identity between the KL-
divergence of a joint probability distribution and the KL-divergence of one of
the marginals of this distribution plus the mean of the KL-divergence of the
conditional distributions (see eq. (B)). From this identity, we could recover
the relative entropy of the marginal distribution as the first term of the
Chain Rules of the expression. On the other hand, it was possible to prove if
$P^{\prime}(a,b|x,y)$ is nonlocal, then the second term is strictly positive.
Concluding therefore that
$E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))>E_{\mathrm{LR}}(P^{\prime}(a,b|x,y))$.

We can now establish one of the main results of this work.

######  Theorem 1 (No-broadcasting for boxes).

It is impossible to broadcast nonlocal boxes using LRns-LOSR transformations.

###### Proof.

Let’s suppose, by contradiction, that there is a $\mathrm{LR_{ns}}$-LOSR
transformation $\mathcal{M}$ that broadcasts a nonlocal box
$P^{\prime}(a,b|x,y)$. Then, by Proposition 1, we have:

$E_{\mathrm{LR}}\left(\mathcal{M}(P^{\prime}(a,b|x,y))\right)\leq
E_{\mathrm{LR}}(P^{\prime}(a,b|x,y))$ (16)

On the other hand, as $\mathcal{M}(P^{\prime}(a,b|x,y))$ is a broadcasting of
$P^{\prime}(a,b|x,y)$, by Proposition 2:

$E_{\mathrm{LR}}(P^{\prime}(a,b|x,y))<E_{\mathrm{LR}}\left(\mathcal{M}(P^{\prime}(a,b|x,y))\right).$
(17)

Therefore, by equations (16) and (17):

$E_{\mathrm{LR}}(P^{\prime}(a,b|x,y))<E_{\mathrm{LR}}(P^{\prime}(a,b|x,y)),$
(18)

which is a contradiction. Thus, it is impossible to broadcast nonlocal boxes
by $\mathrm{LR_{ns}}$-LOSR transformations. ∎

##  III No-broadcasting for steerable assemblages

In this section, we show that steerable assemblages _cannot_ be broadcast by
local operations. To prove this claim, we adopt a strategy essentially
analogous to the one we implemented for boxes. However, several adaptations
are needed. In a bipartite steering scenario, the situation is described by a
collection of ensembles of quantum states on Bob’s side together with a
conditional probability distribution (boxes) on Alice’s. There is, then, right
at the start, a combination of states and probabilities. This combination is
reflected in our proof, for we are putting together the previous section’s
strategies and the strategies from the no-local-broadcasting theorem for
quantum states [28].

We start by recalling the basic ideas behind quantum steering—more extensive
discussions abound, and we refer to [15, 16, 17]. Then, again based on the KL-
divergence, we elaborate on a definition of relative entropy appropriate for
the steering scenario. Following that, we address a set of physically
motivated transformations that will become the basis of our analysis for the
no-go theorem we conclude this section with.

###  III.1 Unsteerable assemblages in the Broadcasting scenario

Similarly to the correlation scenarios, steering scenarios are usually
determined by the number of agents involved in the process, how those agents
are grouped, and also by what resources those agents have access to. In a
situation with multiple agents, one can think of granting a fraction of those
agents tomographically complete measurements, whereas another fraction could
be granted only boxes. The former group can talk about quantum states and
measurements, and the latter can only provide aggregated statistics. Similar
to Bell scenarios, the central purpose of steering theory is to describe
correlations between the parties involved.

For concreteness, consider the bipartite scenario $AB$, involving two parts
(two agents), namely $A$ and $B$. The characterization of the part $B$ is
given by a family of ensembles of quantum states, whereas the description of
the part $A$ is given by conditional probabilities $p(a|x)$. Each input of
part $A$ is associated with an ensemble of the part $B$, and each output is
associated with a member state of such an ensemble. In fact, if part $A$ has
inputs $x\in[r]$ and outputs $a\in[s]$, the physical description of the system
on $B$’s side for the pair $a|x$ is given by a quantum state
$\rho_{a|x}\in\mathcal{L}(\mathcal{H}_{B})$. We can join both descriptions
using the idea of assemblage. An assemblage is a collection of unnormalized
density matrices
$\\{\varrho_{a|x}\\}_{a,x}\subset\mathcal{L}(\mathcal{H}_{B})$, where
$\varrho_{a|x}=p(a|x)\rho_{a|x}$.

We will restrict ourselves to non-signalling assemblages, i.e., those
assemblages where the reduced state of the part $B$ does not depend on the
input choice of $A$. More formally, an assemblage $\\{\varrho_{a|x}\\}$ is
non-signalling if $\sum_{a}\varrho_{a|x}=\rho_{B},\forall x$ [17].

As we mentioned, analogously to Bell scenarios, the central purpose of the
steering framework is to describe correlations between the parties involved.
Due to the nature of the framework, this analysis is performed on the
assemblages, rather than merely on the probability distributions arising from
boxes—the behaviors. An assemblage is said _unsteerable_ if the correlations
between $A$ and $B$ can be generated from a classical latent variable. In this
case, we say that the assemblage admits a local hidden state (LHS) model. More
formally, an assemblage $\\{\varrho_{a|x}\\}$ is _unsteerable_ if each element
can be written as

$\varrho_{a|x}=\sum_{\lambda}r(\lambda)p_{\lambda}(a|x)\rho_{\lambda},$ (19)

where $\rho_{\lambda}$ are density matrices in $\mathcal{L}(\mathcal{H}_{B})$
and $r(\lambda)$, $p_{\lambda}(a|x)$ are probability distributions.
Assemblages that are not unsteerable are said to be _steerable_ [16]. We
should note the similarity between the definition of local boxes, eq. (2), and
unsteerable assemblages, eq. (19). Both definitions follow the same idea, that
correlations between parts can be generated from, and mediated by, a classical
latent variable $\lambda$.

An alternative representation of an assemblage is given by a set of quantum
states
$\\{\rho_{AB}(x)\\}_{x}\subset\mathcal{D}(\mathcal{H}_{E}\otimes\mathcal{H}_{B})$,
given by:

$\displaystyle\rho_{AB}(x)$
$\displaystyle=\sum_{a=1}^{s}\outerproduct{a}{a}\otimes\varrho_{a|x}$
$\displaystyle=\sum_{a=1}^{s}p(a|x)\outerproduct{a}{a}\otimes\rho_{a|x}.$ (20)

Furthermore, if we take into account the probability distribution of the
choice of inputs of the part $A$ we can even associate a unique quantum state
to an assemblage. Assuming that $\pi(x)$ is such a probability distribution,
we define
$\rho_{XAB}\in\mathcal{L}(\mathcal{H}_{E^{\prime}}\otimes\mathcal{H}_{E}\otimes\mathcal{H}_{B})$
to be the quantum state associated with an assemblage $\varrho_{a|x}$ and a
probability distribution $\pi(x)$ as:

$\displaystyle\rho_{XAB}$
$\displaystyle=\sum_{x=1}^{r}\pi(x)\outerproduct{x}{x}\otimes\rho_{AB}(x)$
$\displaystyle=\sum_{x=1}^{r}\sum_{a=1}^{s}\pi(x)p(a|x)\outerproduct{x}{x}\otimes\outerproduct{a}{a}\otimes\rho_{a|x}.$
(21)

The sets of vectors $\\{\ket{x}\\},\\{\ket{a}\\}$ are orthonormal basis of the
auxiliary Hilbert spaces $\mathcal{H}_{E^{\prime}},\mathcal{H}_{E}$. The
states $\ket{x},\ket{a}$ are merely abstract flag states to express the inputs
and outputs of the part $A$, respectively. They do not describe the system
inside the box on $A$’s side [29, 30].

Now we have the ingredients to introduce the central notion of broadcasting
for assemblages. Let’s start with an assemblage
$\\{\varrho^{\prime}_{a|x}\\}\subset\mathcal{L}(\mathcal{H}_{B})$, where
$x\in[r]$ and $a\in[s]$ labels the inputs and outputs on $A$’s side
respectively. Let
$\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}\in\mathcal{L}(\mathcal{H}_{B_{0}}\otimes\mathcal{H}_{B_{1}})$
be another assemblage, where $x_{0},x_{1}\in[r]$, $a_{0},a_{1}\in[s]$ and
$\mathcal{H}_{B_{0}}=\mathcal{H}_{B}=\mathcal{H}_{B_{1}}$. We say that
$\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}$ is a broadcast of
$\\{\varrho^{\prime}_{a|x}\\}$ if the marginals $A_{0}B_{0}$ and $A_{1}B_{1}$
of $\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}$ are equal to
$\\{\varrho^{\prime}_{a|x}\\}$. More formally:

######  Definition 9 (Broacasting for Assemblages).

We say that $\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}$ is a broadcast of
$\\{\varrho^{\prime}_{a|x}\\}$ if:

$\sum_{a_{1}}\tr_{B_{1}}(\varrho_{a_{0}a_{1}|x_{0}x_{1}})=\varrho^{\prime}_{a_{0}|x_{0}},\
\ \forall a_{0},x_{0},x_{1}$ (22a) and
$\sum_{a_{0}}\tr_{B_{0}}(\varrho_{a_{0}a_{1}|x_{0}x_{1}})=\varrho^{\prime}_{a_{1}|x_{1}}\
\ \forall a_{1},x_{0},x_{1}.$ (22b)

Remark: Notice that while the broadcasting for behaviors and for states are
defined via marginalizing and tracing-out larger objects onto smaller ones,
our definition uses both marginalization and partial trace. This combination
should be thought of as reflecting the fact that assemblages are something in-
between behaviors and states.

Analogously to the box broadcasting scenario, Section II.7, the broadcasting
scenario configures a quadripartite scenario, however, the concept of
unsteerability that we will use is related to the bipartition
$A_{0}A_{1}|B_{0}B_{1}$. Again, this is the natural partition, as the resource
is spatially divided with $A_{0},A_{1}$ on one side and $B_{0},B_{1}$ on the
other. With that in mind, and in parallel with the previous case for boxes, we
define the set of Unsteerable-Realistic Non-Signalling assemblages. This set
consists of assemblages that are unsteerable on the partition
$A_{0}A_{1}|B_{0}B_{1}$, and in addition, the decomposition has non-signalling
components between $A_{0}$ and $A_{1}$. The second assumption is also natural
since we are interested in broadcasting assemblages, so the marginals
$A_{0}B_{0}$ and $A_{1}B_{1}$ ought to be well defined (see eq. (22)). More
formally,

######  Definition 10.

An assemblage $\varsigma_{a_{0}a_{1}|x_{0}x_{1}}$ is Unsteerable-Realistic, if
it can be decomposed as:

$\varsigma_{a_{0}a_{1}|x_{0}x_{1}}=\sum_{\lambda}r(\lambda)q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})\sigma_{\lambda}.$
(23)

where $q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})$ are non-signalling for all
$\lambda$. The set of all Unsteerable-Realistic assemblages will be denoted by
$UR_{ns}$.

###  III.2 Relative Entropy of Steering

####  III.2.1 Quantum KL-Divergence

We proved the no-broadcasting theorem for behaviors in an informational
manner. We defined a monotone information measure for some physically
motivated transformations, which signalled the increase of the same resource
if we sent it through a broadcasting channel. As we will see, the strategy for
the steering case follows the same lines. In this sense, we need to decide
upon an appropriate monotone. For obvious reasons we will use the quantum
version of the Kullback-Leibler divergence.

The quantum Kullback-Leibler divergence, also known as quantum relative
entropy, of two density matrices $\rho,\sigma\in\mathcal{D}(\mathcal{H})$ is
given by 111Extreme cases are dealt with as usual. See ref. [32]:

$S_{Q}(\rho||\sigma)=\tr\left[\rho(\log(\rho)-\log(\sigma))\right].$ (24)

The quantum KL-divergence was first studied by Umegaki [32] as the
noncommutative extension of KL-divergence — eq. (6). It also has the same
statistical interpretation as its classical analogue: it tells us how hard it
is to differentiate the state $\rho$ from the state $\sigma$ [33, 34].

Analogously to the classical KL-divergence, quantum KL-divergence is $(i)$
non-negative; $(ii)$ it is zero if, and only if, the quantum states are equal;
$(iii)$ it is not symmetric in $\rho$ and $\sigma$; and $(iv)$ it can
potentially be equal infinity. Another interesting property of quantum KL-
divergence is the contractivity over quantum channel actions [35]. Indeed, for
any completely positive trace-preserving map $\mathcal{E}$, we have that
$S_{Q}(\mathcal{E}(\rho)||\mathcal{E}(\sigma))\leq S_{Q}(\rho||\sigma)$. It is
precisely this property that we will use in the following paragraphs.

####  III.2.2 KL-Divergence for Assemblages

We have seen that for an assemblage $\\{\varrho_{a|x}\\}$ and a probability
distribution $\pi(x)$ we can associate to them a quantum state $\rho_{XAB}$ —
eq. (III.1). With this association, we can make emerge in the steering
scenario concepts developed for quantum states 222We should notice the
resemblance of this reformulation of assemblages with the classical-quantum
hybrid states thoroughly studied by part of the authors here and M. Leifer and
R. Spekkens in refs. [39, 40]. For instance, we can use the KL-divergence of
quantum states along with the classical-quantum state associated to an
assemblage to define the KL-divergence for assemblages [30, 37].

######  Definition 11.

Given
$\\{\varrho_{a|x}\\},\\{\varsigma_{a|x}\\}\subset\mathcal{L}(\mathcal{H}_{B})$,
the KL-divergence for assemblages between $\\{\varrho_{a|x}\\}$ and
$\\{\varsigma_{a|x}\\}$ is given by:

$S_{A}(\\{\varrho_{a|x}\\}||\\{\varsigma_{a|x}\\})=\sup_{\pi(x)}S_{Q}(\rho_{XAB}||\sigma_{XAB}),$
(25)

where $\rho_{XAB}$ and $\sigma_{XAB}$ are classical-quantum states associated
to $\varrho_{a|x}$ and $\varsigma_{a|x}$, respectively, over the same
probability distribution $\pi(x)$.

Since we already have a set of assemblages of interest — the set $UR_{ns}$—
and a divergence, we can introduce a relative entropy in the steering
scenario. Such a definition was initially introduced in [30]. However, in [30]
the set of free operations used is 1W-LOCC, while in our work we focus on LOSR
transformations. This enables us to introduce a simplified version of the
relative entropy introduced in [30]— we refer to [37] for more details.

######  Definition 12.

The Relative Entropy of Steering is defined as:

$\displaystyle E_{A}\left(\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}\right)$
$\displaystyle=\inf_{\varsigma\in
UR_{ns}}S_{A}(\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}||\\{\varsigma_{a_{0}a_{1}|x_{0}x_{1}}\\}),$
(26)

where the infimum is taken over all possible assemblages that accept a
unsteerable-realistic decomposition—see def. 10 and the discussion around it.

###  III.3 LOSR transformation and the no-local-broadcasting theorem for
assemblages

Analogously to the correlation scenario, we are interested in transformations
that do not require communication between parts $A_{0}A_{1}$ and $B_{0}B_{1}$.
Nevertheless, any kind of local operations and classical correlations are, in
principle, allowed to them. In this way, we enter into the LOSR paradigm
again, where the difference from the case of boxes is that Bob now acts on a
quantum state [38]. As we are interested in the (im)possibility of performing
broadcasting by this type of transformation, we will introduce the special
class of maps LOSR between an arbitrary steering scenario $AB$ and its
expanded broadcasting scenario $A_{0}A_{1}B_{0}B_{1}$.

######  Definition 13.

A map $\mathcal{M}$ from $\mathcal{L}(\mathcal{H}_{B})$ to
$\mathcal{L}(\mathcal{H}_{B_{0}}\otimes\mathcal{H}_{B_{1}})$ is a LOSR
transformation between assemblages if it can be decomposed as

$\displaystyle\varrho_{a_{0}a_{1}|x_{0}x_{1}}=\mathcal{M}(\varrho_{a|x})$
$\displaystyle=\sum_{\lambda,c}\sum_{a,x}\,r(\lambda)\,I(c,x|x_{0},x_{1},\lambda)p(a|x)O(a_{0},a_{1}|a,c)\mathcal{E}_{\lambda}(\rho_{a|x}),$
(27)

where $r(\lambda),I(c,x|x_{0},x_{1},\lambda),O(a_{0},a_{1}|a,c)$ are
probability distributions on appropriate finite alphabets, and
$\mathcal{E}_{\lambda}$ are CPTP maps. See Fig.3 for an illustration.

Fig. 3: Illustration of the LOSR transformation of an assemblage. The green
box and black dot in the middle represent the original assemblage with input
$x$ and output $a$ and conditioned quantum state $\rho_{|a,x}$, with the two
wavy lines between the two parts representing possibly stronger than classical
correlations. The red box represents the initial probability distribution $I$,
with a wavy line ($\lambda$) representing a classical correlation with the
CPTP map $\mathcal{E}_{\lambda}$. The blue box represents the final
probability distribution $O$. The final assemblage has inputs $x_{0},x_{1}$,
outputs $a_{0},a_{1}$ and the conditioned quantum state
$\rho_{|a_{0},a_{1},x_{0},x_{1}}$ (colors online).

Similar to the case of boxes, where our allowed transformations preserved the
local set, the LOSR transformations preserves the set of unsteerable
assemblages, i.e, if $\\{\varrho_{a|x}\\}$ is unsteerable, then
$\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}=\\{\mathcal{M}(\varrho_{a|x})\\}$ will
be also unsteerable [38]. Likewise, the set of transformations that preserves
the unsteerable assemblages is a larger set than that of LOSR transformations.
A relevant class of operations that preserve the set of unsteerable
assemblages are the one-way LOCC transformations (1W-LOCC) [30]. These
transformations demand communication. However, such communication is performed
only from Bob to Alice, so steerability cannot be created. Both sets of free
transformations 1W-LOCC and LOSR are natural free operations for the steering
scenario. However, in this work we will focus only on LOSR transformations.

In fact, we will be interested in a proper subset of LOSR. For the problem of
broadcasting, we are interested in the set of unsteerable assemblages whose
decomposition is given by non-signalling components. With this in mind, the
transformations that we will consider, in addition to being LOSR also need to
preserve the set of unsteerable realistic assemblages.

######  Definition 14.

We will call by $UR_{ns}$-LOSR transformations the subset of LOSR maps that
preserves the set of $UR_{ns}$ assemblages.

###  III.4 A no-broadcasting theorem for assemblages

Now we are ready to formulate our main no-go theorem for the case of
assemblages.

######  Theorem 2.

[No-broadcasting for assemblages] It is impossible to broadcast steerable
assemblages using $UR_{ns}$-LOSR transformations.

This will be proven using the relative entropy of steering. A fundamental
property of Relative Entropy of Steering is that it is contractive over LOSR.
This result was demonstrated in [30] and [37]. For the sake of completeness,
we re-state it below:

######  Proposition 3.

If $\mathcal{M}$ is a LOSR transformation, then:

$E_{A}({\mathcal{M}(\varrho_{a|x})})\leq E_{A}({\varrho_{a|x}}).$ (28)

This is the first property that is required to have a contradiction proof just
as in Theorem 1.

The other desired property is the following. If
$\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}$ is a broadcasting of
${\varrho_{a|x}}$, then the relative entropy steering of the former must be
strictly greater than that of latter. The proof of this result follows in a
similar way to Proposition 2. The main difference is that in place of the
chain rule, we will use Proposition 1 from [28]. More formally:

######  Proposition 4.

If $\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}$ is a broadcasting of a steerable
assemblage $\\{\varrho_{a|x}\\}$, then:

$E_{A}(\varrho_{a_{0}a_{1}|x_{0}x_{1}})>E_{A}({\varrho_{a|x}}).$ (29)

The proof of the proposition can be found in Appendix C.

At this point we see that we have similar propositions as in the box case and
we only need to exchange the respective propositions to arrive at showing
Theorem 2, see Appendix X for more details.

##  IV Discussion

In this work, we provided negative answers for the possibility of local
broadcasting for either boxes or assemblages by using fundamental properties
of the relative entropy of nonlocality and the relative entropy of steering.
Answering, therefore, the long-standing conjecture raised Joshi, Grudka and
Horodecki⊗4.

Similarly to the quantum case, we concluded that the local copying of
information from assemblages and boxes is impossible, provided that the object
in question possesses a non-classical resource such as steerability or
nonlocality. Since non-steerable assemblages, and local boxes are locally
broadcastable, we may say that failure of local broadcasting indicates the
non-classicality of the correlations. As we mentioned in the introduction, the
fact we were able to prove alternative forms of no-broadcasting theorems for
other resources underlines the importance of the standard version of the no-
broadcasting theorem—the impossibility of copying unknown information is not
distinctive of the formalism of quantum theory, but of a broader class of
probabilistic models satisfying no-signalling criteria.

Generalized versions of the local no-broadcasting theorem might also exist for
any non-classical finite-dimensional probabilistic model satisfying a no-
signalling criterion. It is also possible that such a generalized local no-
broadcasting theorem is equivalent to the original generalized no-broadcasting
theorem of Barnum, Barrett, Leifer and Wilce. We leave those points as open
questions to be investigated in the future.

###### Acknowledgements.

CV and CD thank the Wigner Centre for the welcoming environment where part of
this project was elaborated. This research was also supported by the Fetzer
Franklin Fund of the John E. Fetzer Memorial Trust and by grant number FQXi-
RFP-IPW-1905 from the Foundational Questions Institute and Fetzer Franklin
Fund, a donor advised fund of Silicon Valley Community Foundation. Financial
support from Brazilian agency Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - CAPES is gratefully acknowledged. This work has also been
supported by the Ministry of Innovation and Technology and the National
Research, Development and Innovation Office (NKFIH) within the Quantum
Information National Laboratory of Hungary and through OTKA Grants FK 135220,
K 124152, and K 124351.

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##  Appendix A Proof of Proposition 1

See 1

###### Proof.

By definition,

$E_{\mathrm{LR}}(\mathcal{M}(P(a,b|x,y))=\inf_{Q\in\mathrm{LR_{ns}}}S_{b}(\mathcal{M}(P(a,b|x,y))||Q(\bm{a},\bm{b}|\bm{x},\bm{y}))$
(30)

As $\mathcal{M}$ is $\mathrm{LR_{ns}}$ preserving, then the image of
$\mathcal{M}$ over the set of local boxes is a subset of $\mathrm{LR_{ns}}$.
Therefore, if we take the infimum of the equation above over the set
$Im(\mathcal{M})$, we will get a upper bound for
$E_{\mathrm{LR}}(\mathcal{M}(P(a,b|x,y))$. Indeed,

$\displaystyle E_{\mathrm{LR}}(\mathcal{M}(P(a,b|x,y))$
$\displaystyle\leq\inf_{Q\in
Im(\mathcal{M})}S_{b}(\mathcal{M}(P(a,b|x,y))||Q(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle=\inf_{Q^{\prime}\in\mathrm{LR_{ns}}}S_{b}(\mathcal{M}(P(a,b|x,y))||\mathcal{M}(Q^{\prime}(a,b|x,y))).$
(31)

Now, by the contractivity of $S_{b}$ over LOSR transformation — eq. (II.8.2),
we have:

$\displaystyle E_{\mathrm{LR}}(\mathcal{M}(P(a,b|x,y))$
$\displaystyle\leq\inf_{Q^{\prime}\in\mathrm{LR_{ns}}}S_{b}(P(a,b|x,y)||Q^{\prime}(a,b|x,y))$
$\displaystyle=E_{\mathrm{LR}}(P(a,b|x,y)).$ (32)

∎

##  Appendix B Proof of Proposition 2

First, let’s introduce some concepts involving probability distributions and
KL-divergence. Given two joint probability distributions $p(a,b),q(a,b)$, let
$p(b|a),q(b|a)$ be the conditional probabilities distributions. An interesting
identity relating the KL-divergence of the joint probability distribution with
the KL-divergence of the marginal and the average KL-divergence of the
conditional is given by:

$S(p(a,b)||q(a,b))=S(p(a)||q(a))+\sum_{a}p(a)S(p(b|a)||q(b|a)).$ (33)

This identity is called by Chain Rule.

Let $P(\bm{a},\bm{b}|\bm{x},\bm{y})$ and $Q(\bm{a},\bm{b}|\bm{x},\bm{y})$ be
non-signalling boxes. If we fix the inputs $\bm{x},\bm{y}$, then we can apply
the Chain Rule for
$S(P(\bm{a},\bm{b}|\bm{x},\bm{y})||Q(\bm{a},\bm{b}|\bm{x},\bm{y}))$, resulting
in:

$\displaystyle
S(P(\bm{a},\bm{b}|\bm{x},\bm{y})||Q(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle=S(P(a_{0},b_{0}|x_{0},y_{0})||Q(a_{0},b_{0}|x_{0},y_{0}))$
$\displaystyle+\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})||Q(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})),$
(34)

where, In eqs. (B) we have used the non-signalling feature for
$P(\bm{a},\bm{b}|\bm{x},\bm{y})$ and $Q(\bm{a},\bm{b}|\bm{x},\bm{y})$.

In order to prove Proposition 2, we will need to prove some preliminaries
results.

######  Lemma 1.

Let $P(\bm{a},\bm{b}|\bm{x},\bm{y})$ be a NS box s.t the marginal
$P(a_{1},b_{1}|x_{1},y_{1})$ is nonlocal. Then, for every $x_{0},y_{0}$, there
exist $a_{0},b_{0}$ such that $P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})$ is
nonlocal and $P(a_{0},b_{0}|x_{0},y_{0})\neq 0$.

###### Proof.

By the definition of a conditional probability distribution:

$P(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})=\frac{P(a_{0},a_{1},b_{0},b_{1}|\bm{x},\bm{y})}{P(a_{0},b_{0}|x_{0},y_{0})}$
(35)

Thus,

$P(a_{0},a_{1},b_{0},b_{1}|\bm{x},\bm{y})=P(a_{0},b_{0}|x_{0},y_{0})P(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y}).$
(36)

In this way, summing the equation above over $a_{0},b_{0}$, we have:

$P(a_{1},b_{1}|x_{1},y_{1})=\sum_{a_{0},b_{0}}P(a_{0},a_{1},b_{0},b_{1}|\bm{x},\bm{y})=\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})P(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y}).$
(37)

Given $x_{0},y_{0}$, let’s suppose, by contradiction, that for every
$a_{0},b_{0}$ with $P(a_{0},b_{0}|x_{0},y_{0})\neq 0$ we have that
$P(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})$ is local. Thus, for each
$a_{0},b_{0}$ with $P(a_{0},b_{0}|x_{0},y_{0})\neq 0$ there exist
$r_{x_{0},y_{0},a_{0},b_{0}}(\lambda)$,
$P_{x_{0},y_{0},a_{0},b_{0},\lambda}(a_{1}|x_{1})$ and
$P_{x_{0},y_{0},a_{0},b_{0},\lambda}(b_{1}|y_{1})$ such that:

$P(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})=\sum_{\lambda}r_{x_{0},y_{0},a_{0},b_{0}}(\lambda)P_{x_{0},y_{0},a_{0},b_{0},\lambda}(a_{1}|x_{1})P_{x_{0},y_{0},a_{0},b_{0},\lambda}(b_{1}|y_{1}).$
(38)

Thus,

$\displaystyle P(a_{1},b_{1}|x_{1},y_{1})$
$\displaystyle=\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})P(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})$
$\displaystyle=\sum_{\lambda}\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})r_{x_{0},y_{0},a_{0},b_{0}}(\lambda)P_{x_{0},y_{0},a_{0},b_{0},\lambda}(a_{1}|x_{1})P_{x_{0},y_{0},a_{0},b_{0},\lambda}(b_{1}|y_{1}).$
(39)

Let $\lambda^{\prime}=(a_{0},b_{0},\lambda)$ and
$r_{x_{0},y_{0}}(\lambda^{\prime})=P(a_{0},b_{0}|x_{0},y_{0})r_{x_{0},y_{0},a_{0},b_{0}}(\lambda)$.
Then it’s easily to see that $r_{x_{0},y_{0}}(\lambda^{\prime})$ is a
probability distribution on the variable $\lambda^{\prime}$, i.e,
$r_{x_{0},y_{0}}(\lambda^{\prime})\geq 0$ and
$\sum_{\lambda^{\prime}}r_{x_{0},y_{0}}(\lambda^{\prime})=1$. We also have:

$P(a_{1},b_{1}|x_{1},y_{1})=\sum_{\lambda^{\prime}}r_{x_{0},y_{0}}(\lambda^{\prime})P_{x_{0},y_{0},\lambda^{\prime}}(a_{1}|x_{1})P_{x_{0},y_{0},\lambda^{\prime}}(b_{1}|y_{1}).$
(40)

Therefore, $P(a_{1},b_{1}|x_{1},y_{1})$ is local, which is a contradiction
with the hypothesis. ∎

######  Lemma 2.

If $Q(\bm{a},\bm{b}|\bm{x},\bm{y})$ be a $LR_{ns}$ box, then
$Q(a_{1},b_{1}|\bm{x},\bm{y},a_{0},a_{1})$ will be local for every choice of
$a_{0},b_{0},x_{0},y_{0}$.

###### Proof.

Let’s start fixing $a_{0},b_{0},x_{0},y_{0}$. By definition,

$Q(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})=\frac{Q(a_{0},a_{1},b_{0},b_{1}|\bm{x},\bm{y})}{Q(a_{0},b_{0}|x_{0},y_{0})}.$
(41)

As $Q(\bm{a},\bm{b}|\bm{x},\bm{y})$ is a $\mathrm{LR_{ns}}$ box, we have:

$Q(\bm{a},\bm{b}|\bm{x},\bm{y})=\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})Q_{\lambda}(b_{0},b_{1}|y_{0},y_{1}),$
(42)

where $Q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})$ and
$Q_{\lambda}(b_{0},b_{1}|y_{0},y_{1})$ are non-signalling. So, we have:

$\displaystyle
Q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})=Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})Q_{\lambda}(a_{0}|x_{0},x_{1})=Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})Q_{\lambda}(a_{0}|x_{0}).$
(43)

and

$\displaystyle
Q_{\lambda}(b_{0},b_{1}|y_{0},y_{1})=Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0})Q_{\lambda}(b_{0}|y_{0},y_{1})=Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0})Q_{\lambda}(b_{0}|y_{0}).$
(44)

Therefore,

$\displaystyle Q(\bm{a},\bm{b}|\bm{x},\bm{y})$
$\displaystyle=\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0})Q_{\lambda}(b_{0}|y_{0})$
(45)
$\displaystyle=\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0})Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0})$
(46)

Then,

$\displaystyle
Q(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})=\sum_{\lambda}\frac{r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0})}{Q(a_{0},b_{0}|x_{0},y_{0})}Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0})$
(47)

Let

$r_{a_{0},b_{0},x_{0},y_{0}}(\lambda)=\frac{r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0})}{Q(a_{0},b_{0}|x_{0},y_{0})}.$
(48)

It follows that $r_{a_{0},b_{0},x_{0},y_{0}}(\lambda)\geq 0$. On the other
hand,

$\displaystyle Q(a_{0},b_{0}|x_{0},y_{0})$
$\displaystyle=\sum_{a_{1},b_{1}}Q(a_{0},a_{1},b_{0},b_{1}|\bm{x},\bm{y})=\sum_{\lambda}\sum_{a_{1},b_{1}}r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0})Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0})$
$\displaystyle=\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0})\left(\sum_{a_{1}}Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})\right)\left(\sum_{b_{1}}Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0})\right)$
$\displaystyle=\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0}).$
(49)

Therefore,

$\sum_{\lambda}r_{a_{0},b_{0},x_{0},y_{0}}(\lambda)=\frac{\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0})}{\sum_{\lambda}r(\lambda)Q_{\lambda}(a_{0}|x_{0})Q_{\lambda}(b_{0}|y_{0})}=1.$
(50)

So, $r_{a_{0},b_{0},x_{0},y_{0}}(\lambda)$ is a probability distribution in
$\lambda$. In this way:

$\displaystyle
Q(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})=\sum_{\lambda}r_{a_{0},b_{0},x_{0},y_{0}}(\lambda)Q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})Q_{\lambda}(b_{1}|y_{0},y_{1},b_{0}).$
(51)

Thus, $Q(a_{1},b_{1}|a_{0},b_{0},\bm{x},\bm{y})$ is local. ∎

######  Lemma 3.

If $Q(\bm{a},\bm{b}|\bm{x},\bm{y})$ is a $LR_{ns}$ box and
$P(\bm{a},\bm{b}|\bm{x},\bm{y})$ is a NS box with $P(a_{1},b_{1}|x_{1},y_{1})$
nonlocal, then

$\max_{x_{1},y_{1}}\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})||Q(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0}))>0,$
(52)

for every $x_{0},y_{0}$.

###### Proof.

Given $x_{0},y_{0}$, by Lemma 1, there exist $a^{\prime}_{0},b^{\prime}_{0}$
such that $P(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0})$ is
nonlocal and $P(a^{\prime}_{0},b^{\prime}_{0}|x_{0},y_{0})\neq 0$. On the
other hand, by Lemma 2,
$Q(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0})$ is local.
Therefore:

$\displaystyle\max_{x_{1},y_{1}}\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})||Q(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0}))$
$\displaystyle\geq\max_{x_{1},y_{1}}P(a^{\prime}_{0},b^{\prime}_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0})||Q(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0}))$
(53)

But, as boxes, $P(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0})$
and $Q(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0})$ are
different, since the first is nonlocal while the second is local. Therefore,

$S(P(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0})||Q(a_{1},b_{1}|\bm{x},\bm{y},a^{\prime}_{0},b^{\prime}_{0}))>0.$

Thus,

$\max_{x_{1},y_{1}}\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})||Q(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0}))>0.$
(54)

∎

######  Lemma 4.

There is always $\bar{Q}(\bm{a},\bm{b}|\bm{x},\bm{y})\in LR_{ns}$ such that
$E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))=S_{\mathrm{b}}(P(\bm{a},\bm{b}|\bm{x},\bm{y})||\bar{Q}(\bm{a},\bm{b}|\bm{x},\bm{y}))$.
In other words, the infimum on the definition of the relative entropy of
nonlocality is always achieved.

###### Proof.

It’s well known that the set of $\mathrm{LR_{ns}}$ boxes is a polytope,
therefore is a compact set. On the other hand, fixing a box
$P(\bm{a},\bm{b}|\bm{x},\bm{y})$, let’s define
$f(Q(\bm{a},\bm{b}|\bm{x},\bm{y})):=S_{\mathrm{b}}(P(\bm{a},\bm{b}|\bm{x},\bm{y})||Q(\bm{a},\bm{b}|\bm{x},\bm{y}))$
is lower semi-continuous in on the set of $\mathrm{LR_{ns}}$ boxes. Therefore,
by the extended Bolzano–Weierstrass theorem, the function $f$ has a minimum in
$\mathrm{LR_{ns}}$. ∎

Given these results, let’s now move on to the main goal of this section, prove
Proposition 2. See 2

###### Proof.

By the definitions,

$\displaystyle E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle=\inf_{Q\in\mathrm{LR_{ns}}}S_{\mathrm{b}}(P(\bm{a},\bm{b}|\bm{x},\bm{y})||Q(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle=S_{\mathrm{b}}(P(\bm{a},\bm{b}|\bm{x},\bm{y})||\bar{Q}(\bm{a},\bm{b}|\bm{x},\bm{y}))$
(55)

where we have used Lemma 4 in the last equation. Then,

$\displaystyle E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle=S_{\mathrm{b}}(P(\bm{a},\bm{b}|\bm{x},\bm{y})||\bar{Q}(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle=\max_{\bm{x},\bm{y}}S(P(\bm{a},\bm{b}|\bm{x},\bm{y})||\bar{Q}(\bm{a},\bm{b}|\bm{x},\bm{y})).$
(56)

Using now (B), we have:

$\displaystyle E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle=\max_{\bm{x},\bm{y}}\left[S(P(a_{0},b_{0}|x_{0},y_{0})||\bar{Q}(a_{0},b_{0}|x_{0},y_{0}))+\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})||\bar{Q}(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0}))\right]$
$\displaystyle=\max_{x_{0},y_{0}}\max_{x_{1},y_{1}}\left[S(P(a_{0},b_{0}|x_{0},y_{0})||\bar{Q}(a_{0},b_{0}|x_{0},y_{0}))+\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})||\bar{Q}(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0}))\right]$
$\displaystyle=\max_{x_{0},y_{0}}\left[S(P(a_{0},b_{0}|x_{0},y_{0})||\bar{Q}(a_{0},b_{0}|x_{0},y_{0}))+\max_{x_{1},y_{1}}\sum_{a_{0},b_{0}}P(a_{0},b_{0}|x_{0},y_{0})S(P(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0})||\bar{Q}(a_{1},b_{1}|\bm{x},\bm{y},a_{0},b_{0}))\right].$
(57)

Applying Lemma 3,

$\displaystyle E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle>\max_{x_{0},y_{0}}S(P(a_{0},b_{0}|x_{0},y_{0})||\bar{Q}(a_{0},b_{0}|x_{0},y_{0}))$
$\displaystyle=S_{b}(P(a_{0},b_{0}|x_{0},y_{0})||\bar{Q}(a_{0},b_{0}|x_{0},y_{0})).$
(58)

It’s trivial to see that $\bar{Q}(a_{0},b_{0}|x_{0},y_{0})$ is a local box,
therefore:

$\displaystyle E_{\mathrm{LR}}(P(\bm{a},\bm{b}|\bm{x},\bm{y}))$
$\displaystyle>\inf_{Q\in\mathrm{LR_{ns}}}S_{\mathrm{b}}(P(a_{0},b_{0}|x_{0},y_{0})||Q(a_{0},b_{0}|x_{0},y_{0}))$
$\displaystyle=E_{\mathrm{LR}}(P^{\prime}(a,b|x,y)),$ (59)

where in the last equation we have used that as
$P(\bm{a},\bm{b}|\bm{x},\bm{y})$ is a broadcast of $P^{\prime}(a,b|x,y)$, then
$P(a_{0},b_{0}|x_{0},y_{0})=P^{\prime}(a_{0},b_{0}|x_{0},y_{0})$. ∎

##  Appendix C Proof of Proposition 4

Let’s first introduce the concept of a measurement map

######  Definition 15.

Given a POVM $\\{F_{i}\\}_{i=1}^{n}\subset\mathcal{L}(\mathcal{H})$ a POVM,
the measurement map associate it is define as
$\mathcal{F}:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathbb{C}^{n})$, with

$\mathcal{F}(X)=\sum_{i=1}^{n}\tr(F_{i}X)\outerproduct{i}{i},$ (60)

where $\\{\ket{i}\\}$ is an orthogonal basis.

It’s easy to see that every measurement map is a CPTP. In this way, as the
quantum relative entropy is contractible over CPTPs maps, then

$S_{Q}(\mathcal{F}(\rho)||\mathcal{F}(\sigma))\leq S_{Q}(\rho||\sigma)\ \
\forall\rho,\sigma\in\mathcal{L}(\mathcal{H}).$ (61)

On the other hand, let $\rho_{XY}$ and $\sigma_{XY}$ be bipartite quantum
states and
$\mathcal{F}:\mathcal{L}(\mathcal{H}_{X})\to\mathcal{L}(\mathbb{C}^{n})$ a
measurement map. In [28], Piani showed the following inequality:

$S_{Q}(\rho_{XY}||\sigma_{XY})\geq
S_{Q}(\mathcal{F}(\rho_{X})||\mathcal{F}(\sigma_{X}))+S_{Q}\left(\rho_{Y}||\sum_{k}\alpha_{k}\sigma_{Y}^{k}\right),$
(62)

where $\alpha_{k}=\tr(F_{k}\rho_{X})$ and
$\sigma_{W}^{k}=\frac{\tr_{Z}((F_{k}\otimes Id)\sigma_{XY})}{\tr((F_{k}\otimes
Id)\sigma_{XY})}$.

Going back to our goal, let $\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}$ and
$\\{\varsigma_{a_{0}a_{1}|x_{0}x_{1}}\\}$ be two non-signalling assemblages in
the a broadcasting scenario. Let
$\pi(x_{0},x_{1})=\pi_{0}(x_{0})\pi_{1}(x_{1})$ be a independent probability
distribution for $A_{0}A_{1}$ choices. In order not to overload the notation,
we will denote $Z=X_{0}A_{0}B_{0}$ and $W=X_{1}A_{1}B_{1}$. The CQ-states
associated to the assemblage $\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\}$ and
$\\{\varsigma_{a_{0}a_{1}|x_{0}x_{1}}\\}$ are, respectively:

$\rho_{ZW}=\sum_{a_{0},a_{1},x_{0},x_{1}}\pi_{0}(x_{0})\pi_{1}(x_{1})p(a_{0},a_{1}|x_{0},x_{1})\outerproduct{x_{0}x_{1}}{x_{0}x_{1}}\otimes\outerproduct{a_{0}a_{1}}{a_{0}a_{1}}\otimes\rho_{a_{0}a_{1}|x_{0}x_{1}}.$
(63)

and

$\sigma_{ZW}=\sum_{a_{0},a_{1},x_{0},x_{1}}\pi_{0}(x_{0})\pi_{1}(x_{1})q(a_{0},a_{1}|x_{0},x_{1})\outerproduct{x_{0}x_{1}}{x_{0}x_{1}}\otimes\outerproduct{a_{0}a_{1}}{a_{0}a_{1}}\otimes\sigma_{a_{0}a_{1}|x_{0}x_{1}}.$
(64)

Additionally, we will consider that $\\{\varsigma_{a_{0}a_{1}|x_{0}x_{1}}\\}$
is a $UR_{ns}$ assemblage, then:

$\sigma_{ZW}=\sum_{\lambda}\sum_{a_{0},a_{1},x_{0},x_{1}}\pi_{0}(x_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})\outerproduct{x_{0}x_{1}}{x_{0}x_{1}}\otimes\outerproduct{a_{0}a_{1}}{a_{0}a_{1}}\otimes\sigma^{\lambda},$
(65)

where $q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})$ is non-signalling for all
$\lambda$.

Now, let $\\{E_{i}\\}_{i=1}^{n}\subset\mathcal{L}(\mathcal{H}_{B_{0}})$ be an
arbitrary Informational Complete POVM (IC-POVM). In this way, we define
$\\{F_{k}\\}\subset\mathcal{L}(\mathbb{C}^{r}\otimes\mathbb{C}^{s}\otimes\mathcal{H}_{B_{0}})$
as
$F_{k}=F_{x_{0},a_{0},i}=\outerproduct{x_{0}}{x_{0}}\otimes\outerproduct{a_{0}}{a_{0}}\otimes
E_{i}$ (we are using the abbreviation $k=(x_{0},a_{0},I)$). It is easy to see
that $\\{F_{x_{0},a_{0},i}\\}$ is a POVM. Let
$\mathcal{F}:\mathcal{L}(\mathbb{C}^{r}\otimes\mathbb{C}^{s}\otimes\mathcal{H}_{B_{0}})\to\mathcal{L}(\mathbb{C}^{r}\otimes\mathbb{C}^{s}\otimes\mathbb{C}^{n})$
the measurement map associated to $F_{k}$, i.e,

$\mathcal{F}(X)=\sum_{x_{0},a_{0},i}\tr[\outerproduct{x_{0}}{x_{0}}\otimes\outerproduct{a_{0}}{a_{0}}\otimes
E_{i})X]\outerproduct{x_{0}}{x_{0}}\otimes\outerproduct{a_{0}}{a_{0}}\otimes\outerproduct{i}{i}.$
(66)

Then, apply eq. (62) for the states $\rho_{ZW},\sigma_{ZW}$ and the
measurement map $\mathcal{F}$, we have:

$S_{Q}(\rho_{ZW}||\sigma_{ZW})\geq
S_{Q}(\mathcal{F}(\rho_{Z})||\mathcal{F}(\sigma_{Z}))+S_{Q}\left(\rho_{W}||\sum_{k}\alpha_{k}\sigma_{W}^{k}\right).$
(67)

We will now demonstrate some auxiliary statements that will help us reach the
desired conclusion.

######  Lemma 5.

Let $\\{\uptau_{a_{0}a_{1}|x_{0}x_{1}}\\}$ a non-signalling assemblage and
$\pi(x_{0},x_{1})$ a probability distribution. If $\tau_{ZW}$ be the CQ-state
associated to $(\\{\uptau_{a_{0}a_{1}|x_{0}x_{1}}\\},\pi(x_{0},x_{1}))$, then
$\tau_{Z}=\tr_{W}(\tau_{ZW})$ is the CQ-state associated to
$(\\{\uptau_{a_{0}|x_{0}}\\},\pi(x_{0}))$.

###### Proof.

By definition of CQ-state,

$\tau_{ZW}=\sum_{a_{0},a_{1},x_{0},x_{1}}\pi(x_{0},x_{1})p(a_{0},a_{1}|x_{0},x_{1})\outerproduct{x_{0}x_{1}}{x_{0}x_{1}}\otimes\outerproduct{a_{0}a_{1}}{a_{0}a_{1}}\otimes\uptau_{a_{0}a_{1}|x_{0}x_{1}}.$
(68)

Thus,

$\displaystyle\tau_{Z}$
$\displaystyle=\tr_{W}(\tau_{ZW})=\tr_{X_{1}A_{1}B_{1}}\left[\sum_{a_{0},a_{1},x_{0},x_{1}}\pi(x_{0},x_{1})\outerproduct{x_{0}x_{1}}{x_{0}x_{1}}\otimes\outerproduct{a_{0}a_{1}}{a_{0}a_{1}}\otimes\uptau_{a_{0}a_{1}|x_{0}x_{1}}\right]$
$\displaystyle=\sum_{a_{0},a_{1},x_{0},x_{1}}\pi(x_{0},x_{1})\outerproduct{x_{0}}{x_{0}}\otimes\outerproduct{a_{0}}{a_{0}}\otimes\tr_{B_{1}}(\uptau_{a_{0}a_{1}|x_{0}x_{1}})$
$\displaystyle=\sum_{a_{0},x_{0},x_{1}}\pi(x_{0},x_{1})\outerproduct{x_{0}}{x_{0}}\otimes\outerproduct{a_{0}}{a_{0}}\otimes\left(\sum_{a_{1}}\tr_{B_{1}}(\uptau_{a_{0}a_{1}|x_{0}x_{1}})\right)$
(69)

As $\\{\uptau_{a_{0}a_{1}|x_{0}x_{1}}\\}$ is a non-signalling assemblage,
$\sum_{a_{1}}\tr_{B_{1}}(\uptau_{a_{0}a_{1}|x_{0}x_{1}})=\uptau_{a_{0}|x_{0}}$.
Consequently,

$\displaystyle\tau_{Z}$
$\displaystyle=\sum_{a_{0},x_{0},x_{1}}(\sum_{x_{1}}\pi(x_{0},x_{1}))\outerproduct{x_{0}}{x_{0}}\otimes\outerproduct{a_{0}}{a_{0}}\otimes\uptau_{a_{0}|x_{0}}$
$\displaystyle=\sum_{a_{0},x_{0},x_{1}}\pi(x_{0})\outerproduct{x_{0}}{x_{0}}\otimes\outerproduct{a_{0}}{a_{0}}\otimes\uptau_{a_{0}|x_{0}}.$
(70)

Therefore, $\tau_{Z}$ is the CQ-state associated to
$(\\{\uptau_{a_{0}|x_{0}}\\},\pi(x_{0}))$. ∎

######  Lemma 6.

The states $\\{\sigma_{W}^{k}\\}$ are CQ-states associated to unsteerable
assemblages and the probability distribution $\pi(x_{1})$.

###### Proof.

Given a fixed $k=(x^{\prime}_{0},a^{\prime}_{0},i)$, by the definition of
$\sigma_{W}^{k}$,

$\sigma_{W}^{k}=\frac{\tr_{Z}((F_{k}\otimes Id)\sigma_{ZW})}{\tr((F_{k}\otimes
Id)\sigma_{ZW})}.$ (71)

So,

$\displaystyle(F_{k}\otimes
Id)\sigma_{ZW}=(F_{x^{\prime}_{0},a^{\prime}_{0},i}\otimes Id)\sigma_{ZW}$
$\displaystyle=\sum_{\lambda}\sum_{a_{0},a_{1},x_{0},x_{1}}\pi_{0}(x_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})(\outerproduct{x^{\prime}_{0}}{x^{\prime}_{0}}\outerproduct{x_{0}}{x_{0}})\otimes\outerproduct{x_{1}}{x_{1}}\otimes(\outerproduct{a^{\prime}_{0}}{a^{\prime}_{0}}\outerproduct{a_{0}}{a_{0}})\otimes\outerproduct{a_{1}}{a_{1}}\otimes
F_{i}\sigma_{B_{0}B_{1}}^{\lambda}$
$\displaystyle=\sum_{\lambda}\sum_{a_{1},x_{1}}\pi_{0}(x^{\prime}_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a^{\prime}_{0},a_{1}|x^{\prime}_{0},x_{1})\outerproduct{x^{\prime}_{0}}{x^{\prime}_{0}}\otimes\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a^{\prime}_{0}}{a^{\prime}_{0}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes
F_{i}\sigma_{B_{0}B_{1}}^{\lambda}$

Thus,

$\displaystyle\tr_{Z}\left[(F_{x^{\prime}_{0},a^{\prime}_{0},i}\otimes
Id)\sigma_{ZW}\right]$
$\displaystyle=\tr_{X_{0}A_{0}B_{0}}((F_{x^{\prime}_{0},a^{\prime}_{0},i}\otimes
Id)\sigma_{ZW})$
$\displaystyle=\sum_{\lambda}\sum_{a_{1},x_{1}}\pi_{0}(x^{\prime}_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a^{\prime}_{0},a_{1}|x^{\prime}_{0},x_{1})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\tr_{B_{0}}(F_{i}\sigma_{B_{0}B_{1}}^{\lambda})$
$\displaystyle=\sum_{\lambda}\sum_{a_{1},x_{1}}\pi_{0}(x^{\prime}_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a^{\prime}_{0},a_{1}|x^{\prime}_{0},x_{1})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\sigma_{B_{1}}^{\lambda,i},$
(73)

where
$\sigma_{B_{1}}^{\lambda,i}=\tr_{B_{0}}(F_{i}\sigma_{B_{0}B_{1}}^{\lambda})$.
As $\\{\varsigma_{a_{0}a_{1}|x_{0}x_{1}}\\}\in UR_{ns}$, the
$q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})$ are non-signalling for all $\lambda$.
Thus,

$\displaystyle
q_{\lambda}(a_{0},a_{1}|x_{0},x_{1})=q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})q_{\lambda}(a_{0}|x_{0},x_{1})=q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})q_{\lambda}(a_{0}|x_{0}).$
(74)

Therefore,

$\displaystyle\tr_{Z}\left[(F_{x^{\prime}_{0},a^{\prime}_{0},i}\otimes
Id)\sigma_{ZW}\right]$
$\displaystyle=\sum_{\lambda}\sum_{a_{1},x_{1}}\pi_{0}(x^{\prime}_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a_{1}|x_{0},x_{1},a_{0})q_{\lambda}(a_{0}|x_{0})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\sigma_{B_{1}}^{\lambda,i}.$
(75)

Then,

$\displaystyle\tr\left[(F_{x^{\prime}_{0},a^{\prime}_{0},i}\otimes
Id)\sigma_{ZW}\right]$
$\displaystyle=\sum_{\lambda}\sum_{a_{1},x_{1}}\pi_{0}(x^{\prime}_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a_{1}|x^{\prime}_{0},x_{1},a^{\prime}_{0})q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})$
$\displaystyle=\sum_{\lambda}\sum_{x_{1}}\pi_{0}(x^{\prime}_{0})\pi_{1}(x_{1})r(\lambda)\left(\sum_{a_{1}}q_{\lambda}(a_{1}|x^{\prime}_{0},x_{1},a^{\prime}_{0})\right)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})$
$\displaystyle=\sum_{\lambda}\pi_{0}(x^{\prime}_{0})\left(\sum_{x_{1}}\pi_{1}(x_{1})\right)r(\lambda)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})$
$\displaystyle=\sum_{\lambda}\pi_{0}(x^{\prime}_{0})r(\lambda)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i}).$
(76)

Thus,

$\displaystyle\sigma_{W}^{x^{\prime}_{0},a^{\prime}_{0},i}$
$\displaystyle=\frac{\tr_{Z}((F_{x^{\prime}_{0},a^{\prime}_{0},i}\otimes
Id)\sigma_{ZW})}{\tr((F_{x^{\prime}_{0},a^{\prime}_{0},i}\otimes
Id)\sigma_{ZW})}$
$\displaystyle=\frac{\sum_{\lambda}\sum_{a_{1},x_{1}}\pi_{0}(x^{\prime}_{0})\pi_{1}(x_{1})r(\lambda)q_{\lambda}(a_{1}|x^{\prime}_{0},x_{1},a^{\prime}_{0})q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\sigma_{B_{1}}^{\lambda,i}}{\sum_{\lambda}\pi_{0}(x^{\prime}_{0})r(\lambda)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})}$
$\displaystyle=\sum_{\lambda}\sum_{a_{1},x_{1}}\frac{r(\lambda)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})}{\sum_{\lambda}r(\lambda)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})}\pi_{1}(x_{1})q_{\lambda}(a_{1}|x^{\prime}_{0},x_{1},a^{\prime}_{0})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\frac{\sigma_{B_{1}}^{\lambda,i}}{\tr(\sigma_{B_{1}}^{\lambda,i})}.$
(77)

Let

$r_{k}(\lambda)=\frac{r(\lambda)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})}{\sum_{\lambda}r(\lambda)q_{\lambda}(a^{\prime}_{0}|x^{\prime}_{0})\tr(\sigma_{B_{1}}^{\lambda,i})}.$

It follows that $r_{k}(\lambda)\geq 0$ and that
$\sum_{\lambda}r_{k}(\lambda)=1$. Let’s also define
$q_{\lambda}^{k}(a_{1}|x_{1})=q_{\lambda}(a_{1}|x^{\prime}_{0},x_{1},a^{\prime}_{0})$
and

$\sigma_{B_{1}}^{\lambda,k}=\frac{\sigma_{B_{1}}^{\lambda,i}}{\tr(\sigma_{B_{1}}^{\lambda,i})}.$

Then,

$\sigma_{W}^{k}=\sum_{\lambda}\sum_{a_{1},x_{1}}\pi(x_{1})r_{k}(\lambda)q_{\lambda}^{k}(a_{1}|x_{1})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\sigma_{B_{1}}^{\lambda,k}.$
(78)

Thus, $\sigma_{W}^{k}$ are CQ-states associated to unsteerable assemblages by
$\pi(x_{1})$. ∎

######  Lemma 7.

Given $\\{\sigma_{W}^{k}\\}_{k}$ a set of CQ-states associated to unsteerable
assemblages by the same probability distribution $\pi(x_{1})$, and let
$\\{\alpha_{k}\\}\subset\mathbb{R}^{+}$ with $\sum_{k}\alpha_{k}=1$. Then,
$\sum_{k}\alpha_{k}\sigma_{W}^{k}$ is also a CQ-state associated to an
unsteerable assemblage by the probability distribution $\pi(x_{1})$.

###### Proof.

By hypotheses,

$\displaystyle\sigma_{W}^{k}=\sum_{\lambda}\sum_{a_{1},x_{1}}\pi(x_{1})r_{k}(\lambda)q_{\lambda}^{k}(a_{1}|x_{1})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\sigma_{B_{1}}^{\lambda,k}.$
(79)

Then,

$\displaystyle\sum_{k}\alpha_{k}\sigma_{W}^{k}=\sum_{\lambda,k}\sum_{a_{1},x_{1}}\pi(x_{1})\alpha_{k}r_{k}(\lambda)q_{\lambda}^{k}(a_{1}|x_{1})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\sigma_{B_{1}}^{\lambda,k}.$
(80)

Let $\lambda^{\prime}=(\lambda,k)$,
$r(\lambda^{\prime})=r(\lambda,k)=\alpha_{k}r_{k}(\lambda)$,
$q_{\lambda^{\prime}}(a_{1}|x_{1})=q_{\lambda}^{k}(a_{1}|x_{1})$ and
$\sigma_{B_{1}}^{\lambda^{\prime}}=\sigma_{B_{1}}^{\lambda,k}$. We can see
that, $r(\lambda^{\prime})\geq 0$ and
$\sum_{\lambda^{\prime}}r(\lambda^{\prime})=\sum_{k}\alpha_{k}(\sum_{\lambda}r_{k}(\lambda))=\sum_{k}\alpha_{k}=1$.
Thus,

$\sum_{k}\alpha_{k}\sigma_{W}^{k}=\sum_{\lambda^{\prime}}\sum_{a_{1},x_{1}}\pi(x_{1})r(\lambda^{\prime})q_{\lambda^{\prime}}(a_{1}|x_{1})\outerproduct{x_{1}}{x_{1}}\otimes\outerproduct{a_{1}}{a_{1}}\otimes\sigma_{B_{1}}^{\lambda^{\prime}}.$
(81)

Therefore, $\sum_{k}\alpha_{k}\sigma_{W}^{k}$ is a CQ-state associated to an
unsteerable assemblage by $\pi(x_{1})$. ∎

######  Lemma 8.

Given $\pi(x)$ a probability distribution s.t $\pi(x)>0\ \forall\ x$, and
$\\{\varrho_{a|x}\\},\\{\varsigma_{a|x}\\}$ assemblages. Let $\\{E_{i}\\}$ an
IC-POVM and $F_{x,a,i}=\outerproduct{x}{x}\otimes\outerproduct{a}{a}\otimes
E_{i}$. Then, $\mathcal{F}(\rho_{XAB})=\mathcal{F}(\sigma_{XAB})$ $\implies$
$\\{\varrho_{a|x}\\}=\\{\varsigma_{a|x}\\}$.

###### Proof.

$\displaystyle\mathcal{F}(\rho_{XAB})=\sum_{x,a,i}\tr(F_{x,a,i}\rho_{XAB})\outerproduct{x}{x}\otimes\outerproduct{a}{a}\otimes\outerproduct{i}{i}.$
(82)

Expanding the trace:

$\displaystyle\tr(F_{x,a,i}\rho_{XAB})$
$\displaystyle=\sum_{x^{\prime},a^{\prime}}\pi(x)p(a|x)\tr\left(\outerproduct{x}{x}\outerproduct{x^{\prime}}{x^{\prime}}\otimes\outerproduct{a}{a}\outerproduct{a^{\prime}}{a^{\prime}}\otimes
E_{i}\rho_{a|x}\right)$ $\displaystyle=\pi(x)p(a|x)\tr(E_{i}\rho_{a|x}).$ (83)

Thus,

$\displaystyle\mathcal{F}(\rho_{XAB})=\sum_{x,a,i}\pi(x)p(a|x)\tr(E_{i}\rho_{a|x})\outerproduct{x}{x}\otimes\outerproduct{a}{a}\otimes\outerproduct{i}{i}.$
(84)

Analogously,

$\displaystyle\mathcal{F}(\sigma_{XAB})=\sum_{x,a,i}\pi(x)q(a|x)\tr(E_{i}\sigma_{a|x})\outerproduct{x}{x}\otimes\outerproduct{a}{a}\otimes\outerproduct{i}{i}.$
(85)

As $\\{\ket{x}\\},\\{\ket{a}\\}$ and $\\{\ket{i}\\}$ are orthonormal basis,
$\mathcal{F}(\rho_{XAB})=\mathcal{F}(\sigma_{XAB})$ implies in:

$\pi(x)p(a|x)\tr(E_{i}\rho_{a|x})=\pi(x)q(a|x)\tr(E_{i}\sigma_{a|x})\ \forall
x,a,i.$ (86)

Dividing both sides by $\pi(x)$ and summing over $i$:

$\displaystyle
p(a|x)\tr(\sum_{i}E_{i}\rho_{a|x})=q(a|x)\tr(\sum_{i}E_{i}\sigma_{a|x})\
\forall x,a$ $\displaystyle
p(a|x)\tr\left(\rho_{a|x}\right)=q(a|x)\tr\left(\sigma_{a|x}\right)\ \forall
x,a$ $\displaystyle p(a|x)=q(a|x)\ \forall x,a.$ (87)

Thus, by eqs. (86) and (C), we have
$\tr(E_{i}\rho_{a|x})=\tr(E_{i}\sigma_{a|x})$ $\forall x,a,i$. As
$\\{E_{i}\\}$ is a IC-POVM, this implies in $\rho_{a|x}=\sigma_{a|x}$ $\forall
x,a$. Thus,

$\varrho_{a|x}=p(a|x)\rho_{a|x}=q(a|x)\sigma_{a|x}=\varsigma_{a|x}\ \forall
x,a.$

Therefore, $\\{\varrho_{a|x}\\}=\\{\varsigma_{a|x}\\}$. ∎

We are now able to show the main result of this section: See 4

###### Proof.

By definition,

$\displaystyle E_{A}(\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\})$
$\displaystyle=\sup_{\pi(x_{0},x_{1})}\inf_{\varsigma\in
UN_{ns}}S_{Q}(\rho_{ZW}||\sigma_{ZW})$
$\displaystyle\geq\sup_{\pi_{0}(x_{0})\pi_{1}(x_{1})}\inf_{\varsigma\in
UN_{ns}}S_{Q}(\rho_{ZW}||\sigma_{ZW}),$ (88)

where in the second line we are taking the supreme over the smaller set of
product probability distributions, i.e,
$\pi(x_{0},x_{1})=\pi_{0}(x_{0})\pi_{1}(x_{1})$. Using eq. (62), we have:

$\displaystyle
E_{A}(\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\})\geq\sup_{\pi_{0}(x_{0})\pi_{1}(x_{1})}\inf_{\varsigma\in
UN_{ns}}\left[S_{Q}(\mathcal{F}(\rho_{Z})||\mathcal{F}(\sigma_{Z}))+S_{Q}\left(\rho_{W}||\sum_{k}\alpha_{k}\sigma_{W}^{k}\right)\right].$
(89)

For each probability distribution $\pi_{0}(x_{0})\pi_{1}(x_{1})$, let
$\bar{\varsigma}_{a_{0}a_{1}|x_{0}x_{1}}$ be the assemblage that achieves the
infimum of the equation above (The existence of this state that reaches the
minimum is guaranteed by arguments totally analogous to those of Lemma 4).
Then,

$\displaystyle E_{A}(\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\})$
$\displaystyle\geq\sup_{\pi_{0}(x_{0})\pi_{1}(x_{1})}\left[S_{Q}(\mathcal{F}(\rho_{Z})||\mathcal{F}(\bar{\sigma}_{Z}))+S_{Q}\left(\rho_{W}||\sum_{k}\alpha_{k}\bar{\sigma}_{W}^{k}\right)\right]$
$\displaystyle=\sup_{\pi_{0}(x_{0})}S_{Q}(\mathcal{F}(\rho_{Z})||\mathcal{F}(\bar{\sigma}_{Z}))+\sup_{\pi_{1}(x_{1})}S_{Q}\left(\rho_{W}||\sum_{k}\alpha_{k}\bar{\sigma}_{W}^{k}\right).$
(90)

Now, by Lemma 5, $\rho_{Z}$ and $\bar{\sigma}_{Z}$ are the CQ-states
associated to $(\\{\varrho_{a|x}\\},\pi_{1}(x))$ and
$(\\{\bar{\varsigma}_{a|x}\\},\pi_{1}(x))$, respectively. On the other hand,
as $\\{\varrho_{a|x}\\}$ is steerable, and $\\{\varsigma_{a|x}\\}$ is
unsteerable, it follows that $\\{\varrho_{a|x}\\}\neq\\{\varsigma_{a|x}\\}$.
Therefore, by Lemma 8, if $\pi_{0}(x)>0$ $\forall$ $x$, then
$\mathcal{F}(\rho_{Z})\neq\mathcal{F}(\bar{\sigma}_{Z})$. But, we also have
that $S_{Q}(\rho||\sigma)=0\iff\rho=\sigma$. Therefore, for positive
probability distributions, we have
$S_{Q}(\mathcal{F}(\rho_{Z})||\mathcal{F}(\bar{\sigma}_{Z}))>0$. In this way,

$\sup_{\pi_{0}(x_{0})}S_{Q}(\mathcal{F}(\rho_{Z})||\mathcal{F}(\bar{\sigma}_{Z}))>0.$
(91)

Applying this fact to eq. (C), we have:

$E_{A}(\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\})>\sup_{\pi_{1}(x_{1})}S_{Q}\left(\rho_{W}||\sum_{k}\alpha_{k}\bar{\sigma}_{W}^{k}\right).$
(92)

However, by Corrolaries 6 and 7, $\sum_{k}\alpha_{k}\bar{\sigma}_{W}^{k}$ is a
CQ-state associated to a unsteerable assemblage. Thus,

$\displaystyle E_{A}(\\{\varrho_{a_{0}a_{1}|x_{0}x_{1}}\\})$
$\displaystyle>\sup_{\pi_{1}(x_{1})}S_{Q}\left(\rho_{W}||\sum_{k}\alpha_{k}\bar{\sigma}_{W}^{k}\right)$
$\displaystyle\geq\sup_{\pi_{1}(x_{1})}\inf_{\varsigma\in
UR_{ns}}S_{Q}(\rho_{W}||\sigma_{W})$ (93)
$\displaystyle=E_{A}(\\{\varrho_{a_{0}|x_{0}}\\})=E_{A}(\\{\varrho^{\prime}_{a|x}\\}).$
(94)

∎

##  Appendix D Proof of Theorem 2

See 2

###### Proof.

Let’s suppose, by contradiction, that there is a $UR_{ns}$-LOSR transformation
$\mathcal{M}$ that broadcasts a steerable assemblage
$\\{\varrho^{\prime}_{a|x}\\}$. So, by Proposition 3, we have:

$E_{A}\left(\mathcal{M}(\\{\varrho^{\prime}_{a|x}\\})\right)\leq
E_{A}(\\{\varrho^{\prime}_{a|x}\\}).$ (95)

On the other hand, as $\mathcal{M}(\\{\varrho^{\prime}_{a|x}\\})$ is a
broadcasting of $\\{\varrho^{\prime}_{a|x}\\}$, by Proposition 4:

$E_{A}(\\{\varrho^{\prime}_{a|x}\\})<E_{A}\left(\mathcal{M}(\\{\varrho^{\prime}_{a|x}\\})\right).$
(96)

Therefore, by eqs. (95) and (96):

$E_{A}(\\{\varrho^{\prime}_{a|x}\\})<E_{A}(\\{\varrho^{\prime}_{a|x}\\}),$
(97)

which is a contradiction. Thus, it is impossible to broadcast a steerable
assemblage by $UR_{ns}$-LOSR transformations. ∎