# No-broadcasting characterizes operational contextuality

Pauli Jokinen<EMAIL_ADDRESS>Mirjam Weilenmann  Department of
Applied Physics, University of Geneva, Switzerland  Martin Plávala
Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-
Straße 3, 57068 Siegen, Germany  Juha-Pekka Pellonpää  Department of Physics
and Astronomy, University of Turku, FI-20014 Turun yliopisto, Finland  Jukka
Kiukas  Department of Mathematics, Aberystwyth University, Aberystwyth SY23
3BZ, United Kingdom  Roope Uola  Department of Applied Physics, University of
Geneva, Switzerland

###### Abstract

Operational contextuality forms a rapidly developing subfield of quantum
information theory. However, the characterization of the quantum mechanical
entities that fuel the phenomenon has remained unknown with many partial
results existing. Here, we present a resolution to this problem by connecting
operational contextuality one-to-one with the no-broadcasting theorem. The
connection works both on the level of full quantum theory and subtheories
thereof. We demonstrate the connection in various relevant cases, showing
especially that for quantum states the possibility of demonstrating
contextuality is exactly characterized by non-commutativity, and for
measurements this is done by a norm-1 property closely related to
repeatability. Moreover, we show how techniques from broadcasting can be used
to simplify known foundational results in contextuality.

##  I Introduction

Contextuality is a fundamental notion that was originally introduced to rule
out (deterministic) hidden variable models for quantum theory [1]. It captures
the idea that measurements in quantum theory generally cannot be considered as
revealing pre-existing classical values independent of the measurement
context. In contrast to Bell inequalities [2] that are concerned with space-
like separated parties, contextuality can be defined in prepare-and-measure
scenarios on a single system. Contextuality has been established as a feature
of quantum theory that has been shown – in a similar vein as the presence of
entanglement – to be a prerequisite for various quantum advantages, including
in quantum computation [3, 4, 5, 6] and communication tasks [7, 8, 9, 10].

While the original notion of contextuality was referring to projective quantum
measurements, this has since been extended to a notion that applies to more
general operational theories and unsharp measurements [11]. This broader form
of contextuality includes two central concepts: _preparation contextuality_
and _measurement contextuality_ , allowing one to investigate more refined
classical models. These have recently experienced a surge in research activity
in both quantum and general operational theories [12], and different types of
contextuality have been further connected to other foundational concepts such
as measurement incompatibility [13, 14, 15], entanglement [16], steering [13,
14], negativity [17] and anomalous weak values [18, 19].

Figure 1: Non-contextuality of an operational theory and the corresponding
quantum mechanical entities. On the left (resp. right) are the properties
requested from an ontological model for states (resp. measurements). The color
green refers to ontological models for full theory, yellow refers to
subtheories, and red refers to models exhibiting contextuality. The binding
quantum mechanical entity is given in the middle. The black connections
represent known results from Refs. [13, 14, 15, 16]. The contributions of this
work are given by the blue connections.

In this work, we ask whether there is a fundamental quantum entity that fully
characterizes generalized contextuality. As our main result, we give a
positive answer to this question by mapping contextuality one-to-one with the
no-broadcasting theorem [20, 21]. The proof of this fact is based on the
realization that non-contextual models can be interpreted as measure-and-
prepare mechanisms, where only classical information passes through. One
direction of the proof follows since classical information is broadcastable,
and the other is based on an averaging argument over broadcasting channels.
This shows that generalized contextuality captures a well-established
operational entity in quantum theory, and that advantages provided by
contextuality can alternatively be attributed to the no-broadcasting theorem.
We further show that for subtheories of quantum theory the characterization is
given by pseudo-broadcasting.

We demonstrate the power of the connection by using results from broadcasting
[20, 21] in the realm of contextuality. First, we completely characterize the
sets of quantum states and measurements that allow for proofs of contextuality
of quantum theory by using the no-broadcasting theorem. This complements a
line of research on characterizing and witnessing different facets of
contextuality using entities of quantum theory [22, 23, 24, 25, 26, 27, 13,
14, 15, 16]. In contrast, we characterize full, i.e. preparation and
measurement, contextuality, cf. Fig. 1. This allows one to immediately obtain
known results showing contextuality without incompatibility [15] and
robustness of contextuality to arbitrary amounts of noise [28], as analogous
results are known for broadcasting [29, 30, 31, 32]. Second, we show that the
connection is beneficial for understanding the relationship between
contextuality and other established notions of classicality given by joint
measurability [33, 34], non-disturbance [35], and Leggett-Garg macrorealism
[36, 37, 38, 39]. Finally, checking directly whether an arbitrary sub-theory
of quantum theory allows for a non-contextual model is in general not a
straightforward task. We demonstrate how pseudo-broadcasting can be used to
derive non-contextuality criteria in such setting.

##  II Operational contextuality

In measurement and preparation non-contextual theories, the probabilities of
the theory can be reproduced in the following way. Let $\\{P\\}$ be the convex
set of preparations and $\\{M\\}$ the convex set of measurements of the theory
with outcomes labelled by $k$. Then the probabilities are reproduced according
to the following ontological model:

$\displaystyle p(k|P,M)=\sum_{\lambda}\mu_{P}(\lambda)\xi_{M,k}(\lambda).$ (1)

Here $\mu_{P}(\lambda)$ are probabilities with
$\sum_{\lambda}\mu_{P}(\lambda)=1$ for all $P$ and $\xi_{M,k}(\lambda)$ are
indicator functions with $\sum_{k}\xi_{M,k}(\lambda)=1$ for all $M$ and
$\lambda$. Furthermore $\mu_{P}$ and $\xi_{M,k}$ depend only on the
operational equivalence classes [11], i.e., classes of preparations (resp.
measurements) that are not distinguishable by any measurements (resp. states).
This motivates the term operational contextuality and implies that $\mu_{P}$
and $\xi_{M,k}$ are affine with respect to (equivalence classes of)
preparations and measurements respectively [40].

We now ask what this notion of non-contextuality means in quantum theory. In
this case, preparation equivalence classes are formally described as quantum
states, i.e. positive semi-definite unit-trace linear operators acting on a
finite-dimensional Hilbert space, denoted by $\rho$. Furthermore, equivalence
classes of measurements, or observables, are described by positive operator
valued measures (POVMs), i.e. collections of positive linear operators
$M=\\{M_{k}\\}_{k}$ with the normalization $\sum_{k}M_{k}=I$. In quantum
theory probabilities are produced via the Born rule $p(k|\rho,M)=\tr{\rho
M_{k}}$. Using basic Hilbert space duality and the affinity requirements of
$\mu_{\rho}$ and $\xi_{M_{k}}$ one sees that Eq. (1) gets the form

$\displaystyle\tr{\rho M_{k}}=\sum_{\lambda}\tr{\rho
G_{\lambda}}\tr{\sigma_{\lambda}M_{k}},$ (2)

where $\\{G_{\lambda}\\}_{\lambda}$ and $\\{\sigma_{\lambda}\\}_{\lambda}$ are
such that $\tr{\rho G_{\lambda}}\geq 0$, $\sum_{\lambda}\tr{\rho
G_{\lambda}}=1$, and $\tr{\sigma_{\lambda}M_{k}}\geq 0$ for all $\lambda$, all
states $\varrho$ and all POVMs $\\{M_{k}\\}_{k}$. In other words, a non-
contextual model for quantum theory is equivalent to Eq. (2) with
$\\{G_{\lambda}\\}_{\lambda}$ being a POVM and
$\\{\sigma_{\lambda}\\}_{\lambda}$ being quantum states.

##  III Broadcasting

Classical information can be copied arbitrarily many times in an exact manner.
In quantum theory the same idea is captured by the concept of broadcasting and
the inability to copy information stored in a quantum state is known as the
no-broadcasting theorem [20, 21]. To define broadcasting, we need the notion
of _quantum channels_. A quantum channel
$\Lambda:\mathcal{L(H)}\to\mathcal{L}(\mathcal{K})$ is a completely positive
trace preserving linear map. Here $\mathcal{H}$ and $\mathcal{K}$ denote
finite dimensional Hilbert spaces and $\mathcal{L(H)}$ is the set of linear
operators acting on $\mathcal{H}$. Let now $\mathcal{T}$ be a subset of states
and $\mathcal{A},\mathcal{B}$ subsets of observables acting on $\mathcal{H}$.
Then the triple $(\mathcal{T},\mathcal{A},\mathcal{B})$ is called
_broadcastable_ if there exists a channel
$\Lambda:\mathcal{L(H)}\to\mathcal{L}(\mathcal{H}\otimes\mathcal{H})$ such
that the following holds for all $\\{A_{k}\\}_{k}\in\mathcal{A}$,
$\\{B_{k}\\}_{k}\in\mathcal{B}$, $\rho\in\mathcal{T}$, and $\forall k$:

$\displaystyle\tr{A_{k}\tr_{2}{\Lambda(\rho)}}$
$\displaystyle=\tr{A_{k}\rho},$ (3)
$\displaystyle\tr{B_{k}\tr_{1}{\Lambda(\rho)}}$
$\displaystyle=\tr{B_{k}\rho}.$ (4)

Here $\tr_{i}(\cdot)$ for $i=1,2$ denotes the partial trace over subsystem
$i$. One can directly extend this definition to a notion of $1\mapsto
n$_-broadcasting_. The tuple
$(\mathcal{T},\mathcal{A}_{1},\dots,\mathcal{A}_{n})$ is $1\mapsto
n$-broadcastable, if there exists a channel
$\Lambda:\mathcal{L(H)}\to\mathcal{L}(\mathcal{H}^{\otimes n})$ such that for
all $i\in\\{1,\dots,n\\}$ and for all $\\{A_{i,k}\\}_{k}\in\mathcal{A}_{i}$
the condition $\tr{A_{i,k}\tr_{i^{c}}\Lambda(\rho)}=\tr{A_{i,k}\rho}$ holds
for all $\rho\in\mathcal{T}$. Here $\tr_{i^{c}}(\cdot)$ denotes the partial
trace over the complement of $\\{i\\}$ i.e. the subsystems not labeled by the
index $i$. Note that we may assume above that $\mathcal{T}$ is convex, by
passing to the convex hull if necessary, since (3) and (4) are linear with
respect to $\rho$.

##  IV Connection between operational non-contextuality and broadcasting

Quantum theory is contextual. It is natural to what the operational
ingredients that are responsible for this phenomenon are. In this section, we
give an answer to this question. We first state the main observation, the
validity of which follows from the two special cases presented in Theorem 1
for states and Theorem 3 for measurements.

######  Observation.

Any proof of contextuality of quantum theory is equivalent to a proof of no-
broadcasting.

It is apparent from Eq. (2), that a subset of states that does not allow for a
proof of contextuality of quantum theory, has to be a subset of the fixed
points of some measure-and-prepare or entanglement breaking channel (EBC)
$\Lambda$ given by $\rho=\sum_{\lambda}\tr{\rho
G_{\lambda}}\sigma_{\lambda}=:\Lambda(\rho)$ [41]. This formulation encodes
the intuition of non-contextual models being measure-and-prepare models
allowing only classical information $\lambda$ to pass through from the
measurement $\\{G_{\lambda}\\}_{\lambda}$ to the preparation
$\\{\sigma_{\lambda}\\}_{\lambda}$. Using this, we can prove the following
theorem. Here
$\mathrm{Fix}(\Lambda)=\\{\rho\in\mathcal{S(H)}\,|\,\Lambda(\rho)=\rho\\}$,
where $\mathcal{S(H)}$ denotes the set of quantum states in the Hilbert space
$\mathcal{H}$, and $\mathcal{O}$ denotes the set of all observables.

######  Theorem 1.

Let $\mathcal{T}\subset\mathcal{S(H)}$. Then there is an EBC $\Lambda$ such
that $\mathcal{T}\subset\mathrm{Fix}(\Lambda)$ if and only if
$(\mathcal{T},\mathcal{O},\mathcal{O})$ is broadcastable.

The proof of this theorem, which is based on an averaging argument over
repeated broadcasting channels, is presented in the Appendix A. The following
Corollary is a direct application of the no-broadcasting theorem [21].

######  Corollary 2.

A set of quantum states $\mathcal{T}$ does not allow for a proof of
contextuality of quantum theory if and only if $\mathcal{T}$ is commutative.

The problem of identifying sets of measurements allowing for a proof of
contextuality of quantum theory also reduces to a fixed point problem. One
sees from Eq. (2) that a non-contextual model entails that
$M_{k}=\sum_{\lambda}\tr{\sigma_{\lambda}M_{k}}G_{\lambda}$. In other words,
the measurements need to be fixed points of the Heisenberg picture
$\Lambda^{*}$ of an EBC $\Lambda$.

Before stating the result for measurements, we need to define _instruments_
and _repeatability_ of measurements. An instrument is a collection of trace-
non-increasing completely positive linear maps $\\{\mathcal{I}_{k}\\}_{k}$
such that $\sum_{k}\mathcal{I}_{k}$ is a quantum channel. An instrument
$\\{\mathcal{I}_{k}\\}_{k}$ is said to implement the measurement
$\\{A_{k}\\}_{k}$ if $\mathcal{I}_{k}^{*}(I)=A_{k}$ for all $k$. If there
exists an instrument $\\{\mathcal{I}_{k}\\}_{k}$ implementing
$\\{A_{k}\\}_{k}$ such that $\mathcal{I}_{k}^{*}(A_{k})=A_{k}$ for all $k$,
then $\\{A_{k}\\}_{k}$ is called repeatable. In the following $\|\cdot\|$
denotes the operator norm, and classical post-processing of a POVM
$\\{G_{\lambda}\\}_{\lambda}$ refers to a POVM
$A_{k}=\sum_{\lambda}p(k|\lambda)G_{\lambda}$, where $p(\cdot|\lambda)$ is a
probability distribution for each $\lambda$.

######  Theorem 3.

For a set of observables $\mathcal{A}\subset\mathcal{O}$, the following
statements are equivalent.

  1. 1.

$\mathcal{A}$ does not allow for a proof of contextuality of quantum theory.

  2. 2.

$(\mathcal{S(H)},\mathcal{A},\mathcal{A})$ is broadcastable.

  3. 3.

Every $\\{A_{k}\\}_{k}\in\mathcal{A}$ is a classical post-processing of a
single norm-1 POVM $\\{G_{\lambda}\\}_{\lambda}$, i.e. $\|G_{\lambda}\|=1$,
$\forall\lambda$.

  4. 4.

Every $\\{A_{k}\\}_{k}\in\mathcal{A}$ is a classical post-processing of a
single repeatable measurement.

The proof is presented in the Appendix B. We note that the equivalence between
conditions $2$ and $3$ is the no-broadcasting theorem for measurements [32],
which is here utilized to give more structure for measurements fulfilling
condition $1$.

To understand condition $3$ better, we investigate a few example cases. First,
commutative sets of measurements fulfill the conditions of Theorem 3. This is
due to a theorem by von Neumann [42, Theorem 11.3] stating that commutative
self-adjoint operators are functions of a common self-adjoint operator.

Second, the conditions of Theorem 3 do not imply commutativity, as there are
non-commuting and broadcastable measurements [32]. An explicit example is
given by a norm-1 measurement. These are measurements that have a projective
part in one subspace and possibly a non-projective part in the orthogonal
complement. Intuitively, the deterministic or projective part allows one to
pick the classical information $\lambda$ that a non-contextual model can pass
through. More precisely, let $\mathcal{H}$ be a 5-dimensional Hilbert space
with the orthonormal basis $\\{\ket{i}\\}_{i=0}^{4}$. Define $\ket{+}$ and
$\ket{-}$: $\ket{\pm}=\frac{1}{\sqrt{2}}(\ket{0}\pm\ket{1})$. Let $a\in(0,1)$
and define the following three-valued POVM:

$\displaystyle E_{1}$ $\displaystyle:=|2\rangle\langle 2|+a|0\rangle\langle
0|+(1-a)|+\rangle\langle+|$ $\displaystyle E_{2}$
$\displaystyle:=|3\rangle\langle 3|+a|1\rangle\langle 1|$ $\displaystyle
E_{3}$ $\displaystyle:=|4\rangle\langle 4|+(1-a)|-\rangle\langle-|$

This POVM is clearly noncommutative, since for example $E_{2}E_{3}\neq
E_{3}E_{2}$. Define then the EB-channel $\Lambda$ as
$\Lambda^{*}(A):=\sum_{z=1}^{3}\tr{A|z+1\rangle\langle z+1|}E_{z}$. Then we
have

$\displaystyle\Lambda^{*}(E_{k})=\sum_{z=1}^{3}\tr{E_{k}|z+1\rangle\langle
z+1|}E_{z}=\sum_{z=1}^{3}\delta_{kz}E_{z}=E_{k}$

Therefore there exist measurements that are non-commutative but do not allow
for a proof of contextuality of quantum theory.

Finally, there are special sets of measurements for which the conditions of
Theorem 3 imply commutativity. One example is given by rank-1 measurements. As
stated above, a POVM $\\{G_{\lambda}\\}_{\lambda}$ is of norm 1 if and only if
$G_{\lambda}=P_{\lambda}+F_{\lambda}$ for all $\lambda$, where
$\\{P_{\lambda}\\}_{\lambda}$ is a projection valued observable acting on a
subspace (with $P_{\lambda}\neq 0$ for all $\lambda$) and
$\\{F_{\lambda}\\}_{\lambda}$ a POVM acting on the orthogonally complemented
subspace [43]. Moreover, if a rank-1 POVM $\\{A_{k}\\}_{k}$ is a
postprocessing of a norm-1 $\\{G_{\lambda}\\}_{\lambda}$, then $F_{\lambda}=0$
for all $\lambda$ and $\\{P_{\lambda}\\}_{\lambda}$ is a rank-1 basis
measurement, i.e.
$P_{\lambda}=|\varphi_{\lambda}\rangle\langle\varphi_{\lambda}|$ for some
orthonormal basis vectors $\varphi_{\lambda}$ [44]. Now
$A_{k}=p_{k}P_{\lambda_{k}}$ where $p_{k}\in[0,1]$ so $\\{A_{k}\\}_{k}$ is
commutative. This is summarized in the following Corollary.

######  Corollary 4.

Suppose $\\{A_{k}\\}_{k}$ is a rank-1 POVM. Then it fulfills the equivalent
conditions of Theorem 3 if and only if it is commutative.

##  V Relation to other notions of classicality

Non-contextuality can be seen as a dividing line between classical and quantum
behaviour, in that it asks whether a given theory is simplex embeddable [45].
In a simplex theory, states have a unique decomposition into extreme points.
Our Theorem 3 helps one to relate contextuality to other notions of non-
classicality.

As the first example, we take joint measurability [33, 34]. In Ref. [13] it
was shown that a set of measurements allows for a proof of preparation
contextuality of quantum theory if and only if the set is not jointly
measurable, i.e. not a post-processing of any single POVM. In contrast,
Theorem 3 requires such single POVM to be norm-1. This complements an example
given in Ref. [15] of POVMs that allow for a proof of full contextuality, i.e.
do not fulfill the conditions of Theorem 3, but are jointly measurable. There
are indeed considerably more measurements allowing for a proof of full
contextuality than those allowing for a proof of preparation contextuality:
Norm-1 POVMs can not have more outcomes than the dimension of the underlying
Hilbert space. Combining this with a dimension counting argument shows that
any subset of measurements not allowing for a proof of full contextuality of
quantum theory has zero volume. On the contrary, it is well known that the
jointly measurable subset of measurements has a non-zero volume [46].

As another example, the notion of inherent measurement disturbance goes back
to Heisenberg’s microscope, and has a clear operational formulation, cf. Refs.
[35, 47, 42]. Here, we follow Ref. [35]: a measurement $\\{A_{a}\\}$ does not
disturb a measurement $\\{B_{b}\\}$ if there is an instrument
$\\{\mathcal{I}_{a}\\}$ implementing $\\{A_{a}\\}$ such that
$\sum_{a}\mathcal{I}_{a}^{*}(B_{b})=B_{b}$ for all outcomes $b$. In the qubit
case, non-disturbance reduces to commutativity [35], but in qutrits and beyond
there are non-commuting and non-disturbing measurements [35]. An example is
given by the following pair of binary qutrit POVMs [35]:
$A_{1}=\frac{1}{4}\big{(}2|0\rangle\langle 0|+|2\rangle\langle
2|+\sqrt{2}(|0\rangle\langle 2|+|2\rangle\langle 0|)\big{)}$ and
$B_{1}=\frac{1}{2}\big{(}2|0\rangle\langle 0|+|2\rangle\langle 2|\big{)}$,
with $A_{2}=I-A_{1}$ and $B_{2}=I-B_{1}.$ Taking an optimal non-disturbing
implementation $\\{\mathcal{I}_{a}\\}_{a}$ of $\\{A_{a}\\}_{a}$, we define the
POVM $\\{\mathcal{I}_{a}^{*}(B_{b})\\}_{a,b}$. This POVM does not fulfill the
conditions of Theorem 3, as it is not a post-processing of a norm-1
measurement: If the pair given by $\\{A_{a}\\}$ and $\\{B_{b}\\}$ were a post-
processing of a norm-1 POVM $\\{G_{\lambda}\\}$, the linear independence of
$\\{A_{2},B_{1},B_{2}\\}$ would imply that
$\dim\mathrm{lin}(\\{G_{\lambda}\\})\geq 3$. Since
$G_{\lambda}=P_{\lambda}+F_{\lambda}$, where $\\{P_{\lambda}\\}$ is a
projection valued measure, i.e. a POVM consisting of projections, this implies
that $\dim\mathrm{lin}(\\{G_{\lambda}\\})=3$ and furthermore that
$G_{\lambda}=P_{\lambda}$, as the underlying Hilbert space is
$\mathbb{C}^{3}$. This in turn would imply that $A$ and $B$ are classical
post-processings of a common projection valued measure, i.e. commutative. This
is a contradiction. Hence, operational non-contextuality in quantum theory is
more restrictive than non-disturbance.

The above example can be directly applied to temporal correlations. It is
well-known that a temporal correlation scenario consisting of non-disturbing
measurements can be described by a macrorealistic hidden variable model for
all input quantum states [48]. These are models similar to Bell’s local models
with the locality assumption replaced by a non-invasiveness assumption.
Therefore, the sequential POVM $\\{\mathcal{I}_{a}^{*}(B_{b})\\}_{a,b}$ does
allow for a proof of contextuality, but not for a violation of macrorealism.

##  VI Pseudo-broadcasting and conditions for contextuality in a subtheory

One can also define operational contextuality for other theories than quantum.
We are here interested in theories that consist of some convex subset of
quantum states and a convex set of effects $E$, i.e. operators with $0\leq
E\leq I$, that includes the identity operator and yes-no questions, i.e., if
an effect $E$ is part of the theory, then also $I-E$ is. Such a theory is
called measurement and preparation non-contextual if

$\displaystyle\tr{\rho
E}=\sum_{\lambda}\tr{\rho\tilde{G}_{\lambda}}\tr{\tilde{\sigma}_{\lambda}E}$
(5)

for all states $\rho$ and effects $E$ of the theory. Here
$\\{\tilde{G}_{\lambda}\\}_{\lambda}$ and
$\\{\tilde{\sigma}_{\lambda}\\}_{\lambda}$ satisfy
$\tr{\rho\tilde{G}_{\lambda}}\geq 0$ and
$\sum_{\lambda}\tr{\rho\tilde{G}_{\lambda}}=1$ for all states of the theory,
and $\tr{\tilde{\sigma}_{\lambda}E}\geq 0$ and
$\tr{\tilde{\sigma}_{\lambda}}=1$ for all $\lambda$ and all effects $E$ of the
theory. Note that this is different from full quantum theory, where positivity
is required for all quantum states and measurements. Hence, non-contextuality
in subtheories is a weaker notion.

One can also relax broadcasting by requiring that the broadcasting map
$\Lambda$ in Eq. (3) and Eq. (4) is not necessarily completely positive, but
only by requiring that it preserves positivity of all probabilities that can
be produced from $(\mathcal{T},\mathcal{A}_{1},\dots,\mathcal{A}_{n})$. That
is, we say that $(\mathcal{T},\mathcal{A}_{1},\dots,\mathcal{A}_{n})$ is
$1\mapsto n$_-pseudo-broadcastable_ if there is a trace-preserving map
$\Xi:\mathcal{L(H)}\to\mathcal{L}(\mathcal{H}^{\otimes n})$ such that for all
$i\in\\{1,\dots,n\\}$, $\rho\in\mathcal{T}$, and
$\\{A_{i,k}\\}_{k}\in\mathcal{A}_{i}$ it holds that

$\displaystyle\tr{A_{i,k}\tr_{i^{c}}\Xi(\rho)}$
$\displaystyle=\tr{A_{i,k}\rho},$ (6)
$\displaystyle\tr{\Xi(\rho)\otimes_{i=1}^{n}A_{i,k}}$ $\displaystyle\geq 0.$
(7)

Clearly (7) is a relaxation of the complete positivity of the broadcasting
channel $\Lambda$ and one can find examples that are pseudo-broadcastable for
all $n$ but not broadcastable, see the Appendix C for an example of a qubit
symmetric informationally complete POVM having such property. We will be
mostly interested in the case when $\mathcal{A}_{1}=\ldots=\mathcal{A}_{n}$,
we will denote the tuple as $(\mathcal{T},\mathcal{A})$, where $n$ will be
understood from context.

The combined notion of measurement and preparation non-contextuality of a
subtheory can be seen to be equivalent to $1\mapsto n$-pseudobroadcasting for
all $n\in\mathbb{N}$.

######  Theorem 5.

A subtheory of quantum theory characterized by the allowed states
$\mathcal{T}$ and allowed measurements $\mathcal{A}$ is measurement and
preparation non-contextual if and only if $(\mathcal{T},\mathcal{A})$ is
$1\mapsto n$-pseudo-broadcastable for all $n\in\mathbb{N}$.

The proof, based on recent results in monogamy of ordered vector spaces and
general probabilistic theories is relegated to the Appendix D.

Figure 2: The regions for pseudobroadcastability of the preparations and
measurements considered in [49]. $\mu$ and $\eta$ are unitless parameters of
the dephasing channels acting on the preparations and depolarizing channel
acting on the measurements, respectively. The green region where the scenario
is $1\mapsto 3$-pseudobroadcastable coincides with the region where the
inequality presented in [49] is violated up to numerical precision of our
calculations.

One can also use the result of Theorem 5 to prove non-contextuality of sets of
preparations and measurements: for finite sets of preparations and
measurements and for fixed $n\in\mathbb{N}$, checking whether there exists
$1\mapsto n$-pseudobroadcasting linear map is a semidefinite programming (SDP)
problem since the conditions (6) and (7) are linear in the linear map $\Xi$.
We use this to investigate contextuality of the preparations and measurements
used in [49]. These consist of six operators, all of which lie on the $x-z$
plane of the Bloch sphere and are separated by the angle $\pi/3$. We will
moreover assume that a dephasing channel with parameter $\mu$ is acting on the
preparations and a depolarizing channel with parameter $\eta$ is acting on the
measurements, we will use $\mu$ and $\eta$ as parameters with respect to which
we will investigate whether the scenario is contextual or not. The results are
depicted in Fig. 2 and the details are relegated to the Appendix E. For $n=2$
we obtain a region of the parameter space where the scenario is $1\mapsto
2$-pseudobroadcastable and thus, according to Theorem 5, outside of this
region the scenario must be contextual. For $n=3$ we obtain a smaller region
of the parameter space where the scenario is $1\mapsto 3$-pseudobroadcastable,
this parameter space coincides up to numerical precision with the region where
the inequality presented in [49] is violated. This is likely not a coincidence
since the scenario is generated by $3$ measurements and we conjecture that for
a scenario generated by $k$ measurements $1\mapsto k$-pseudobroadcasting
coincides with non-contextuality. An intuition for why this conjecture may be
true is given in the Appendix D after the proof of Theorem 5 as this intuition
is based on the techniques used in the respective proof.

It is noteworthy that while our method agrees with the inequality presented in
[49], our method can be easily applied to other similar scenarios and solved
numerically without the need for finding additional contextuality
inequalities. This example also shows a scenario which is $1\mapsto
2$-pseudobroadcastable but not $1\mapsto 3$-pseudobroadcastable, thus showing
that unlike in the case of broadcasting, $1\mapsto(n+1)$-pseudobroadcasting is
not implied by $1\mapsto n$-pseudobroadcasting.

##  VII Conclusions

By proving a one-to-one connection between non-contextuality and broadcasting
in quantum theory, we have provided an operational, information theoretic
characterization of the former. This not only shows that non-contextuality
connects to another well-known concept in quantum information theory, but it
also allows one to transfer insights between the two fields. As an example,
there have been various efforts to understand the interplay between known
quantum properties and proofs of contextuality [22, 23, 24, 25, 26, 27, 13,
14, 15, 16], but the characterization of full contextuality was missing until
now. This was reached here for both quantum states and measurements by
utilizing the no-broadcasting theorem.

We further showed that our results help one to relate non-contextuality to
other notions of classicality, such as joint measurability, non-disturbance
and Leggett-Garg macrorealism. Our results suggest that non-contextuality is
the most restrictive notion of these four in quantum theory. This complements
the recent results of Ref. [39] on connections between macrorealism and non-
contextuality, and shows that measurements not allowing for a proof of
contextuality of quantum theory need to have measure zero.

Finally, we showed that our results are not restricted to quantum theory, but
they also apply to subtheories by using the notion of pseudo-broadcasting.
This notion can be easily decided numerically, hence, providing a method for
finding witnesses for contextuality in subtheories.

Whether the results provided in this article carry over to continuous variable
systems is left as an intriguing open question. The solution may rely on the
used extension of non-contextuality to the continuous realm. One possible
extension can be found by drawing inspiration from our results and known
results on broadcasting in infinite-dimensional systems [50]. However, it is
beyond the scope of this work to investigate how this would relate to the
existing approaches to contextuality in such systems [51, 52, 53, 54, 55, 56,
57].

###### Acknowledgements.

We are thankful to Arindam Mitra for discussions to Jef Pauwels and Ismaël
Septembre for providing feedback on an earlier version of the manuscript. PJ
and RU acknowledge support from the Swiss National Science Foundation
(Ambizione PZ00P2-202179). MW is grateful for the support from the Swiss
National Science Foundation (Ambizione PZ00P2_208779). MP acknowledges support
from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation,
project numbers 447948357 and 440958198), the Sino-German Center for Research
Promotion (Project M-0294), the German Ministry of Education and Research
(Project QuKuK, BMBF Grant No. 16KIS1618K), the DAAD, and the Alexander von
Humboldt Foundation.

## References

  * Kochen and Specker [1967] S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics 17, 59 (1967).
  * Bell [1966] J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966).
  * Raussendorf [2013] R. Raussendorf, Contextuality in measurement-based quantum computation, Phys. Rev. A 88, 022322 (2013).
  * Howard  _et al._ [2014] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the ‘magic’for quantum computation, Nature 510, 351 (2014).
  * Delfosse  _et al._ [2015] N. Delfosse, P. Allard Guerin, J. Bian, and R. Raussendorf, Wigner function negativity and contextuality in quantum computation on rebits, Phys. Rev. X 5, 021003 (2015).
  * Bermejo-Vega  _et al._ [2017] J. Bermejo-Vega, N. Delfosse, D. E. Browne, C. Okay, and R. Raussendorf, Contextuality as a resource for models of quantum computation with qubits, Phys. Rev. Lett. 119, 120505 (2017).
  * Saha and Chaturvedi [2019] D. Saha and A. Chaturvedi, Preparation contextuality as an essential feature underlying quantum communication advantage, Phys. Rev. A 100, 022108 (2019).
  * Spekkens  _et al._ [2009] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Preparation contextuality powers parity-oblivious multiplexing, Phys. Rev. Lett. 102, 010401 (2009).
  * Schmid and Spekkens [2018] D. Schmid and R. W. Spekkens, Contextual advantage for state discrimination, Phys. Rev. X 8, 011015 (2018).
  * Cubitt  _et al._ [2010] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, Improving zero-error classical communication with entanglement, Phys. Rev. Lett. 104, 230503 (2010).
  * Spekkens [2005] R. W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements, Phys. Rev. A 71, 052108 (2005).
  * Selby  _et al._ [2023a] J. H. Selby, D. Schmid, E. Wolfe, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality, Phys. Rev. A 107, 062203 (2023a).
  * Tavakoli and Uola [2020] A. Tavakoli and R. Uola, Measurement incompatibility and steering are necessary and sufficient for operational contextuality, Phys. Rev. Res. 2, 013011 (2020).
  * Plávala [2022] M. Plávala, Incompatibility in restricted operational theories: connecting contextuality and steering, J. Phys. A 55, 174001 (2022).
  * Selby  _et al._ [2023b] J. H. Selby, D. Schmid, E. Wolfe, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Contextuality without incompatibility, Phys. Rev. Lett. 130, 230201 (2023b).
  * Plávala and Gühne [2024] M. Plávala and O. Gühne, Contextuality as a precondition for quantum entanglement, Phys. Rev. Lett. 132, 100201 (2024).
  * Spekkens [2008] R. W. Spekkens, Negativity and contextuality are equivalent notions of nonclassicality, Phys. Rev. Lett. 101, 020401 (2008).
  * Pusey [2014] M. F. Pusey, Anomalous weak values are proofs of contextuality, Phys. Rev. Lett. 113, 200401 (2014).
  * Kunjwal  _et al._ [2019] R. Kunjwal, M. Lostaglio, and M. F. Pusey, Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts, Phys. Rev. A 100, 042116 (2019).
  * Barnum  _et al._ [1996] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Noncommuting mixed states cannot be broadcast, Phys. Rev. Lett. 76, 2818 (1996).
  * Barnum  _et al._ [2006] H. Barnum, J. Barrett, M. Leifer, and A. Wilce, Cloning and broadcasting in generic probabilistic theories, arXiv:quant-ph/0611295 (2006).
  * Liang  _et al._ [2011] Y.-C. Liang, R. W. Spekkens, and H. M. Wiseman, Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity, Phys. Rep. 506, 1 (2011).
  * Yu and Oh [2013] S. Yu and C. H. Oh, Quantum contextuality and joint measurement of three observables of a qubit, arXiv:1312.6470 (2013).
  * Kunjwal and Ghosh [2014] R. Kunjwal and S. Ghosh, Minimal state-dependent proof of measurement contextuality for a qubit, Phys. Rev. A 89, 042118 (2014).
  * Kunjwal [2014] R. Kunjwal, A note on the joint measurability of povms and its implications for contextuality, arXiv:1403.0470 (2014).
  * Lostaglio [2018] M. Lostaglio, Quantum fluctuation theorems, contextuality, and work quasiprobabilities, Phys. Rev. Lett. 120, 040602 (2018).
  * Xu and Cabello [2019] Z.-P. Xu and A. Cabello, Necessary and sufficient condition for contextuality from incompatibility, Phys. Rev. A 99, 020103 (2019).
  * Rossi  _et al._ [2023] V. P. Rossi, D. Schmid, J. H. Selby, and A. B. Sainz, Contextuality with vanishing coherence and maximal robustness to dephasing, Phys. Rev. A 108, 032213 (2023).
  * Heinosaari [2016] T. Heinosaari, Simultaneous measurement of two quantum observables: Compatibility, broadcasting, and in-between, Phys. Rev. A 93, 042118 (2016).
  * Heinosaari  _et al._ [2019] T. Heinosaari, L. Leppäjärvi, and M. Plávala, No-free-information principle in general probabilistic theories, Quantum 3, 157 (2019).
  * Mitra [2021] A. Mitra, Layers of classicality in the compatibility of measurements, Phys. Rev. A 104, 022206 (2021).
  * Heinosaari  _et al._ [2023] T. Heinosaari, A. Jenčová, and M. Plávala, Dispensing of quantum information beyond no-broadcasting theorem – is it possible to broadcast anything genuinely quantum?, J. Phys. A 56, 135301 (2023).
  * Heinosaari  _et al._ [2016] T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, J. Phys. A 49, 123001 (2016).
  * Gühne  _et al._ [2023] O. Gühne, E. Haapasalo, T. Kraft, J.-P. Pellonpää, and R. Uola, Colloquium: Incompatible measurements in quantum information science, Rev. Mod. Phys. 95, 011003 (2023).
  * Heinosaari and Wolf [2010] T. Heinosaari and M. M. Wolf, Nondisturbing quantum measurements, J. Math. Phys. 51, 092201 (2010).
  * Leggett and Garg [1985] A. J. Leggett and A. Garg, Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?, Phys. Rev. Lett. 54, 857 (1985).
  * Emary  _et al._ [2013] C. Emary, N. Lambert, and F. Nori, Leggett–garg inequalities, Rep. Prog. Phys. 77, 016001 (2013).
  * Vitagliano and Budroni [2023] G. Vitagliano and C. Budroni, Leggett-garg macrorealism and temporal correlations, Phys. Rev. A 107, 040101 (2023).
  * Schmid [2024] D. Schmid, A review and reformulation of macroscopic realism: resolving its deficiencies using the framework of generalized probabilistic theories, Quantum 8, 1217 (2024).
  * Müller and Garner [2023] M. P. Müller and A. J. Garner, Testing quantum theory by generalizing noncontextuality, Phys. Rev. X 13, 041001 (2023).
  * Horodecki  _et al._ [2003] M. Horodecki, P. W. Shor, and M. B. Ruskai, Entanglement breaking channels, Rev. Math. Phys. 15, 629–641 (2003).
  * Busch  _et al._ [2016] P. Busch, P. Lahti, J.-P. Pellonpää, and K.Ylinen,  _Quantum Measurement (Theoretical and Mathematical Physics)_ (Springer, 2016).
  * Haapasalo and Pellonpää [2017] E. Haapasalo and J.-P. Pellonpää, Optimal quantum observables, J. Math. Phys. 58, 122104 (2017).
  * Pellonpää [2014] J.-P. Pellonpää, On coexistence and joint measurability of rank-1 quantum observables, J. Phys. A 47, 052002 (2014).
  * Schmid  _et al._ [2021] D. Schmid, J. H. Selby, E. Wolfe, R. Kunjwal, and R. W. Spekkens, Characterization of noncontextuality in the framework of generalized probabilistic theories, PRX Quantum 2, 010331 (2021).
  * Reeb  _et al._ [2013] D. Reeb, D. Reitzner, and M. M. Wolf, Coexistence does not imply joint measurability, J. Phys. A 46, 462002 (2013).
  * Busch  _et al._ [2014] P. Busch, P. Lahti, and R. F. Werner, Colloquium: Quantum root-mean-square error and measurement uncertainty relations, Rev. Mod. Phys. 86, 1261 (2014).
  * Uola  _et al._ [2019] R. Uola, G. Vitagliano, and C. Budroni, Leggett-garg macrorealism and the quantum nondisturbance conditions, Phys. Rev. A 100, 042117 (2019).
  * Mazurek  _et al._ [2016] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, An experimental test of noncontextuality without unphysical idealizations, Nat. Commun. 7, ncomms11780 (2016).
  * Kuramochi [2020] Y. Kuramochi, Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements, arXiv:2002.03504 (2020).
  * Plastino and Cabello [2010] A. R. Plastino and A. Cabello, State-independent quantum contextuality for continuous variables, Phys. Rev. A 82, 022114 (2010).
  * McKeown  _et al._ [2011] G. McKeown, M. G. A. Paris, and M. Paternostro, Testing quantum contextuality of continuous-variable states, Phys. Rev. A 83, 062105 (2011).
  * Su  _et al._ [2012] H.-Y. Su, J.-L. Chen, C. Wu, S. Yu, and C. H. Oh, Quantum contextuality in a one-dimensional quantum harmonic oscillator, Phys. Rev. A 85, 052126 (2012).
  * Asadian  _et al._ [2015] A. Asadian, C. Budroni, F. E. S. Steinhoff, P. Rabl, and O. Gühne, Contextuality in phase space, Phys. Rev. Lett. 114, 250403 (2015).
  * Laversanne-Finot  _et al._ [2017] A. Laversanne-Finot, A. Ketterer, M. R. Barros, S. P. Walborn, T. Coudreau, A. Keller, and P. Milman, General conditions for maximal violation of non-contextuality in discrete and continuous variables, J. Phys. A 50, 155304 (2017).
  * Barbosa  _et al._ [2022] R. S. Barbosa, T. Douce, P.-E. Emeriau, E. Kashefi, and S. Mansfield, Continuous-variable nonlocality and contextuality, Commun. Math. Phys. 391, 1047–1089 (2022).
  * Booth  _et al._ [2022] R. I. Booth, U. Chabaud, and P.-E. Emeriau, Contextuality and wigner negativity are equivalent for continuous-variable quantum measurements, Phys. Rev. Lett. 129, 230401 (2022).
  * Chiribella [2011] G. Chiribella, On quantum estimation, quantum cloning and finite quantum de finetti theorems, in  _Theory of Quantum Computation, Communication, and Cryptography_ , edited by W. van Dam, V. M. Kendon, and S. Severini (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 9–25.
  * Busch  _et al._ [1996] P. Busch, P. Lahti, and P. Mittelstaedt,  _The Quantum Theory of Measurement (Lecture Notes in Physics)_ (Springer, 1996).
  * Plávala [2023] M. Plávala, General probabilistic theories: An introduction, Phys. Rep. 1033, 1 (2023).
  * Aubrun  _et al._ [2022] G. Aubrun, A. Müller-Hermes, and M. Plávala, Monogamy of entanglement between cones, arXiv:2206.11805 (2022).

##  Appendix A Proof of Theorem 1

($\Rightarrow$) Suppose that $\mathcal{T}\subset\mathrm{Fix}(\Lambda)$ for
some EB-channel $\Lambda$ with $\Lambda(\rho):=\sum_{\lambda}\tr{\rho
G_{\lambda}}\sigma_{\lambda}$ for all $\rho$. Then one can define the joint
channel $\Phi(\rho):=\sum_{\lambda}\tr{\rho
G_{\lambda}}\sigma_{\lambda}\otimes\sigma_{\lambda}$ such that
$\mathrm{tr}_{i}[\Phi(\rho)]=\rho$, $i=1,2$ for all
$\rho\in\mathrm{Fix}(\Lambda)$, i.e. the broadcasting condition holds.

($\Leftarrow$) Suppose $(\mathcal{T},\mathcal{O},\mathcal{O})$ is
broadcastable with the channel $\Lambda$. Then $\mathcal{T}$ is $1\to
n$-broadcastable for all $n\geq 2$. This can be seen by induction as follows.
Let $\Lambda_{2}=\Lambda$ and assume that $\mathcal{T}$ is $1\to
n$-broadcastable with the channel
$\Lambda_{n}:\mathcal{T(H)}\to\mathcal{T}(\mathcal{H}^{\otimes n})$. Define
then the channels $i_{n}^{*}:\mathcal{L(H)}\to\mathcal{L}(\mathcal{H}^{\otimes
n-1})$, $i_{n}^{*}(A):=A\otimes\left(\bigotimes_{k=1}^{n-2}I\right)$ and
$\Lambda_{n+1}:=(\Lambda_{n}\otimes i_{n})\circ\Lambda_{n}$. Now if
$\rho\in\mathcal{T}$, we have the following for all $A\in\mathcal{L(H)}$ and
$k\in\\{1,\dots,n\\}$, where $k^{c}$ denotes the complement of the set
$\\{k\\}$:

$\displaystyle\tr{A\mathrm{tr}_{k^{c}}[\Lambda_{n+1}(\rho)]}$
$\displaystyle=\tr{(\Lambda_{n}^{*}(I\otimes\cdots\otimes
A\otimes\cdots\otimes I)\otimes I\otimes\cdots\otimes I)\Lambda_{n}(\rho)}$
$\displaystyle=\tr{\Lambda_{n}^{*}(I\otimes\cdots\otimes A\otimes\cdots\otimes
I)\rho}=\tr{A\rho}.$

If $k=n+1$, then

$\displaystyle\tr{A\mathrm{tr}_{(n+1)^{c}}[\Lambda_{n+1}(\rho)]}$
$\displaystyle=\tr{(I\otimes A\otimes I\otimes\cdots\otimes
I)\Lambda_{n}(\rho)}$
$\displaystyle=\tr{A\mathrm{tr}_{2^{c}}[\Lambda_{n}(\rho)]}=\tr{A\rho}.$

Thus the set $\mathcal{T}$ is $1\to n+1$-broadcastable. By induction we see
that $\mathcal{T}$ is $1\to n$-broadcastable for all $n\in\mathbb{N}$.

Let us denote the $1\to n$ broadcasting channel still by $\Lambda_{n}$.
Furthermore, let $S_{n}$ be the $n$:th symmetric group with the unitary
representation $\pi\mapsto U_{\pi}$ where $U_{\pi}:\mathcal{H}^{\otimes
n}\to\mathcal{H}^{\otimes n}$ with
$U_{\pi}\left(\bigotimes_{i=1}^{n}\varphi_{i}\right)=\bigotimes_{i=1}^{n}\varphi_{\pi(i)}$.
Then for all $n\in\mathbb{N}$ we define the channel
$\tilde{\Lambda}_{n}:\mathcal{T(H)}\to\mathcal{T}(\mathcal{H}^{\otimes n})$ by

$\displaystyle\tilde{\Lambda}_{n}(\rho)=\frac{1}{n!}\sum_{\pi\in
S_{n}}U_{\pi}\Lambda_{n}(\rho)U_{\pi}^{*}.$

Let then $k\in\\{1,\dots,n\\}$. Now

$\displaystyle\mathrm{tr}_{k^{c}}[\tilde{\Lambda}_{n}(\rho)]$
$\displaystyle=\frac{1}{n!}\sum_{\pi\in
S_{n}}\mathrm{tr}_{k^{c}}[U_{\pi}\Lambda_{n}(\rho)U_{\pi}^{*}]$
$\displaystyle=\frac{1}{n!}\sum_{\pi\in
S_{n}}\mathrm{tr}_{\pi(k)^{c}}[\Lambda_{n}(\rho)]$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\mathrm{tr}_{k^{c}}[\Lambda_{n}(\rho)]=:\Phi_{n}(\rho).$

Here if $\rho\in\mathcal{T}$, then it is a fixed point of $\Phi_{n}$ by the
$1\to n$-broadcasting condition. Furthermore, $\Phi_{n}$ is $n$-self
compatible with the joint channel $\tilde{\Lambda}_{n}$. Now since the set of
quantum channels is compact in the diamond norm, we have that there is a
convergent subsequence $(\Phi_{n_{m}})_{m}$ of $(\Phi_{n})_{n}$. Furthermore
each $\Phi_{n}$ is especially the first marginal of the symmetric broadcast
channel $\tilde{\Lambda}_{n}$. Therefore by [58, Theorem 5] we see that for
every $n$ there exists an entanglement breaking channel
$\Lambda_{n}^{EB}:\mathcal{T(H)}\to\mathcal{T(H)}$ such that

$\displaystyle\|\Phi_{n}-\Lambda_{n}^{EB}\|_{\Diamond}\leq\frac{2d^{2}}{n}.$

Here $d=\dim\mathcal{H}$ and $\|\cdot\|_{\Diamond}$ denotes the diamond norm.
Now as the set of entanglement breaking channels is also compact in the
diamond norm, we have that the sequence $(\Lambda_{n_{m}}^{EB})_{m}$ has a
convergent subsequence $(\Lambda_{n_{m_{j}}}^{EB})_{j}$. Let the limit of
$(\Lambda_{n_{m_{j}}}^{EB})_{j}$ be $\Lambda^{EB}$ and the limit of
$(\Phi_{n_{m}})_{m}$ be $\Phi$. Now

$\displaystyle\|\Phi-\Lambda^{EB}\|_{\Diamond}$
$\displaystyle\leq\|\Phi-\Phi_{n_{m_{j}}}\|_{\Diamond}+\|\Phi_{n_{m_{j}}}-\Lambda_{n_{m_{j}}}^{EB}\|_{\Diamond}+\|\Lambda_{n_{m_{j}}}^{EB}-\Lambda^{EB}\|_{\Diamond}$
$\displaystyle\leq\|\Phi-\Phi_{n_{m_{j}}}\|_{\Diamond}+\frac{2d^{2}}{n_{m_{j}}}+\|\Lambda_{n_{m_{j}}}^{EB}-\Lambda^{EB}\|_{\Diamond}.$

Letting $j\to\infty$ we see that $\Phi=\Lambda^{EB}$.

Finally we show that $\mathcal{T}\subset\mathrm{Fix}(\Phi)$. Let
$\rho\in\mathcal{T}$. Now

$\displaystyle\|\Phi(\rho)-\rho\|_{1}=\|\Phi(\rho)-\Phi_{n_{m}}(\rho)\|_{1}\to
0,$

as $m\to\infty$. Therefore $\rho$ is a fixed point of the entanglement
breaking channel $\Phi$.

##  Appendix B Proof of Theorem 3

The equivalence $2\Leftrightarrow 3$ is Theorem 2 of [32] and
$3\Leftrightarrow 4$ is a known equivalence [59]. Let us prove the implication
$1\Rightarrow 2$. Let $\\{A_{k}\\}_{k}\in\mathcal{A}$. Then all $A_{k}$ are
fixed points of a Heisenberg picture EB-channel $\Lambda^{*}$ with
$\Lambda^{*}(A)=\sum_{\lambda}\tr{\sigma_{\lambda}A}G_{\lambda}$. Here
$\\{\sigma_{\lambda}\\}_{\lambda}$ is a family of states and $G$ a POVM.
Define then the channel
$\Phi(\rho)=\sum_{\lambda}\tr{G_{\lambda}\rho}\sigma_{\lambda}\otimes\sigma_{\lambda}$.
Then we have that
$\tr{A_{k}\tr_{i}{\Phi(\rho)}}=\tr{A_{k}\Lambda(\rho)}=\tr{A_{k}\rho}$ for all
$\rho\in\mathcal{S(H)}$ and $i=1,2$. Thus the broadcasting condition holds.

Finally we need to prove the implication $3\Rightarrow 1$. This can be seen
very directly from the sufficiency part of the proof of Theorem 2 in [32] as
the marginal channels in the proof are obviously entanglement breaking. For
completeness, the argument is as follows. Suppose that $E$ is a post-
processing of a norm-1 POVM $G$. Then for every $G_{\lambda}$ there is a unit
vector $\ket{\lambda}$ such that $G_{\lambda}\ket{\lambda}=\ket{\lambda}$.
Since
$\sum_{\lambda^{\prime}}\innerproduct{\lambda^{\prime}}{G_{\lambda}|\lambda^{\prime}}=1$,
we see that
$\innerproduct{\lambda^{\prime}}{G_{\lambda}|\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}$.
Therefore we define the EB-channel

$\displaystyle\Lambda^{*}(A):=\sum_{\lambda^{\prime}}\innerproduct{\lambda^{\prime}}{A|\lambda^{\prime}}G_{\lambda^{\prime}}$

Since
$\innerproduct{\lambda^{\prime}}{G_{\lambda}|\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}$,
all $G_{\lambda}$ are fixed points by construction. Therefore any post-
processing of the POVM $G$ is also a fixed point by linearity.

##  Appendix C There exist tuples $(\mathcal{T},\mathcal{A})$ that are not
broadcastable, but are $1\mapsto n$-pseudo-broadcastable

Let $\\{\ket{i}\\}_{i=0,1}$ be an orthonormal basis for the qubit Hilbert
space $\mathcal{H}=\mathbb{C}^{2}$ and let $\\{E_{i}\\}_{i=1}^{4}$ be the
symmetric, informationally complete POVM (SIC-POVM) in qubit. In other words
$E_{i}=\frac{1}{2}|\varphi_{i}\rangle\langle\varphi_{i}|$ with

$\displaystyle\ket{\varphi_{1}}$ $\displaystyle:=\ket{0}$ (8)
$\displaystyle\ket{\varphi_{2}}$
$\displaystyle:=\frac{1}{\sqrt{3}}\ket{0}+\sqrt{\frac{2}{3}}\ket{1}$ (9)
$\displaystyle\ket{\varphi_{3}}$
$\displaystyle:=\frac{1}{\sqrt{3}}\ket{0}+\sqrt{\frac{2}{3}}e^{i\frac{2\pi}{3}}\ket{1}$
(10) $\displaystyle\ket{\varphi_{4}}$
$\displaystyle:=\frac{1}{\sqrt{3}}\ket{0}+\sqrt{\frac{2}{3}}e^{i\frac{4\pi}{3}}\ket{1}.$
(11)

These are indeed symmetric in the sense that their Hilbert-Schmidt inner
product is of special form:
$\tr{|\varphi_{i}\rangle\langle\varphi_{i}||\varphi_{j}\rangle\langle\varphi_{j}|}=\frac{2\delta_{ij}+1}{3}$.

We will now show that the single POVM theory
$(\mathcal{S(H)},\\{\\{E_{i}\\}_{i=1}^{4}\\})$ is not broadcastable, but is
$1\mapsto n$ pseudo-brodacastable for all $n\in\mathbb{N}$. The fact that
$(\mathcal{S(H)},\\{\\{E_{i}\\}_{i=1}^{4}\\})$ is not broadcastable follows
immediately from the fact there is no universal broadcasting. This is since
$\mathrm{lin}\\{E_{1},E_{2},E_{3},E_{4}\\}=\mathcal{L}(\mathbb{C}^{2})$ by
informational completeness.

Let us then show that $(\mathcal{S(H)},\\{\\{E_{i}\\}_{i=1}^{4}\\})$ is
$1\mapsto n$ pseudo-broadcastable. For this, we define the $1\mapsto n$
pseudo-broadcasting map
$\Xi_{n}:\mathcal{L}(\mathbb{C}^{2})\to\mathcal{L}((\mathbb{C}^{2})^{\otimes
n})$ as follows. For all $\rho\in\mathcal{L}(\mathbb{C}^{2})$ we let

$\displaystyle\Xi_{n}(\rho):=\sum_{i=1}^{4}\tr{E_{i}\rho}(3|\varphi_{i}\rangle\langle\varphi_{i}|-I)^{\otimes
n}.$ (12)

This is obviously trace-preserving and indeed fulfills the conditions of a
valid $1\mapsto n$ pseudo-broadcasting map, which can be seen as follows.

$\displaystyle\tr{(E_{i_{1}}\otimes E_{i_{2}}\otimes\cdots
E_{i_{n}})\Xi_{n}(\rho)}$ (13)
$\displaystyle=\frac{1}{2^{n}}\sum_{j=1}^{4}\tr{\rho
E_{j}}\prod_{k=1}^{n}(3\tr{|\varphi_{j}\rangle\langle\varphi_{j}||\varphi_{i_{k}}\rangle\langle\varphi_{i_{k}}|}-1)$
(14) $\displaystyle=\frac{1}{2^{n}}\sum_{j=1}^{4}\tr{\rho
E_{j}}\prod_{k=1}^{n}2\delta_{i_{k}j}\geq 0$ (15)

Therefore positivity holds. Furthemore, by the equation above we also get

$\displaystyle\tr{(I\otimes I\otimes\cdots\otimes E_{i_{j}}\otimes\cdots
I)\Xi_{n}(\rho)}$ (16) $\displaystyle=\sum_{i_{1}\dots i_{j-1}i_{j+1}\dots
i_{n}}\tr{(E_{i_{1}}\otimes E_{i_{2}}\otimes\cdots E_{i_{n}})\Xi_{n}(\rho)}$
(17) $\displaystyle=\frac{1}{2^{n}}\sum_{k=1}^{4}\tr{\rho
E_{k}}2^{n}\delta_{i_{j}k}=\tr{\rho E_{i_{j}}}$ (18)

Thus the pseudo-broadcasting condition holds. Since $n\in\mathbb{N}$ was
arbitrary, $(\mathcal{S(H)},\\{\\{E_{i}\\}_{i=1}^{4}\\})$ is $1\mapsto n$
pseudo-broadcastable for all $n\in\mathbb{N}$.

##  Appendix D Proof of Theorem 5

If $(\mathcal{T},\mathcal{A})$ is non-contextual, then there are $G_{\lambda}$
and $\sigma_{\lambda}$ such that for any $\rho\in\mathcal{T}$ and
$M\in\mathcal{A}$ we have $\tr{\rho M}=\sum_{\lambda}\tr{\rho
G_{\lambda}}\tr{\sigma_{\lambda}M}$, note that we can, without loss of
generality, choose the operators $G_{\lambda}$ and $\sigma_{\lambda}$ such
that $\tr{\sigma_{\lambda}}=1$. Let $n\in\mathbb{N}$, we now construct
$\Xi_{n}:\mathcal{L(H)}\to\mathcal{L}(\mathcal{H}^{\otimes n})$ as

$\Xi_{n}(\rho)=\sum_{\lambda}\tr{\rho G_{\lambda}}\sigma_{\lambda}^{\otimes
n}.$ (19)

Since $\tr{\sigma_{\lambda}}=1$ we have

$\tr{\tr_{i^{c}}[\Xi_{n}(\rho)]M}=\sum_{\lambda}\tr{\rho
G_{\lambda}}\tr{\sigma_{\lambda}M}=\tr{\rho M}$ (20)

for all $\rho\in\mathcal{T}$ and $M\in\mathcal{A}$. Moreover

$\tr{\Xi_{n}(\rho)\otimes_{i=1}^{n}M_{i}}=\sum_{\lambda}\tr{\rho
G_{\lambda}}\prod_{i=1}^{n}\tr{\sigma_{\lambda}M_{i}}\geq 0$ (21)

since $\tr{\sigma_{\lambda}M_{i}}\geq 0$ for all
$M_{1},\ldots,M_{n}\in\mathcal{A}$ and $\tr{\rho G_{\lambda}}\geq 0$. Thus
$\Xi_{n}$ is the $1\mapsto n$-pseudo-broadcasting map for
$(\mathcal{T},\mathcal{A})$.

Now assume that $(\mathcal{T},\mathcal{A})$ is $1\mapsto n$-pseudo-
bradcastable for all $n\in\mathbb{N}$. We will use $\mathcal{S}(\mathcal{A})$
to denote the state space over $\mathcal{A}$, that is
$\mathcal{S}(\mathcal{A})$ is the set of linear functionals on
$\mathrm{span}(\mathcal{A})$ such that for
$\varphi\in\mathcal{S}(\mathcal{T})$ we have $\varphi(M)\geq 0$ for all
$M\in\mathcal{A}$ and $\varphi(I)=1$. Note that to every
$\varphi\in\mathcal{S}(\mathcal{T})$ one can find at least one operator
$\sigma\in\mathcal{L(H)}$ such that $\varphi(M)=\tr{\sigma M}$ since
$\mathrm{span}(\mathcal{A})\subset\mathcal{L(H)}$. We will also need to define
maximal tensor product of the state spaces $\mathcal{S}(\mathcal{A})$: we
define
$\mathcal{S}(\mathcal{A})\hat{\otimes}\mathcal{S}(\mathcal{A})=\mathcal{S}(\mathcal{A})^{\hat{\otimes}2}$
to be the set of linear functionals on $\mathrm{span}(\mathcal{A})^{\otimes
2}$ such that for $\varphi_{2}\in\mathcal{S}(\mathcal{T})^{\hat{\otimes}2}$ we
have $\varphi(M_{1}\otimes M_{2})\geq 0$ for all $M_{1},M_{2}\in\mathcal{A}$
and $\varphi(I\otimes I)=1$. Analogically, we define
$\mathcal{S}(\mathcal{A})^{\hat{\otimes}k}$ for $k\in\mathbb{N}$ to be the set
linear functionals on $\mathrm{span}(\mathcal{A})^{\otimes k}$ such that for
$\varphi_{k}\in\mathcal{S}(\mathcal{T})^{\hat{\otimes}k}$ we have
$\varphi(\otimes_{i=1}^{k}M_{i})\geq 0$ for all
$M_{1},\ldots,M_{k}\in\mathcal{A}$ and $\varphi(I^{\otimes k})=1$.

Let $\Xi_{1}:\mathcal{T}\to\mathcal{S}(\mathcal{A})$ be the linear map defined
as $(\Xi_{1}(\rho))(M)=\tr{\rho M}$ for all $\rho\in\mathcal{T}$ and
$M\in\mathcal{A}$. We will now argue that for every $n\in\mathbb{N}$ the map
$\Xi_{1}$ has a $n$-copy extension, that is, for every $n\in\mathbb{N}$ there
is a map $\Psi_{n}:\mathcal{T}\to\mathcal{S}(\mathcal{T})^{\hat{\otimes}n}$
such that for any $\rho\in\mathcal{T}$ and $M\in\mathcal{A}$ we have

$(\Psi_{n}(\rho))(M\otimes I^{\otimes(n-1)})=(\Psi_{n}(\rho))(I\otimes
M\otimes I^{\otimes(n-2)})=\ldots=(\Psi_{n}(\rho))(I^{\otimes(n-1)}\otimes
M)=(\Xi_{1}(\rho))(M)=\tr{\rho M}.$ (22)

The $n$-copy extension of $\Xi_{1}$ are the $1\mapsto n$-pseudo-broadcasting
maps $\Xi_{n}$, that is, we can take $\Psi_{n}=\Xi_{n}$: the equation above
corresponds to the conditions $\tr{M\tr_{i^{c}}\Xi_{n}(\rho)}=\tr{\rho M}$ for
all $i\in\\{1,\dots,n\\}$ and
$\Psi_{n}(\rho)\in\mathcal{S}(\mathcal{A})^{\hat{\otimes}n}$ is equivalent to
$\tr{\Xi_{n}(\rho)\otimes_{i=1}^{n}M_{i}}\geq 0$ for all
$M_{1},\ldots,M_{n}\in\mathcal{A}$ by definition. It now follows from the
identification of linear maps with elements of the appropriate tensor product
[60, Proposition 6.9] that existence of $n$-copy extensions for all
$n\in\mathbb{N}$ implies that the map $\Xi_{1}$ is separable [61], i.e.,
measure-and-prepare. That is, there are operators $G_{\lambda}$ such that
$\tr{\rho G_{\lambda}}\geq 0$ for all $\rho\in\mathcal{T}$ and all $\lambda$
and $\varphi_{\lambda}\in\mathcal{S}(\mathcal{A})$ such that

$\Xi_{1}(\rho)=\sum_{\lambda}\tr{\rho G_{\lambda}}\varphi_{\lambda}.$ (23)

By taking any of the operators $\sigma_{\lambda}$ corresponding to the
functionals $\varphi_{\lambda}$ via
$\varphi_{\lambda}(M)=\tr{\sigma_{\lambda}M}$ for any $M\in\mathcal{A}$, we
get

$\tr{\rho M}=(\Xi_{1}(\rho))(M)=\sum_{\lambda}\tr{\rho
G_{\lambda}}\varphi_{\lambda}(M)=\sum_{\lambda}\tr{\rho
G_{\lambda}}\tr{\sigma_{\lambda}M}.$ (24)

It was proved in [61] that if $\mathcal{S}(\mathcal{A})$ is a Cartesian
product of $k$ simplexes, then one only needs the existence of $n$-copy
extensions for $n\leq k$ to prove that $\Xi_{1}$ is separable and thus
measure-and-prepare. One can also see that $\mathcal{S}(\mathcal{A})$ is a
Cartesian product of $k$ simplexes if $\mathcal{A}$ is generated by $k$
independent measurements. This leads to a conjecture that if $\mathcal{A}$ is
generated by $k$ measurements, then $1\mapsto k$-pseudobroadcasting is
equivalent to non-contextuality. This is only a conjecture since even if
$\mathcal{A}$ is generated by $k$ measurements, $\mathcal{S}(\mathcal{A})$ in
general is only a subset of a Cartesian product of $k$ simplexes. We leave
resolving this issue and proving or disproving the conjecture for future work.

##  Appendix E $1\mapsto n$-pseudobroadcasting in the scenario investigated in
[49]

Consider the qubit Hilbert space, $\dim(\mathcal{H})=2$. Let $t\in\\{1,2,3\\}$
and $b\in\\{0,1\\}$ then we will consider a scenario with preparations
$\sigma_{t,b}$ and measurement effects $M_{t,b}$ given as

$\displaystyle\sigma_{1,0}=M_{1,0}=\dfrac{1}{2}\left(I+\sigma_{Z}\right),$
$\displaystyle\sigma_{1,1}=M_{1,1}=\dfrac{1}{2}\left(I-\sigma_{Z}\right),$
(25)
$\displaystyle\sigma_{2,0}=M_{2,0}=\dfrac{1}{2}\left(I+\dfrac{\sqrt{3}}{2}\sigma_{X}-\dfrac{1}{2}\sigma_{Z}\right),$
$\displaystyle\sigma_{2,1}=M_{2,1}=\dfrac{1}{2}\left(I-\dfrac{\sqrt{3}}{2}\sigma_{X}+\dfrac{1}{2}\sigma_{Z}\right),$
(26)
$\displaystyle\sigma_{3,0}=M_{3,0}=\dfrac{1}{2}\left(I-\dfrac{\sqrt{3}}{2}\sigma_{X}-\dfrac{1}{2}\sigma_{Z}\right),$
$\displaystyle\sigma_{3,1}=M_{3,1}=\dfrac{1}{2}\left(I+\dfrac{\sqrt{3}}{2}\sigma_{X}+\dfrac{1}{2}\sigma_{Z}\right),$
(27)

where $\sigma_{X}$, $\sigma_{Z}$ are the usual Pauli operators. We will
consider the dephasing channel acting on the preparations,

$P_{\mu}(\rho)=\mu\rho+(1-\mu)\sum_{0}^{1}\bra{i}\rho\ket{i}\ket{i}\bra{i},$
(28)

where $\ket{0},\ket{1}$ is the standard computational basis of eigenvectors of
$\sigma_{Z}$, and the depolarizing channel acting on the measurement effects

$D_{\eta}(E)=\eta E+(1-\eta)\dfrac{I}{2}.$ (29)

Our task is to determine for which pair of parameters $\mu,\eta$ are the
preparations $P_{\mu}(\sigma_{t,b})$a and measurement effects
$D_{\eta}(M_{t,b})$ contextual. Using the same approach as in [49], the
scenario is contextual if the non-contextuality inequality presented in [49]
is violated, the non-contextuality inequality reads

$\sum_{t=1}^{3}\sum_{b=0}^{1}\tr(\sigma_{t,b}M_{t,b})\leq 5.$ (30)

By explicit calculations, one can show that this inequality is violated when

$\eta\geq\dfrac{4}{3(1+\mu)}.$ (31)

Another options is to check whether the scenario is $1\mapsto
2$-pseudobraodcastable for given $\mu,\eta$. This can be done via numerically
solving the following semidefinite program (SDP):

$\begin{split}\text{find}\quad&\Xi\in\mathcal{L}(\mathcal{H}^{\otimes 3})\\\
\text{such that}\quad&\tr(\Xi[P_{\mu}(\sigma_{t,b})\otimes
D_{\eta}(M_{t^{\prime},b^{\prime}})\otimes
D_{\eta}(M_{t^{\prime\prime},b^{\prime\prime}})])\geq 0\\\
&\tr(\Xi[P_{\mu}(\sigma_{t,b})\otimes
D_{\eta}(M_{t^{\prime},b^{\prime}})\otimes
I])=\tr[P_{\mu}(\sigma_{t,b})D_{\eta}(M_{t^{\prime},b^{\prime}})]\\\
&\tr(\Xi[P_{\mu}(\sigma_{t,b})\otimes I\otimes
D_{\eta}(M_{t^{\prime},b^{\prime}})])=\tr[P_{\mu}(\sigma_{t,b})D_{\eta}(M_{t^{\prime},b^{\prime}})]\\\
\text{for
all}\quad&t,t^{\prime},t^{\prime\prime}\in\\{1,2,3\\},b,b^{\prime},b^{\prime\prime}\in\\{0,1\\}\end{split}$
(32)

If the SDP (32) is not feasible, then the scenario is not $1\mapsto
2$-pseudobraodcastable and thus contextual. In a similar fashion one gets the
SDP for $1\mapsto 3$-pseudobraodcasting, this reads

$\begin{split}\text{find}\quad&\Xi\in\mathcal{L}(\mathcal{H}^{\otimes 4})\\\
\text{such that}\quad&\tr(\Xi[P_{\mu}(\sigma_{t,b})\otimes
D_{\eta}(M_{t^{\prime},b^{\prime}})\otimes
D_{\eta}(M_{t^{\prime\prime},b^{\prime\prime}})\otimes
D_{\eta}(M_{t^{\prime\prime\prime},b^{\prime\prime\prime}})])\geq 0\\\
&\tr(\Xi[P_{\mu}(\sigma_{t,b})\otimes
D_{\eta}(M_{t^{\prime},b^{\prime}})\otimes I\otimes
I])=\tr[P_{\mu}(\sigma_{t,b})D_{\eta}(M_{t^{\prime},b^{\prime}})]\\\
&\tr(\Xi[P_{\mu}(\sigma_{t,b})\otimes I\otimes
D_{\eta}(M_{t^{\prime},b^{\prime}})\otimes
I])=\tr[P_{\mu}(\sigma_{t,b})D_{\eta}(M_{t^{\prime},b^{\prime}})]\\\
&\tr(\Xi[P_{\mu}(\sigma_{t,b})\otimes I\otimes I\otimes
D_{\eta}(M_{t^{\prime},b^{\prime}})])=\tr[P_{\mu}(\sigma_{t,b})D_{\eta}(M_{t^{\prime},b^{\prime}})]\\\
\text{for
all}\quad&t,t^{\prime},t^{\prime\prime},t^{\prime\prime\prime}\in\\{1,2,3\\},b,b^{\prime},b^{\prime\prime},b^{\prime\prime\prime}\in\\{0,1\\}\end{split}$
(33)