# Contextual advantage for state-dependent cloning

Matteo Lostaglio  ICFO-Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The
Netherlands  Gabriel Senno  ICFO-Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, Castelldefels (Barcelona),
08860, Spain

###### Abstract

A number of noncontextual models exist which reproduce different subsets of
quantum theory and admit a no-cloning theorem. Therefore, if one chooses
noncontextuality as one’s notion of classicality, no-cloning cannot be
regarded as a nonclassical phenomenon. In this work, however, we show that
there are aspects of the phenomenology of quantum state cloning which are
indeed nonclassical according to this principle. Specifically, we focus on the
task of state-dependent cloning and prove that the optimal cloning fidelity
predicted by quantum theory cannot be explained by any noncontextual model. We
derive a noise-robust noncontextuality inequality whose violation by quantum
theory not only implies a quantum advantage for the task of state-dependent
cloning relative to noncontextual models, but also provides an experimental
witness of noncontextuality.

An important guiding principle for quantum theorists is the identification of
genuine nonclassical effects certified by rigorous theorems. Given a quantum
phenomenon, the relevant question is: Are there classical models able to
reproduce the observed operational data? Here we investigate this question in
the context of a cloning experiment.

The no-cloning theorem [1, 2, 3] is widely regarded as a central result in
quantum theory. Informally, the theorem states the impossibility of copying
quantum information, and is contrasted with the fact that classical
information, on the other hand, can be perfectly copied. More precisely, there
is no machine (formally, a quantum channel) that can take two distinct and
nonorthogonal states $\\{\ket{\psi_{1}},\ket{\psi_{2}}\\}$ sent at random as
inputs and output the corresponding copies
$\\{\ket{\psi_{1}}\otimes\ket{\psi_{1}},\ket{\psi_{2}}\otimes\ket{\psi_{2}}\\}$
[4].

While no-cloning is often regarded as an intrinsically quantum feature, one
would like to back that claim by a precise theorem stating what operational
features cannot be explained within classical models. The theorem should hence
define a precise notion of ‘classicality’ and show that such notion leads to
operational predictions incompatible with the relevant quantum statistics [5].
At the operational level, we can schematically think of an experiment as a set
of black-boxes each corresponding to certain sets of operational instructions.
At the ontological level we look for theoretical explanations of the empirical
data within the framework of _ontological models_. This is a very broad class
of models involving an arbitrary set of physical states evolving according to
some laws and ultimately determining (the probabilities of) the measurement
outcomes. This analysis forces us to look for any plausible alternative
explanation of the empirical data collected in a quantum experiment before we
certify it as “nonclassical”. But, which ontological models should be deemed
“classical"?

Clearly, the broader the chosen notion of classicality is, the stronger the
resulting no-go theorem is. Since the scenario of quantum cloning does not
feature space-like separated measurements, we need a different notion of
‘classicality’ than the ubiquitous Bell’s locality. Hence, in this work we
identify nonclassical features as those that cannot be explained within any
noncontextual model, in the generalized sense introduced in Ref. [6]. It is a
known fact that, with respect to this broad notion, no-cloning by itself
should not be regarded as a nonclassical phenomenon. There are, in fact,
several examples of noncontextual models for subsets of quantum theory with a
no-cloning theorem [7, 8]. The mechanism behind no-cloning in noncontextual
theories is simple: non-orthogonal quantum states $\ket{\psi_{1}}$,
$\ket{\psi_{2}}$ correspond to overlapping probability distributions
$\mu_{1}(\lambda)$, $\mu_{2}(\lambda)$ over the posited set of physical states
$\lambda$ and there is no deterministic nor stochastic process mapping
$\\{\mu_{1},\mu_{2}\\}$ to $\\{\mu_{1}\otimes\mu_{1},\mu_{2}\otimes\mu_{2}\\}$
[9]. The existence of these models proves that no-cloning cannot be
interpreted as a nonclassical phenomenon when the notion of classicality is
taken to be that of noncontextuality.111Crucially, cloning should be
distinguished from the notion of broadcasting. Broadcasting only requires the
creation of a joint distribution with marginals $\mu_{i}$ and can be done
perfectly by a generalized CNOT. Hence, we need to look more closely at the
phenomenology of quantum cloning if we are to identify aspects of it that are
nonclassical according to the principle of noncontextuality.

In this work, we identify a strongly nonclassical aspect in the ultimate
limits of imperfect cloning. The question of what is the best fidelity with
which a given set of quantum states can be cloned has been widely studied
since the pivotal work of Bužek and Hillary in 1996 [10] (for a review on
quantum cloning, see, _e.g._ , Ref. [11]). We find that the optimal fidelity
predicted by quantum theory for the cloning of two distinct, non orthogonal
pure states cannot be reproduced by any noncontextual model which complies
with the operational phenomenology featured in a quantum cloning experiment.
Specifically, contextuality provides an advantage to the maximum copying
fidelity. Our result directly links contextuality to a quantum advantage [5,
12, 13].

##  1 Noncontextual ontological models of operational theories

At the operational level, we can schematically think of an experiment as a set
of black-boxes each corresponding to certain sets of operational
instructions.222While empirical data is always to some degree _theory-laden_ ,
the word “operational” here signifies that we are striving towards the ideal
of the most low-level instructions we can imagine (e.g. press this button,
write down an outcome when a corresponding light flashes etc.). This is to be
opposed to high-level instructions that refer to theoretical entities, such as
“lower the potential barrier in which the electron is trapped”. We can
distinguish three kinds of black-boxes:

  1. 1.

A preparation black-box $P_{s}$ initialises the system;

  2. 2.

A transformation black-box $T$ takes in a system prepared according to $P_{s}$
and transforms it into some new preparation, denoted $T(P_{s})$.

  3. 3.

A measurement black-box $M_{s^{\prime}}$ takes a preparation $P_{s}$ as input
and returns an outcome $x$ with probability $p(x|P_{s},M_{s^{\prime}})$.

  4. 4.

An experiment consists of collecting the statistics
$p(x|T(P_{s}),M_{s^{\prime}})$ for various choices of the black boxes $P_{s}$,
$T$ and $M_{s^{\prime}}$.

The set of $P_{s}$, $T$, $M_{s^{\prime}}$ and corresponding observed
statistics $p(x|T(P_{s}),M_{s^{\prime}})$ are the defining elements of an
_operational theory_. Noncontextuality is a restriction on the ontological
models that try to explain the statistics of some operational theory. An
ontological model for an operational theory is one which [14]:

  1. 1.

Makes every preparation $P_{s}$ correspond to sampling from a probability
distribution $\mu_{s}(\lambda)$ over some set of ontic variables $\lambda$.
$\lambda$s are referred to as ‘hidden variables’ in the context of Bell
nonlocality and they form a (measurable) set $\Lambda$.

  2. 2.

Represents transformations by matrices $T(\lambda^{\prime}|\lambda)$ of
transition probabilities ($T(\lambda^{\prime}|\lambda)\geq 0$, $\int
d\lambda^{\prime}T(\lambda^{\prime}|\lambda)=1$ $\forall\lambda$) acting on
the corresponding probability density.

  3. 3.

Represents a measurement $M_{s^{\prime}}$ by a response function
$\xi_{s^{\prime}}(x|\lambda)$ giving the probability of outcome $x$ given that
the hidden variable takes the value $\lambda$
($\xi_{s^{\prime}}(x|\lambda)\geq 0$, $\sum_{x}\xi_{s^{\prime}}(x|\lambda)=1$
$\forall\lambda$).

An ontological model then defines its predictions as

$p(x|T(P_{s}),M_{s^{\prime}})=\int d\lambda
d\lambda^{\prime}\mu_{s}(\lambda)T(\lambda^{\prime}|\lambda)\xi_{s^{\prime}}(x|\lambda^{\prime}).$
(1)

Two operational procedures (be them preparations, measurements or
transformations) are said to be _operationally equivalent_ if they cannot be
distinguished by any experiment. Noncontextuality, in the generalized form
introduced in [6], is a restriction to ontological models requiring that _if
two procedures are operationally equivalent, they must be represented by the
same object in the ontological model_. This notion can be seen as an extension
of the traditional one of Kochen-Specker [15, 6].

In this work we will be concerned with operational equivalences only at the
level of preparations. Two preparations $P_{s}$ and $P_{s^{\prime}}$ are
operationally equivalent if they cannot be distinguished by any measurements:

$p(x|P_{s},M)=p(x|P_{s^{\prime}},M),\quad\forall M,$

which, for short, we will denote by $P_{s}\simeq P_{s^{\prime}}$. The
assumption of (preparation) noncontextuality is then

$P_{s}\simeq
P_{s^{\prime}}\Rightarrow\mu_{s}(\lambda)=\mu_{s}(\lambda^{\prime}).$ (2)

This principle can be understood as an ‘identity of the indiscernibles’ and,
together with locality, it can be seen as a successful methodological
principle for theory construction [16]. Examples of noncontextual ontological
models include classical Hamiltonian mechanics, Hamiltonian mechanics with a
resolution limit on phase space [7] and Spekken’s toy model [8].

Figure 1: _Cloning experiment._ Top: black-box of the cloning protocol; one of
two preparation procedures $P_{x}$, $x=a,b$ is performed with equal
probability, the resultant state is sent through a cloning machine
(independent of $x$), which respectively prepares $P_{\gamma}$,
$\gamma=\alpha,\beta$; a test measurement $M_{xx}$ for the target preparation
$P_{xx}$ is performed and passed with probability $P(M_{aa}|P_{\alpha})$ (or
$P(M_{bb}|P_{\beta})$). Bottom: ontological description of the same
experiment, where preparing $P_{x}$ corresponds to sampling $\lambda$ with
probability $\mu_{x}(\lambda)$, the cloning machine maps
$\lambda\mapsto\lambda^{\prime}$ with probability
$T(\lambda^{\prime}|\lambda)$ and $M_{xx}$ gives a ‘pass’ outcome with
probability $\xi_{xx}(1|\lambda^{\prime})$.

##  2 Operational features of quantum cloning - ideal scenario

We now describe the operational features of optimal state-dependent quantum
cloning which, as we will show, are impossible to explain with noncontextual
models (see also Fig. 1). We will make the assumption that certain perfect
correlations are observed, but we will later remove these idealizations. For
all two-outcome measurements $M_{s}$ we will use the shortcuts
$p(x=1|P,M_{s})\equiv p(M_{s}|P)$ and
$\xi_{s}(1|\lambda^{\prime})\equiv\xi_{s}(\lambda^{\prime})$.

Let $P_{a}$ and $P_{b}$ denote the experimental procedures followed to prepare
the states $\ket{a}$ and $\ket{b}$ to be cloned. As an operational signature
of the fact that $\ket{a}$ and $\ket{b}$ are two pure and, in general,
nonorthogonal states, we consider the ‘test measurements’ $M_{a}$, $M_{b}$,
with outcomes $x\in\\{0,1\\}$, giving the operational statistics
$p(M_{a}|P_{a})=p(M_{b}|P_{b})=1$. In the quantum formalism, this statistics
is reproduced by performing the projective measurements
$\\{\left|a\right\rangle\\!\left\langle
a\right|,\mathbbold{1}-\left|a\right\rangle\\!\left\langle a\right|\\}$ and
$\\{\left|b\right\rangle\\!\left\langle
b\right|,\mathbbold{1}-\left|b\right\rangle\\!\left\langle b\right|\\}$ (with
$x=1$ corresponding to the first outcome). We will use the notation
$c_{ab}:=p(M_{b}|P_{a})$, which is called ‘confusability’ in Ref. [5], for the
probability of observing the first outcome of the $M_{b}$ measurement when the
system is initialized according to $P_{a}$. Clearly, in the ideal quantum
experiment one observes $c_{ab}=|\\!\left\langle
a\vphantom{b}\middle|b\vphantom{a}\right\rangle\\!|^{2}$.

The two preparations $P_{a}$, $P_{b}$ go through a cloning machine $T$, which
outputs new preparations $P_{\alpha}=T(P_{a})$, $P_{\beta}=T(P_{b})$. In
quantum theory, the optimal-state dependent cloning operation is a unitary $U$
and, hence, the preparations $P_{\alpha}$ and $P_{\beta}$ correspond to pure
states $\ket{\alpha}:=U\ket{a\,0}$, $\ket{\beta}:=U\ket{b\,0}$ respectively,
with $\ket{0}$ the initial state of some ancillary register. Operationally,
and similarly to the discussion above, the purity of the outputs implies that
we can perform test measurements $M_{\alpha}$, $M_{\beta}$ satisfying
$p(M_{\alpha}|P_{\alpha})=1$, $p(M_{\beta}|P_{\beta})=1$ (again, by performing
the measurements described in the quantum formalism as
$\\{\left|\alpha\right\rangle\\!\left\langle\alpha\right|,\mathbbold{1}-\left|\alpha\right\rangle\\!\left\langle\alpha\right|\\}$
and
$\\{\left|\beta\right\rangle\\!\left\langle\beta\right|,\mathbbold{1}-\left|\beta\right\rangle\\!\left\langle\beta\right|\\}$).

The experiment ends by testing what the fidelity between the output and the
ideal clone is. To do so, given the ideal clones $P_{aa}$, $P_{bb}$ we
introduce test-measurements $M_{aa}$, $M_{bb}$ and assume one observes the
statistics $p(M_{aa}|P_{aa})=p(M_{bb}|P_{bb})=1$,
$p(M_{bb}|P_{aa})=|\\!\left\langle
aa\vphantom{bb}\middle|bb\vphantom{aa}\right\rangle\\!|^{2}=|\\!\left\langle
a\vphantom{b}\middle|b\vphantom{a}\right\rangle\\!|^{4}.$ In a quantum
experiment this is realized by preparing states $\ket{aa}$, $\ket{bb}$ and
performing the projective measurements
$\\{\left|aa\right\rangle\\!\left\langle
aa\right|,\mathbbold{1}-\left|aa\right\rangle\\!\left\langle aa\right|\\}$,
$\\{\left|bb\right\rangle\\!\left\langle
bb\right|,\mathbbold{1}-\left|bb\right\rangle\\!\left\langle bb\right|\\}$.

Then, denoting by $c_{\alpha aa}:=P(M_{aa}|P_{\alpha})$, $c_{\beta
bb}:=P(M_{bb}|P_{\beta})$, the (global) cloning fidelity is operationally
defined to be

$F_{g}:=\frac{1}{2}c_{\alpha aa}+\frac{1}{2}c_{\beta bb},$

i.e., the average probability that the imperfect clones $P_{\alpha}$ and
$P_{\beta}$ pass the corresponding test measurements for the ideal clones,
$M_{aa}$ and $M_{bb}$ respectively. In quantum theory, the optimal cloning
unitary achieves [17]

$\displaystyle F^{\rm Q,opt}_{g}$
$\displaystyle:=\frac{1}{4}\left[\sqrt{(1+c_{ab})(1+\sqrt{c_{ab}})}\right.$
$\displaystyle\qquad\qquad\left.+\sqrt{(1-c_{ab})(1-\sqrt{c_{ab}})}\right]^{2},$
(3)

with $c_{ab}=|\\!\left\langle
a\vphantom{b}\middle|b\vphantom{a}\right\rangle\\!|^{2}$.

This brief summary captures the main operational features of the traditional
‘optimal state-dependent cloning’ and highlights the main issue with this
approach: it leaves no room to leverage operational equivalences to further
study its potential non-classical aspects. To fix that, we follow Ref. [5] and
exploit another operational consequence of the purity of $\ket{a}$, $\ket{b}$:
the existence of preparations $P_{a^{\perp}}$, $P_{b^{\perp}}$ satisfying
$p(M_{a}|P_{a^{\perp}})=p(M_{b}|P_{b^{\perp}})=0$ and such that the mixture
$P_{a}/2+P_{a^{\perp}}/2$ (tossing a fair coin and following either $P_{a}$ or
$P_{a^{\perp}}$) is operationally equivalent to the mixture
$P_{b}/2+P_{b^{\perp}}/2$: $P_{a}/2+P_{a^{\perp}}/2\simeq
P_{b}/2+P_{b^{\perp}}/2$. In the idealized quantum experiment one observes
this operational statistics by preparing pure states $\ket{a^{\perp}}$,
$\ket{b^{\perp}}$ in the span of {$\ket{a}$, $\ket{b}$} and satisfying
$\\!\left\langle
a\vphantom{a^{\perp}}\middle|a^{\perp}\vphantom{a}\right\rangle\\!=\\!\left\langle
b\vphantom{b^{\perp}}\middle|b^{\perp}\vphantom{b}\right\rangle\\!=0$ as well
as $\frac{1}{2}\left|a\right\rangle\\!\left\langle
a\right|+\frac{1}{2}\left|a^{\perp}\right\rangle\\!\left\langle
a^{\perp}\right|=\frac{1}{2}\left|b\right\rangle\\!\left\langle
b\right|+\frac{1}{2}\left|b^{\perp}\right\rangle\\!\left\langle
b^{\perp}\right|$. The same discussion can be repeated for each of the pairs
$\\{(a,b),(\alpha,aa),(\beta,bb)\\}$.

To conclude, here is an operational account (without any reference to quantum
theory) of the features that we demand are observed in the idealized scenario
of the cloning experiment: there exists $P_{s},P_{s^{\perp}},M_{s}$ such that

  1. O1

$p(M_{s}|P_{s})=1$, $p(M_{s}|P_{s^{\perp}})=0$ for $s=a,b,\alpha,\beta,aa,bb$.

  2. O2

$\frac{1}{2}P_{s}+\frac{1}{2}P_{s^{\perp}}\simeq\frac{1}{2}P_{s^{\prime}}+\frac{1}{2}P_{s^{{}^{\prime}\perp}}$,
for all $(s,s^{\prime})$ in $\\{(a,b),(\alpha,aa),(\beta,bb)\\}$.

##  3 Optimal cloning is contextual - ideal scenario

In any ontological model, a cloning experiment is described as follows (see
Fig. 1). A preparation device randomly prepares either $P_{a}$ or $P_{b}$,
i.e., it samples a $\lambda$ from either the distribution $\mu_{a}(\lambda)$
or $\mu_{b}(\lambda)$. This state is sent into the cloning machine that maps
$\lambda$ into some new $\lambda^{\prime}$ with probability
$T(\lambda^{\prime}|\lambda)$. For example, if $\lambda=(x_{1},p_{1})$ one
could have
$\lambda^{\prime}=(x^{\prime}_{1},p^{\prime}_{1},x^{\prime}_{2},p^{\prime}_{2})$.
This $\lambda^{\prime}$ is sent into a testing device doing the measurement
$M_{aa}$ if $P_{a}$ was prepared, or $M_{bb}$ if $P_{b}$ was prepared. Upon
receiving $\lambda^{\prime}$, the device gives an outcome $x$ with probability
$\xi_{aa}(\lambda^{\prime})$ or $\xi_{bb}(\lambda^{\prime})$.

The assumption of noncontextuality (more precisely, preparation
noncontextuality [6]) and linearity applied to the operational equivalences in
O2 requires that any noncontextual ontological model must satisfy (see Eq.
(2))

$\frac{1}{2}\mu_{s}(\lambda)+\frac{1}{2}\mu_{s^{\perp}}(\lambda)=\frac{1}{2}\mu_{s^{\prime}}(\lambda)+\frac{1}{2}\mu_{s^{{}^{\prime}\perp}}(\lambda),$
(4)

for all $(s,s^{\prime})$ in $\\{(a,b),(\alpha,aa),(\beta,bb)\\}$ and
$\lambda\in\Lambda$. Our main result is that no noncontextual ontological
model can reproduce the operational features listed O1-O2 and match the
optimal cloning fidelity predicted by quantum theory. More precisely:

######  Theorem 1 (Optimal cloning fidelity in noncontextual models).

Let $P_{\alpha}=T(P_{a})$, $P_{\beta}=T(P_{a})$ be the achieved outputs of a
cloning process with inputs $P_{a}$, $P_{b}$ and target outputs $P_{aa}$,
$P_{bb}$. Suppose one observes the operational features O1-O2. Then, for any
noncontextual model we have that

$\displaystyle F_{g}\leq F^{\rm NC}_{g}=\leavevmode\nobreak\
1-\frac{c_{ab}}{2}+\frac{c_{aa,bb}}{2}.$ (5)

###### Proof.

The first part of the proof essentially follows the argument given in Ref. [6]
Sec. VIIIA and reproduced in Ref. [5] Sec. IVA, slightly adapted to use the
fewer assumptions of the statement. We have that

$1=p(M_{k}|P_{k})=\int_{S_{k}}d\lambda\mu_{k}(\lambda)\xi_{k}(\lambda),\quad
k=s,s^{\prime},$

where $S_{k}$ denotes the support of $\mu_{k}$. From this equation, it follows
that $\xi_{k}(\lambda)=1$ almost everywhere on $S_{k}$ (that is, modulo sets
of measure zero). Furthermore,

$0=p(M_{k}|P_{k^{\perp}})=\int_{S_{k^{\perp}}}\mu_{k^{\perp}}(\lambda)\xi_{k}(\lambda),\quad
k=s,s^{\prime},$

from which it follows that $\xi_{k}(\lambda)=0$ almost everywhere on
$S_{k^{\perp}}$. Hence, $S_{k}\cap S_{k^{\perp}}=\emptyset$ modulo sets of
zero measure.

The operational equivalence of assumption 1 implies that in a noncontextual
model

$\mu_{s}(\lambda)+\mu_{s^{\perp}}(\lambda)=\mu_{s^{\prime}}(\lambda)+\mu_{s^{\prime\perp}}(\lambda),\quad\forall\lambda\in\Lambda.$
(6)

Since $S_{s}\cap S_{s^{\perp}}=S_{s^{\prime}}\cap
S_{s^{\prime\perp}}=\emptyset$ modulo a set of zero measure, this implies
$\mu_{s}(\lambda)=\mu_{s^{\prime}}(\lambda)$ for almost all $\lambda\in
S_{s}\cap S_{s^{\prime}}$. Hence, using the facts above, the $\ell_{1}$ norm
distance between $\mu_{s}$ and $\mu_{s^{\prime}}$ reads
($\|\mu_{s}-\mu_{s^{\prime}}\|:=\int
d\lambda|\mu_{s}(\lambda)-\mu_{s^{\prime}}(\lambda)|$).

$\displaystyle\|\mu_{s}-\mu_{s^{\prime}}\|$
$\displaystyle=\int_{\Lambda\backslash
S_{s}}d\lambda\mu_{s^{\prime}}(\lambda)+\int_{\Lambda\backslash
S_{s^{\prime}}}d\lambda\mu_{s}(\lambda)$ $\displaystyle=2-2\int_{S_{s}\cap
S_{s^{\prime}}}d\lambda\mu_{s}(\lambda)$ $\displaystyle=2-2\int_{S_{s}\cap
S_{s^{\prime}}}d\lambda\mu_{s}(\lambda)\xi_{s^{\prime}}(\lambda).$

Note that the last integral can be extended to $\Lambda$. In fact, by
contradiction suppose that $\xi_{s^{\prime}}(\lambda)\neq 0$ for some nonzero
measure set $X\subseteq S_{s}\backslash S_{s^{\prime}}$. Then, from Eq. (6),
it follows that, for almost all $\lambda\in X$,
$0<\mu_{s}(\lambda)=\mu_{s^{\prime\perp}}(\lambda)$. However, as we discussed
$\xi_{s^{\prime}}(\lambda)=0$ almost everywhere on $S_{s^{\prime\perp}}$,
which gives the desired contradiction. Hence the integral can be extended to
$S_{s}\cup S_{s^{\prime}}$ and, trivially, to all $\Lambda$. In conclusion,

$\|\mu_{s}-\mu_{s^{\prime}}\|=2-2\int_{\Lambda}d\lambda\mu_{s}(\lambda)\xi_{s^{\prime}}(\lambda)=2(1-c_{ss^{\prime}}),$
(7)

where $c_{ss^{\prime}}=p(M_{s^{\prime}}|P_{s})$. Using the triangle
inequality,

$\|\mu_{aa}-\mu_{bb}\|\leq\|\mu_{aa}-\mu_{\alpha}\|+\|\mu_{\alpha}-\mu_{\beta}\|+\|\mu_{\beta}-\mu_{bb}\|.$

By definition, $\mu_{\alpha}(\lambda)=\int
d\lambda^{\prime}T(\lambda|\lambda^{\prime})\mu_{a}(\lambda^{\prime})$, for a
stochastic matrix $T(\lambda|\lambda^{\prime})$. Similarly,
$\mu_{\beta}(\lambda)=\int
d\lambda^{\prime}T(\lambda|\lambda^{\prime})\mu_{b}(\lambda^{\prime})$, with
the same stochastic matrix. Since $\int d\lambda
T(\lambda|\lambda^{\prime})=1$ and $T(\lambda|\lambda^{\prime})\geq 0$, one
can readily verify from the convexity of the absolute value that
$\|\mu_{\alpha}-\mu_{\beta}\|\leq\|\mu_{a}-\mu_{b}\|$ (data processing
inequality), which implies

$\|\mu_{aa}-\mu_{bb}\|\leq\|\mu_{\alpha}-\mu_{aa}\|+\|\mu_{a}-\mu_{b}\|+\|\mu_{\beta}-\mu_{bb}\|.$
(8)

We can apply Eq. (7) to each of the couples $(s,s^{\prime})$ on the right hand
side of Eq. (8), obtaining

$\|\mu_{aa}-\mu_{bb}\|\leq 2(1-c_{\alpha aa})+2(1-c_{ab})+2(1-c_{\beta bb}).$
(9)

Let us now show $\|\mu_{aa}-\mu_{bb}\|\geq 2(1-c_{aa,bb})$. First, notice that

$\displaystyle\|\mu_{aa}-\mu_{bb}\|=\int\limits_{\begin{subarray}{c}S_{aa}\backslash
S_{bb}\end{subarray}}d\lambda\mu_{aa}(\lambda)+\int\limits_{\begin{subarray}{c}S_{bb}\backslash
S_{aa}\end{subarray}}d\lambda\mu_{bb}(\lambda)\leavevmode\nobreak\ +$
$\displaystyle\int\limits_{\begin{subarray}{c}R_{1}\end{subarray}}d\lambda(\mu_{aa}(\lambda)-\mu_{bb}(\lambda))+\int\limits_{\begin{subarray}{c}R_{2}\end{subarray}}d\lambda(\mu_{bb}(\lambda)-\mu_{aa}(\lambda)),$

with $R_{1}:=\\{\lambda\in S_{aa}\cap
S_{bb}:\mu_{aa}(\lambda)\geq\mu_{bb}(\lambda)\\}$ and $R_{2}:=(S_{aa}\cap
S_{bb})\backslash R_{1}$. Next,

$\displaystyle\|\mu_{aa}-\mu_{bb}\|$
$\displaystyle=2\left(1-\int\limits_{\begin{subarray}{c}R_{1}\end{subarray}}d\lambda\mu_{bb}(\lambda)-\int\limits_{\begin{subarray}{c}R_{2}\end{subarray}}d\lambda\mu_{aa}(\lambda)\right)$
$\displaystyle\geq 2-2\int\limits_{\begin{subarray}{c}R_{1}\cup
R_{2}=S_{aa}\cap S_{bb}\end{subarray}}d\lambda\mu_{aa}(\lambda)$
$\displaystyle=2-2\int\limits_{\begin{subarray}{c}S_{aa}\cap
S_{bb}\end{subarray}}d\lambda\mu_{aa}(\lambda)\xi_{bb}(\lambda)$
$\displaystyle\geq 2(1-c_{aa,bb})$

where the first inequality follows from
$\mu_{aa}(\lambda)\geq\mu_{bb}(\lambda)\leavevmode\nobreak\ \forall\lambda\in
R_{1}$ and the second equality follows from $\xi_{bb}(\lambda)=1$ almost
everywhere in $S_{bb}$. Finally, substituting this in Eq. (9) and rearranging
the terms gives

$\frac{1}{2}c_{\alpha aa}+\frac{1}{2}c_{\beta bb}\leq
1-\frac{c_{ab}}{2}+\frac{c_{aa,bb}}{2}$

and since $F_{g}=\frac{1}{2}c_{\alpha aa}+\frac{1}{2}c_{\beta bb}$ the global
cloning achieved by non-contextual ontological models that comply with the
operational features O1-O2 is upper bounded as in Eq. (5). ∎

In Fig. 2 we compare the optimal quantum cloning (global) fidelity of Eq. (2)
with the maximum noncontextual cloning fidelity of Eq. (5), taking into
account that, in quantum experiments, one observes $c_{aa,bb}=c^{2}_{ab}$. One
can see, for any $0<c_{ab}<1$, that quantum mechanics achieves higher copying
fidelities than what is allowed by the principle of noncontextuality. Hence,
the phenomenology of optimal cloning cannot be reproduced within noncontextual
ontological models. Contextuality provides an advantage for the maximum
copying fidelity.333Of course, when $c_{ab}=0$ \- as it is for classical,
i.e., orthogonal, states - both the quantum and the noncontextual fidelities
are $1$.

Figure 2: Maximum tradeoff between cloning fidelity $F_{g}$ and confusability
$c_{ab}$ allowed for noncontextual models (blue line, Eq. (5)) versus optimal
tradeoff achievable in quantum theory (red line, Eq. (2)).

Interestingly, the above derivation also gives an alternative, simple proof of
the main result of Ref. [5]. In fact, an intermediate technical result in the
proof of Theorem 1 is that in the presence of the operational features O1-O2,
noncontextual models must have a direct relation between the experimentally
accessible confusabilities $c_{ss^{\prime}}=p(M_{s^{\prime}}|P_{s})$ and the
$\ell_{1}$ distance between the corresponding probability distributions:

$\|\mu_{s}-\mu_{s^{\prime}}\|=2(1-c_{ss^{\prime}}).$ (10)

(This was implicitly shown in Ref. [5] Sec. IVA, but using infinitely many
extra operational assumptions. That is, they assume O2 for all pairs of
orthogonal states).

Since the maximum probability $s_{ab}$ of distinguishing two preparations
$P_{a}$ and $P_{b}$ is at most $1/2+$ $\|\mu_{a}-\mu_{b}\|/4$, it immediately
follows $s_{ab}\leq 1-c_{ab}/2$, which is the optimal state discrimination
probability in noncontextual models, as given in Ref. [5]. Conversely, it is
not immediately obvious how the techniques of Ref. [5] could be adapted to
obtain our result on cloning, due to our use of the data processing inequality
in Theorem 1.

We also note that the noncontextual bound on cloning is tight. Denote by
$S_{s}$ the support of $\mu_{s}$. Consider a model in which
$\mu_{ss}=\mu_{s}\mu_{s}$ and $\xi_{s}(\lambda)=1$ if $\lambda\in S_{s}$ and
zero otherwise. A cloning strategy that saturates the bound is as follows: if
the input $\lambda$ is in $S_{a}\backslash S_{b}$, output
$(\lambda,\lambda^{\prime})$, with $\lambda^{\prime}$ sampled according to
$\mu_{a}$; otherwise, output $(\lambda,\lambda^{\prime})$ with
$\lambda^{\prime}$ sampled according to $\mu_{b}$. Notice that this sets
$\mu_{\beta}=\mu_{b}\mu_{b}$ and, hence, $c_{\beta bb}=1$ ($\mu_{b}$ is copied
perfectly). On the other hand,
$\mu_{\alpha}(\lambda,\lambda^{\prime})=\mu_{a}(\lambda)\mu_{a}(\lambda^{\prime})$
for $\lambda\in S_{a}\backslash S_{b}$ and
$\mu_{\alpha}(\lambda,\lambda^{\prime})=\mu_{a}(\lambda)\mu_{b}(\lambda^{\prime})$
for $\lambda\in S_{a}\cap S_{b}$ and, hence,

$\displaystyle c_{\alpha aa}$ $\displaystyle=\int d\lambda
d\lambda^{\prime}\mu_{\alpha}(\lambda,\lambda^{\prime})\xi_{aa}(\lambda,\lambda^{\prime})$
$\displaystyle=\int_{S_{a}\times S_{a}}d\lambda
d\lambda^{\prime}\mu_{\alpha}(\lambda,\lambda^{\prime})$
$\displaystyle=\int_{(S_{a}\setminus S_{b})\times S_{a}}d\lambda
d\lambda^{\prime}\mu_{a}(\lambda)\mu_{a}(\lambda^{\prime})\quad+$
$\displaystyle\qquad\qquad\int_{(S_{a}\cap S_{b})\times S_{a}}d\lambda
d\lambda^{\prime}\mu_{a}(\lambda)\mu_{b}(\lambda^{\prime})$
$\displaystyle=(1-c_{ab})+c_{ab}\cdot c_{ba}=1-c_{ab}+c_{ab}^{2},$

where, in the last equality, we use the operational fact that $c_{ab}=c_{ba}$.
Finally, this gives $F_{g}=\frac{1}{2}(1-c_{ab}+c_{ab}^{2})+\frac{1}{2}=F^{\rm
NC}_{g}$. In Appendix A we complete this strategy with a concrete choice of
$\mu_{a}$, $\mu_{b}$, $\mu_{{aa}^{\perp}}$, $\mu_{{bb}^{\perp}}$,
$\mu_{{\alpha}^{\perp}}$ and $\mu_{{\beta}^{\perp}}$ complying with O1 and
satisfying Eq. (4) for all the operational equivalences in O2.

This optimal strategy seems to suggest the following intuition behind the
theorem: our assumption of preparation noncontextuality on the input
preparations $P_{a}$, $P_{b}$ imply that the distributions $\mu_{a}(\lambda)$
and $\mu_{b}(\lambda)$ overlap “too much” (formally, it implies maximal
$\psi$-epistemicity, $c_{ab}=\int_{S_{b}}d\lambda\mu_{a}(\lambda)$ [18]),
hence the cloning performance turns out worse than in quantum mechanics.
Furthermore, noncontextuality implies that $\mu_{a}$ and $\mu_{b}$ coincide on
their overlap, which implies a direct relation between $c_{ab}$ and the
$\ell_{1}$ norm $\|\mu_{a}-\mu_{b}\|$. Crucially the latter cannot be
increased by the cloning machine, since $\|\cdot\|$ decreases under post-
processing.

However, this mechanism can only be part of the story. First, the cloning
performance is not monotonically decreasing with increasing overlap, since for
$c_{ab}=1$ one can clone perfectly. Second, cloning is defined as the creation
of two independent copies of the preparations $P_{a}$ or $P_{b}$, but these do
not necessarily correspond to two independent copies $\mu_{a}\mu_{a}$,
$\mu_{b}\mu_{b}$ (this assumption, which we do not make, is called
_preparation independence_ [19]). Nevertheless, we showed that a no-go theorem
results from the observed overlaps $c_{ab}$, $c_{aa,bb}$ and noncontextuality
assumptions only as a consequence of information processing inequalities and
the triangle inequality.

We note in passing that our proof technique can be abstracted and applied to
other tasks as follows:

  1. 1.

First, given a set of observed overlaps $\\{c_{ss^{\prime}}\\}$,
noncontextuality applied to the operational equivalences
$\frac{1}{2}P_{s}+\frac{1}{2}P_{s^{\perp}}\simeq\frac{1}{2}P_{s^{\prime}}+\frac{1}{2}P_{s^{{}^{\prime}\perp}}$
gives the equations (10).

  2. 2.

Second, verify if the equations (10) are compatible with triangle and data
processing inequalities and the performance of quantum protocol under
consideration (in this case, state-dependent cloning).

In fact, the same proof technique can be extended to nonideal scenarios (with
Eq. (10) replaced by Eq. (12)), as we now see.

##  4 Optimal cloning is contextual - beyond idealizations

Theorem 1 is a no-go result for noncontextual ontological models aimed at
explaining the phenomenology of state-dependent quantum cloning. However, the
inequality derived in Eq. (5) is not a proper noncontextuality inequality
because the operational features considered refer to an idealized experiment.
In any real experiment, on the other hand, one will need to confront the
following nonidealities:

  * •

The correlations in O1 will only approximatively hold in data collected in a
real experiment.

  * •

O2 will only be approximatively realized.

Theorem 2 below extends Theorem 1 beyond the ideal limit, allowing for the
observation of nonperfect correlations in O1, such as those generated by a
cloning experiment carried out with nonideal preparations and test
measurements. As we will discuss later, there are general techniques to deal
with the idealization in O2, so that the problem of deriving an experimentally
testable statement reduces to the elimitation of the idealization in O1.
Specifically, we want to weaken it to

  1. O1ni

$p(M_{s}|P_{s})\geq 1-\epsilon_{s},\quad
p(M_{s}|P_{s^{\perp}})\leq\epsilon_{s}$ for $s=a,b,\alpha,\beta,aa,bb$,

where ‘ni’ stands for ‘non-ideal’.

######  Theorem 2 (Optimal cloning fidelity in noncontextual models – noise-
robust version).

With the notation of Thm. 1, suppose that one observes the operational
features O1ni and O2. Then, for any noncontextual model we have that

$\displaystyle F_{g}\leq F^{\rm
NC,ni}_{g}=1-\frac{c_{ab}}{2}+\frac{c_{aa,bb}}{2}+\emph{Err}.$ (11)

where $\emph{Err}=\frac{1}{2}({\epsilon_{b}+2\epsilon_{bb}+\epsilon_{aa}})$.

Note that, while we gave an independent and simpler proof of Theorem 1, we can
now see it as a corollary of the result above once all error terms are set of
zero. Another interesting case is when all error terms are equal,
$\epsilon_{b}=\epsilon_{bb}=\epsilon_{aa}:=\epsilon$, which gives $F^{\rm
NC,ni}_{g}=1-\frac{c_{ab}}{2}+\frac{c_{aa,bb}}{2}+2\epsilon$. In fact, we can
give a slightly stronger and symmetric bound than the above. For the specific
form, see Appendix B.

The proof of Theorem 2 follows the same lines as that of Theorem 1. The key
addition is to extend Eq. (10) to the noisy setting. Specifically, we show
that in the presence of the operational features O1ni-O2, noncontextual models
must satisfy

$|\|\mu_{s}-\mu_{s^{\prime}}\|-2(1-c_{ss^{\prime}})|\leq
2\epsilon_{s^{\prime}},$ (12)

and similarly if we exchange $s$ and $s^{\prime}$. In other words, the
relation of Eq. (10) holds approximatively, and we can bound its violation
with the experimentally accessible noise level. The proof of this result is
more involved than in the ideal scenario, so we postpone the derivation to
Appendix B.

Eq. (12) imposes a strict relation, in any noncontextual model and beyond the
ideal scenario, between the $\ell_{1}$ distance of two epistemic states and
their operationally accessible confusability. Hence, we anticipate that these
relations will be of broader use to identify quantum advantages beyond state-
dependent cloning. For instance, following the same reasoning given after
Theorem 1, these inequalities provide an alternative and intuitive derivation
of the tight noise-robust noncontextual bound on state discrimination of Ref.
[5], $s_{ab}\leq\frac{1}{2}+\frac{1}{4}\|\mu_{a}-\mu_{b}\|\leq
1-\frac{c_{ab}-\epsilon_{b}}{2}$.

###  4.1 An explicit noise model

Having derived a noise-robust version of our noncontextual bound, the next
step is to investigate whether quantum mechanics violates it. We consider a
standard noise model in which the ideal quantum preparations, measurements and
unitary transformation are all thwarted by a depolarizing channel
$\mathcal{N}_{v}$ with noise level $v\in[0,1]$:

$\displaystyle\mathcal{N}_{v}(\rho)=(1-v)\leavevmode\nobreak\
\rho+v\mathbb{I}/{4}.$

A direct calculation (see Appendix C) shows that this sets
$\epsilon=v(31-21v+9v^{2})/16$ in Eq. (11). If one uses the unitary
transformation that is optimal for state-dependent cloning in the noiseless
setting, one gets a quantum strategy whose global average fidelity reads

$F_{g}^{\rm Q,noisy}(v):=(1-v)^{3}F_{g}^{\rm Q,opt}+\frac{1}{4}v(3-3v+v^{2})$
(13)

which coincides with the optimal for $v=0$. For $v>0$, however, and unlike in
the ideal case, the tradeoff between $c_{ab}$ and $F_{g}$ is not necessarily
above the noncontextual bound. For example, for $v=0.015$ a violation can be
observed only for $c_{ab}\in[0.318,0.718]$, see Fig. 3. Nevertheless, a
preliminary comparison with the experimental results of Ref. [20] suggests
that the required low level of noise is not beyond current experiments. In
fact, in terms of the parameter
$C_{s}=1/2\,p(M_{s}|P_{s})+1/2\,p(M_{s^{\perp}}|P_{s^{\perp}})$ defined in
Ref. [20] ($C_{s}=1$ in the ideal scenario), $v=0.015$ corresponds to
$C_{s}\approx 0.9851$ for $s=a,b$ and $C_{s}\approx 0.9667$ for $s=aa,bb$, and
Ref. [20] experimentally realized $C_{s}=0.9969$.

Figure 3: _Noise-resistance of the quantum advantage in cloning._ This plot
shows the maximum value of the noise parameter $v$ of a depolarizing channel
(affecting preparations, measurements and transformation) for which the
quantum value of the cloning fidelity (Eq. (13)) is above the noncontextual
bound, as a function of the confusability between the inputs $c_{ab}$.

###  4.2 Remaining assumptions

As we mentioned, the only remaining idealization is the operational
verification of O2. Let us suppose that, in an experiment, after doing
tomography,444We will later discuss the assumption that can one access a
tomographically complete set of measurements. one determines that the actual
experimental realizations of the ideal preparations are
$P^{(1)}_{a},P^{(1)}_{a^{\perp}},P^{(1)}_{b},P^{(1)}_{b^{\perp}},P^{(1)}_{aa},P^{(1)}_{{aa}^{\perp}},P^{(1)}_{bb},P^{(1)}_{{bb}^{\perp}},P^{(1)}_{\alpha},P^{(1)}_{{\alpha}^{\perp}}$,
$P^{(1)}_{\beta}$ and $P^{(1)}_{{\beta}^{\perp}}$. These ’primary’
preparations will, in general, not respect the required operational
equivalences in O2, due to unavoidable imperfections in the experimental
realisation. Luckily, there are general considerations to tackle this
idealization [20].

The first thing to notice is that if one can experimentally achieve a set of
preparations $P^{(1)}_{s}$, then one can also prepare any convex combination
of them, i.e. any preparation in the convex hull $\mathcal{C}$ of the
preparations $P^{(1)}_{s}$. By the linearity of Eq. (1), one can then compute
the measurement statistics of all the preparations in $\mathcal{C}$.
Therefore, as put forward in Sec. IV of Ref. [20], to go ahead with the
experimental verification of Theorem 2 one only needs to find ‘secondary’
preparations $P^{(2)}_{s}$ in $\mathcal{C}$ whose measurement statistics
satisfy the operational equivalences in O2; that is, we only need

  1. O2ni

O2 is satisfied for some preparations $P^{(2)}_{s}$ in the convex hull
$\mathcal{C}$ of the experimental preparations $P^{(1)}_{s}$.

This post-processing hence allows one to apply Theorem 2 even if the collected
data does not satisfy O2. One can think of the secondary preparations as noisy
versions of the primary preparations. Hence, the price one pays in this
construction is that the corresponding noise parameters
$\epsilon^{\prime}_{s}=p(M_{s}|P^{(2)}_{s})$ in O1ni will in general be
larger. Note that, even if $\epsilon^{\prime}_{s}$ is too large compared to
$\epsilon_{s}$ to see any violation in Theorem 2, one can get around this
issue by adding extra experimental preparations $P^{(1)}_{\rm extra}$ to
enlarge $\mathcal{C}$, as explicitly done in Ref. [20]. To summarize, there
are good general tools to deal with imperfections in the operational
equivalences O2.

As a final remark, it is useful to briefly talk about loopholes. These are all
those assumptions that cannot be _conclusively_ tested by any experimental
means. In a nonlocality experiment, for example, these include the assumption
that the two sides cannot communicate and the ability to choose the
measurement freely, i.e., independently of any other variable relevant to the
experiment. In a contextuality experiment the notion of operational
equivalence relies on the knowledge of a tomographically complete set of
measurements. However, if quantum theory is not correct, the tomographically
complete set of a post-quantum theory may contain extra unknown measurements
(just like a future theory may allow signalling). Recent work has shown that
the problem can be mitigated by the addition of extra (known) measurements and
preparations (see Ref. [21]), but this goes beyond the scope of the present
work.

##  5 Conclusions and open questions.

We have shown that the operational statistics observed in the optimal state-
dependent quantum cloning is incompatible with the predictions of every
noncontextual ontological model. In particular, for given overlap, the
noncontextual global cloning fidelity is strictly smaller than the quantum
prediction. A similar result continues to hold in more realistic experiments
which are unavoidably affected by noise (while the effect can be ‘washed out’
by excessive experimental imperfections). This identifies contextuality as the
resource for optimal state-dependent quantum cloning.

From a foundational point of view, it would be relevant to explore whether the
relation between contextuality and cloning fidelity, that we proved for
optimal state-dependent cloning, extends to the other types of imperfect
cloning studied in the literature, mainly phase-covariant and/or universal
cloning, as well as to probabilistic cloning [11]. From an applications’ point
of view, one important open question is if our noncontextual bound can be used
to prove a contextual advantage for quantum information processing tasks which
rely on optimal quantum state-dependent cloning (_e.g._ , [22, 23]).

Finally, it may be possible to use the connection between $\ell_{1}$ norm and
confusability developed here to understand what aspects of other quantum
information primitives, such as quantum teleportation, are truly nonclassical.

_Acknowledgements._ We are grateful to Joseph Bowles for useful comments on a
draft of this manuscript. We acknowledge financial support from the the
European Union’s Marie Sklodowska-Curie individual Fellowships (H2020-MSCA-
IF-2017, GA794842), Spanish MINECO (Severo Ochoa SEV-2015-0522 and project
QIBEQI FIS2016-80773-P), Fundacio Cellex and Generalitat de Catalunya (CERCA
Programme and SGR 875).

## References

  * Wootters and Zurek [1982] William K Wootters and Wojciech H Zurek.  A single quantum cannot be cloned.  _Nature_ , 299(5886):802, 1982.  doi: 10.1038/299802a0.
  * Dieks [1982] D Dieks.  Communication by EPR devices.  _Physics Letters A_ , 92(6):271–272, 1982.  doi: 10.1016/0375-9601(82)90084-6.
  * Park [1970] James L Park.  The concept of transition in quantum mechanics.  _Foundations of Physics_ , 1(1):23–33, 1970.  doi: 10.1007/BF00708652.
  * Yuen [1986] Horace P Yuen.  Amplification of quantum states and noiseless photon amplifiers.  _Physics Letters A_ , 113(8):405–407, 1986.  doi: 10.1016/0375-9601(86)90660-2.
  * Schmid and Spekkens [2018] David Schmid and Robert W Spekkens.  Contextual advantage for state discrimination.  _Physical Review X_ , 8(1):011015, 2018.  doi: 10.1103/PhysRevX.8.011015.
  * Spekkens [2005] Robert W Spekkens.  Contextuality for preparations, transformations, and unsharp measurements.  _Physical Review A_ , 71(5):052108, 2005.  doi: 10.1103/PhysRevA.71.052108.
  * Bartlett et al. [2012] Stephen D Bartlett, Terry Rudolph, and Robert W Spekkens.  Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction.  _Physical Review A_ , 86(1):012103, 2012.  doi: 10.1103/PhysRevA.86.012103.
  * Spekkens [2007] Robert W Spekkens.  Evidence for the epistemic view of quantum states: A toy theory.  _Physical Review A_ , 75(3):032110, 2007.  doi: 10.1103/PhysRevA.75.032110.
  * Daffertshofer et al. [2002] A Daffertshofer, AR Plastino, and A Plastino.  Classical no-cloning theorem.  _Physical review letters_ , 88(21):210601, 2002\.  doi: 10.1103/PhysRevLett.88.210601.
  * Bužek and Hillery [1996] Vladimir Bužek and Mark Hillery.  Quantum copying: Beyond the no-cloning theorem.  _Physical Review A_ , 54(3):1844, 1996.  doi: 10.1103/PhysRevA.54.1844.
  * Scarani et al. [2005] Valerio Scarani, Sofyan Iblisdir, Nicolas Gisin, and Antonio Acín.  Quantum cloning.  _Rev. Mod. Phys._ , 77:1225–1256, Nov 2005.  doi: 10.1103/RevModPhys.77.1225.
  * Saha et al. [2019] Debashis Saha, Paweł Horodecki, and Marcin Pawłowski.  State independent contextuality advances one-way communication.  _New Journal of Physics_ , 21(9):093057, 2019\.  doi: 10.1088/1367-2630/ab4149.
  * Tavakoli and Uola [2020] Armin Tavakoli and Roope Uola.  Measurement incompatibility and steering are necessary and sufficient for operational contextuality.  _Phys. Rev. Research_ , 2:013011, Jan 2020.  doi: 10.1103/PhysRevResearch.2.013011.
  * Leifer [2014] Matthew Saul Leifer.  Is the quantum state real? an extended review of $\psi$-ontology theorems.  _Quanta_ , 3(1):67–155, 2014.  doi: 10.12743/quanta.v3i1.22.
  * Kochen and Specker [1975] Simon Kochen and Ernst P Specker.  The problem of hidden variables in quantum mechanics.  In _The logico-algebraic approach to quantum mechanics_ , pages 293–328. Springer, 1975.  doi: 10.1007/978-94-010-1795-4_17.
  * Spekkens [2019] Robert W Spekkens.  The ontological identity of empirical indiscernibles: Leibniz’s methodological principle and its significance in the work of Einstein.  2019\.  URL https://arxiv.org/abs/1909.04628v1.
  * Bruß et al. [1998] Dagmar Bruß, David P DiVincenzo, Artur Ekert, Christopher A Fuchs, Chiara Macchiavello, and John A Smolin.  Optimal universal and state-dependent quantum cloning.  _Physical Review A_ , 57(4):2368, 1998.  doi: 10.1103/PhysRevA.57.2368.
  * Leifer and Maroney [2013] Matthew S Leifer and Owen JE Maroney.  Maximally epistemic interpretations of the quantum state and contextuality.  _Physical review letters_ , 110(12):120401, 2013\.  doi: 10.1103/PhysRevLett.110.120401.
  * Pusey et al. [2012] Matthew F Pusey, Jonathan Barrett, and Terry Rudolph.  On the reality of the quantum state.  _Nature Physics_ , 8(6):475–478, 2012.  doi: 10.1038/nphys2309.
  * Mazurek et al. [2016] Michael D. Mazurek, Matthew F. Pusey, Ravi Kunjwal, Kevin J. Resch, and Robert W. Spekkens.  An experimental test of noncontextuality without unphysical idealizations.  _Nat. Commun._ , 7:11780, Jun 2016.  doi: 10.1038/ncomms11780.
  * Pusey et al. [2019] Matthew F. Pusey, Lídia del Rio, and Bettina Meyer.  Contextuality without access to a tomographically complete set, 2019\.  URL https://arxiv.org/abs/1904.08699v1.
  * Scarani et al. [2009] Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J Cerf, Miloslav Dušek, Norbert Lütkenhaus, and Momtchil Peev.  The security of practical quantum key distribution.  _Reviews of modern physics_ , 81(3):1301, 2009\.  doi: 10.1103/RevModPhys.81.1301.
  * Deuar and Munro [2000] P Deuar and WJ Munro.  Quantum copying can increase the practically available information.  _Physical Review A_ , 62(4):042304, 2000.  doi: 10.1103/PhysRevA.62.042304.

##  Appendix A Noncontextual model saturating the bound in Theorem 1

To complement the cloning strategy given in the main text, in this section we
give a concrete choice of distributions $\mu_{aa}$, $\mu_{bb}$,
$\mu_{{aa}^{\perp}}$, $\mu_{{bb}^{\perp}}$ and $\mu_{{\alpha}^{\perp}}$
satisfying the operational features targeted by Theorem 1. The supports of all
these distributions, which we set to be subsets of $[0,2]\times[0,2]$, are
plotted in Figure 4. All the distributions are constantly $1$ on their
support. Notice that since the cloning map given in the main text makes
$\mu_{\beta}\equiv\mu_{bb}$, it follows that to satisfy the operational
equivalence for $(\mu_{\beta},\mu_{bb})$ we must have
$\mu_{{\beta}^{\perp}}\equiv\mu_{{bb}^{\perp}}$. We let the reader verify, by
inspecting the plots, that the remaining requirements implied by the
operational features O1 and O2 are satisfied by these distributions (and the
choice of response functions made in the main text).

(a) $S_{aa}$

(b) $S_{{aa}^{\perp}}$

(c) $S_{{bb}}$

(d) $S_{{bb}^{\perp}}$

(e) $S_{{\alpha}^{\perp}}$

Figure 4: Supports of distributions $\mu_{aa}$, $\mu_{bb}$,
$\mu_{{aa}^{\perp}}$, $\mu_{{bb}^{\perp}}$ and $\mu_{{\alpha}^{\perp}}$
satisfying the restrictions imposed on noncotextual models by the requirements
of Theorem 1. The distributions are $1$-valued in the filled regions (i.e. in
their support).

##  Appendix B Generalization and proof of Theorem 2

In this section we will prove a slightly stronger and more symmetric bound on
the noncontextual cloning fidelity $F^{\rm NC}_{g}$ from which the bound in
Thm. 2 in the main text follows straightforwardly as a corollary.

######  Theorem 3.

With the notation of Thm. 1, suppose that one observes the operational
features O1ni and O2:

  1. O1ni

$p(M_{s}|P_{s})\geq 1-\epsilon_{s},\quad
p(M_{s}|P_{s^{\perp}})\leq\epsilon_{s}$ for $s=a,b,\alpha,\beta,aa,bb,$

  2. O2

$\frac{1}{2}P_{s}+\frac{1}{2}P_{s^{\perp}}\simeq\frac{1}{2}P_{s^{\prime}}+\frac{1}{2}P_{s^{{}^{\prime}\perp}}$,
for all $(s,s^{\prime})$ in $\\{(a,b),(\alpha,aa),(\beta,bb)\\}$.

Then, for any noncontextual model we have that

$\displaystyle F^{\rm NC}_{g}\leq
1+\frac{\min\\{\epsilon_{b}-c_{ab},\epsilon_{a}-c_{ba}\\}}{2}+\frac{\min\\{c_{aa,bb}+\epsilon_{bb},c_{bb,aa}+\epsilon_{aa}\\}}{2}+\frac{\epsilon_{aa}+\epsilon_{bb}}{2}.$
(14)

For the proof of Theorem 3, we make use of the following lemma relating the
$\ell_{1}$ distance of two epistemic states in any ontological model
satisfying the hypothesis of the theorem and their operationally accessible
confusability.

######  Lemma 4.

Let $P_{s}$, $P_{s^{\prime}}$ be preparations. Suppose there exists
preparations $P_{s^{\perp}}$, $P_{s^{\prime\perp}}$ and a two outcome
measurement $M_{s}$ such that

  1. 1.

$\frac{1}{2}P_{s}+\frac{1}{2}P_{s^{\perp}}\simeq\frac{1}{2}P_{s^{\prime}}+\frac{1}{2}P_{s^{\prime\perp}},$

  2. 2.

$p(M_{k}|P_{k})\geq 1-\epsilon_{k}$, $p(M_{k}|P_{k^{\perp}})\leq\epsilon_{k}$,
$k=s,s^{\prime}.$

Then, in a noncontextual ontological model,

$2\max\\{1-c_{ss^{\prime}}-\epsilon_{s^{\prime}},1-c_{s^{\prime}s}-\epsilon_{s}\\}\leq\|\mu_{s}-\mu_{s^{\prime}}\|\leq
2\min\\{1-c_{ss^{\prime}}+\epsilon_{s^{\prime}},1-c_{s^{\prime}s}+\epsilon_{s}\\},$
(15)

###### Proof.

We denote by $S_{s}$ the support of $\mu_{s}$. Define a partition $S_{s}\cup
S_{s^{\prime}}=\sqcup_{i=1}^{4}R_{i}$, as summarized in Figure 5:

  * •

$R_{1}=S_{s}\backslash(S_{s}\cap S_{s^{\prime}})$,
$R_{4}=S_{s^{\prime}}\backslash(S_{s}\cap S_{s^{\prime}})$.

  * •

$R_{2}=\\{\lambda\in S_{s}\cap
S_{s^{\prime}}|\mu_{s}(\lambda)\geq\mu_{s^{\prime}}(\lambda)\\}$,
$R_{3}=\\{\lambda\in S_{s}\cap
S_{s^{\prime}}|\mu_{s}(\lambda)<\mu_{s^{\prime}}(\lambda)\\}$.

Figure 5: Sketch of the relevant regions in the proof of Lemma 4.

Then,

$\displaystyle\|\mu_{s}-\mu_{s^{\prime}}\|$ $\displaystyle=\int
d\lambda|\mu_{s}(\lambda)-\mu_{s^{\prime}}(\lambda)|$
$\displaystyle=\int_{R_{1}}d\lambda\mu_{s}(\lambda)+\int_{R_{4}}d\lambda\mu_{s^{\prime}}(\lambda)+\int_{R_{2}}d\lambda[\mu_{s}(\lambda)-\mu_{s^{\prime}}(\lambda)]+\int_{R_{3}}d\lambda[\mu_{s^{\prime}}(\lambda)-\mu_{s}(\lambda)]$
$\displaystyle=2-\int_{R_{2}\cup
R_{3}}d\lambda[\mu_{s}(\lambda)+\mu_{s^{\prime}}(\lambda)]+\int_{R_{2}}d\lambda[\mu_{s}(\lambda)-\mu_{s^{\prime}}(\lambda)]+\int_{R_{3}}d\lambda[\mu_{s^{\prime}}(\lambda)-\mu_{s}(\lambda)]$
$\displaystyle=2-2\int_{R_{3}}d\lambda\mu_{s}(\lambda)-2\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda).$
(16)

Consider,

$\displaystyle
c_{ss^{\prime}}-\int_{R_{3}}d\lambda\mu_{s}(\lambda)-\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)$
$\displaystyle=\int_{R_{1}\cup R_{2}\cup
R_{3}}d\lambda\mu_{s}(\lambda)\xi_{s^{\prime}}(\lambda)-\int_{R_{3}}d\lambda\mu_{s}(\lambda)-\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)$
$\displaystyle\leq\int_{R_{1}\cup
R_{2}}d\lambda\mu_{s}(\lambda)\xi_{s^{\prime}}(\lambda)-\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)$
$\displaystyle=\int_{R_{1}\cup
R_{2}}d\lambda[\mu_{s^{\prime}}(\lambda)+\mu_{s^{{}^{\prime}\bot}}(\lambda)-\mu_{s^{\bot}}(\lambda)]\xi_{s^{\prime}}(\lambda)-\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)$
$\displaystyle=-\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)\xi_{s^{{}^{\prime}\perp}}(\lambda)+\int_{R_{1}\cup
R_{2}}d\lambda[\mu_{s^{{}^{\prime}\perp}}(\lambda)-\mu_{s^{\perp}}(\lambda)]\xi_{s^{\prime}}(\lambda)$
$\displaystyle\leq\int_{R_{1}\cup
R_{2}}d\lambda\mu_{s^{{}^{\prime}\perp}}(\lambda)\xi_{s^{\prime}}(\lambda)\leq\int
d\lambda\mu_{s^{{}^{\prime}\perp}}(\lambda)\xi_{s^{\prime}}(\lambda)=p(M_{s^{\prime}}|P_{s^{{}^{\prime}\perp}})\leq\epsilon_{s^{\prime}},$

where we used $\xi_{s^{\prime}}\leq 1$ in the first inequality and assumption
1 and non-contextuality in the second equality. In the third equality, we used
$\int_{R_{1}}d\lambda\mu_{s^{\prime}}(\lambda)\xi_{s^{\prime}}(\lambda)=0$ and
in the final inequality we used assumption 2. Then, using Eq. (16),

$\displaystyle\|\mu_{s}-\mu_{s^{\prime}}\|\leq
2(1-c_{ss^{\prime}}+\epsilon_{s^{\prime}}).$

Furthermore, recalling that $\xi_{s^{{}^{\prime}\perp}}=1-\xi_{s^{\prime}}$,

$\displaystyle\int_{R_{3}}d\lambda\mu_{s}(\lambda)+\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)-c_{ss^{\prime}}$
$\displaystyle=\int_{R_{3}}d\lambda\mu_{s}(\lambda)+\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)-\int_{R_{1}\cup
R_{2}\cup R_{3}}d\lambda\mu_{s}(\lambda)\xi_{s^{\prime}}(\lambda)$
$\displaystyle\leq\int_{R_{3}}d\lambda\mu_{s^{\prime}}(\lambda)\xi_{s{{}^{\prime\perp}}}(\lambda)+\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)\xi_{s{{}^{\prime\perp}}}(\lambda)-\int_{R_{1}}d\lambda\mu_{s}\xi_{s^{\prime}}(\lambda)$
$\displaystyle\leq\int_{R_{3}}d\lambda\mu_{s^{\prime}}(\lambda)\xi_{s{{}^{\prime\perp}}}(\lambda)+\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)\xi_{s{{}^{\prime\perp}}}(\lambda)$
$\displaystyle\leq\int
d\lambda\mu_{s^{\prime}}(\lambda)\xi_{s^{{}^{\prime}\perp}}(\lambda)=p(M_{s^{{}^{\prime}\perp}}|P_{s^{\prime}})\leq\epsilon_{s^{\prime}}.$

where in the first inequality we used that
$\mu_{s}(\lambda)\leq\mu_{s^{\prime}}(\lambda)$ in $R_{3}$ and
$\mu_{s}(\lambda)\geq\mu_{s^{\prime}}(\lambda)$ in $R_{2}$. In the final
inequality, we used assumption 2. Hence, we have that

$\displaystyle
c_{ss^{\prime}}-\int_{R_{3}}d\lambda\mu_{s}(\lambda)-\int_{R_{2}}d\lambda\mu_{s^{\prime}}(\lambda)\geq-\epsilon_{s^{\prime}}$

and, using Eq. (16), that

$\displaystyle\|\mu_{s}-\mu_{s^{\prime}}\|\geq
2(1-c_{ss^{\prime}}+\epsilon_{s^{\prime}}).$ (17)

Finally, noting that
$\|\mu_{s}-\mu_{s^{\prime}}\|=\|\mu_{s^{\prime}}-\mu_{s}\|$ and that the above
derivation is symmetric under the exchange of $s$ with $s^{\prime}$ we arrive
to the desired result

$\displaystyle
2\max\\{1-c_{ss^{\prime}}-\epsilon_{s^{\prime}},1-c_{s^{\prime}s}-\epsilon_{s}\\}\leq\|\mu_{s}-\mu_{s^{\prime}}\|\leq
2\min\\{1-c_{ss^{\prime}}+\epsilon_{s^{\prime}},1-c_{s^{\prime}s}+\epsilon_{s}\\},$
(18)

Notice that for the lower bound in Eq. (17) (and, hence, the left hand side of
Eq. (18)) we did not use assumption 1 of operational equivalence.

∎

Given the above we can now prove Theorem 3:

###### Proof of Theorem 3.

In the first part we proceed as in the ideal case. From the triangle
inequality and the contractivity of the $\ell_{1}$ norm under stochastic
processes (which gives $\|\mu_{\alpha}-\mu_{\beta}\|\leq\|\mu_{a}-\mu_{b}\|$),
one can show that the following equation holds (see Eq. (8)):

$\|\mu_{aa}-\mu_{bb}\|\leq\|\mu_{\alpha}-\mu_{aa}\|+\|\mu_{a}-\mu_{b}\|+\|\mu_{\beta}-\mu_{bb}\|.$
(19)

Using both upper and lower bounds for the $\ell_{1}$ distance derived in Lemma
4, this implies

$\displaystyle 2\max\\{1-c_{aa,bb}-\epsilon_{bb},1-c_{bb,aa}-\epsilon_{aa}\\}$
$\displaystyle\leq 2(1-c_{\alpha
aa})+2\epsilon_{aa}+2\min\\{1-c_{ab}+\epsilon_{b},1-c_{ba}+\epsilon_{a}\\}$
$\displaystyle+2(1-c_{\beta bb})+2\epsilon_{bb},$

which can be rearranged to give the claimed bound on $F^{\rm NC}_{g}$. ∎

##  Appendix C Quantum violation of noise-contextual bound under depolarizing
noise

###  C.1 Introducing noise

We will assume that all experimental procedures in the ideal quantum cloning
experiment (that is, preparations, measurements and transformations) are
affected by a depolarizing channel $\mathcal{N}_{v}$ with noise level
$v\in[0,1]$:

$\displaystyle\mathcal{N}_{v}(\rho)=(1-v)\leavevmode\nobreak\
\rho+v\frac{\mathbb{I}}{4}.$

Therefore, for $x\in\\{a,b\\}$, the ideal input preparations transform as

$\ket{x0}\mapsto\mathcal{N}_{v}(\left|x0\right\rangle\\!\left\langle
x0\right|)=(1-v)\left|x0\right\rangle\\!\left\langle
x0\right|+v\frac{\mathbb{I}}{4},$

so that the actual input preparations become

$\rho_{x}:=\operatorname{Tr}_{2}\left[\mathcal{N}_{v}(\left|x0\right\rangle\\!\left\langle
x0\right|)\right]=(1-v)\left|x\right\rangle\\!\left\langle
x\right|+v\frac{\mathbb{I}_{ab}}{2},$

with $\mathbb{I}_{ab}$ the projector over the ${\rm
span}(\\{\ket{a},\ket{b}\\})$. The ideal cloning transformation $\mathcal{U}$
becomes $\mathcal{N}_{v}\circ\mathcal{U}=(1-v)\leavevmode\nobreak\
\mathcal{U}+v\mathcal{D}$, where $\mathcal{D}(\rho)=\mathbb{I}/4$ for all
$\rho$. Hence, the actual outcomes $\chi\in\\{\alpha,\beta\\}$ correspondent
to input $x\in\\{a,b\\}$ become

$\displaystyle\rho_{\chi}$
$\displaystyle:=\mathcal{N}_{v}\circ\mathcal{U}(\rho_{x})=\left[(1-v)\mathcal{U}+v\mathcal{D}\right]\left((1-v)\left|x0\right\rangle\\!\left\langle
x0\right|+v\frac{\mathbb{I}}{4}\right)=(1-v)^{2}\left|\chi\right\rangle\\!\left\langle\chi\right|+(1-(1-v)^{2})\frac{\mathbb{I}}{4}.$

The actual target copies would become
$\mathcal{N}_{v}(\left|xx\right\rangle\\!\left\langle xx\right|)$. While this
is the minimal amount of noise in this preparation required by our model, not
all operational equivalences are satisfied under it. A simple (albeit likely
not optimal) way to fix this issue is to let the noise act for a second step;
hence, define

$\rho_{xx}:=\mathcal{N}_{v}\circ\mathcal{N}_{v}(\left|xx\right\rangle\\!\left\langle
xx\right|)=(1-v)^{2}\left|xx\right\rangle\\!\left\langle
xx\right|+(1-(1-v)^{2})\frac{\mathbb{I}}{4}.$

Finally, for the ideal measurements, they transform as (for $x\in\\{a,b\\}$,
$\chi\in\\{\alpha,\beta\\}$),

$\displaystyle\\{\left|x\right\rangle\\!\left\langle
x\right|,\mathbb{I}_{ab}-\left|x\right\rangle\\!\left\langle x\right|\\}$
$\displaystyle\mapsto$ $\displaystyle M_{x}:=$
$\displaystyle\left\\{(1-v)\left|x\right\rangle\\!\left\langle
x\right|+v\frac{\mathbb{I}_{ab}}{2},(1-v)(\mathbb{I}_{ab}-\left|x\right\rangle\\!\left\langle
x\right|)+v\frac{\mathbb{I}_{ab}}{2}\right\\},$
$\displaystyle\\{\left|xx\right\rangle\\!\left\langle
xx\right|,\mathbb{I}-\left|xx\right\rangle\\!\left\langle xx\right|\\}$
$\displaystyle\mapsto$ $\displaystyle M_{xx}:=$
$\displaystyle\left\\{(1-v)\left|xx\right\rangle\\!\left\langle
xx\right|+v\frac{\mathbb{I}}{4},(1-v)(\mathbb{I}-\left|xx\right\rangle\\!\left\langle
xx\right|)+v\frac{\mathbb{I}}{4}\right\\},$
$\displaystyle\\{\left|\chi\right\rangle\\!\left\langle\chi\right|,\mathbb{I}-\left|\chi\right\rangle\\!\left\langle\chi\right|\\}$
$\displaystyle\mapsto$ $\displaystyle M_{\chi}:=$
$\displaystyle\left\\{(1-v)\left|\chi\right\rangle\\!\left\langle\chi\right|+v\frac{\mathbb{I}}{4},(1-v)(\mathbb{I}-\left|\chi\right\rangle\\!\left\langle\chi\right|)+v\frac{\mathbb{I}}{4}\right\\}.$

###  C.2 Orthogonal preparations and operational equivalences

We now introduce the orthogonal preparations, necessary for the satisfaction
of the operational equivalences. We start with the ones pertaining to the pair
of input preparations $(a,b)$. For, $x\in\\{a,b\\}$, let

$\rho_{x^{\perp}}:=(1-v)\left|x^{\perp}\right\rangle\\!\left\langle
x^{\perp}\right|+v\frac{\mathbb{I}_{ab}}{2},$

with $\ket{x^{\perp}}\in{\rm span}(\\{\ket{a},\ket{b}\\})$ and
$\\!\left\langle
x\vphantom{x^{\perp}}\middle|x^{\perp}\vphantom{x}\right\rangle\\!=0$. Note
that these are naturally thought as the noisy version of the perfect
orthogonal preparations,
$\rho_{x^{\perp}}=\mathcal{N}_{v}(\left|x^{\perp}0\right\rangle\\!\left\langle
x^{\perp}0\right|)$. Now, let us check that the operational equivalence is
satisfied,

$\displaystyle\frac{1}{2}\rho_{a}+\frac{1}{2}\rho_{a^{\perp}}$
$\displaystyle=(1-v)\frac{\left|a\right\rangle\\!\left\langle
a\right|+\left|a^{\perp}\right\rangle\\!\left\langle
a^{\perp}\right|}{2}+v\frac{\mathbb{I}_{ab}}{2}=\frac{\mathbb{I}_{ab}}{2}=(1-v)\frac{\left|b\right\rangle\\!\left\langle
b\right|+\left|b^{\perp}\right\rangle\\!\left\langle
b^{\perp}\right|}{2}+v\frac{\mathbb{I}_{ab}}{2}=\frac{1}{2}\rho_{b}+\frac{1}{2}\rho_{b^{\perp}}.$

Next, we consider the pair of preparations $s\in(\alpha,aa)$. Let

$\displaystyle\rho_{s^{\perp}}:=(1-v)^{2}\left|s\right\rangle\\!\left\langle
s\right|+(1-(1-v)^{2})\frac{\mathbb{I}}{4},$

with $\ket{s^{\perp}}\in{\rm span}(\\{\ket{\alpha},\ket{aa}\\})$, and
$\\!\left\langle
s\vphantom{s^{\perp}}\middle|s^{\perp}\vphantom{s}\right\rangle\\!=0$.
$\rho_{s^{\perp}}$ can be seen as the state resulting from preparing
$\ket{s^{\perp}}$ and letting the noise channel act for two steps, i.e.,
$\rho_{s^{\perp}}=\mathcal{N}_{v}\circ\mathcal{N}_{v}(\left|s^{\perp}\right\rangle\\!\left\langle
s^{\perp}\right|)$.

We can now see that with this choice of states the operational equivalences
are satisfied:

$\displaystyle\frac{1}{2}\rho_{\alpha}+\frac{1}{2}\rho_{\alpha^{\perp}}$
$\displaystyle=(1-v)^{2}\frac{\left|\alpha\right\rangle\\!\left\langle\alpha\right|+\left|\alpha^{\perp}\right\rangle\\!\left\langle\alpha^{\perp}\right|}{2}+(1-(1-v)^{2})\frac{\mathbb{I}}{4}=(1-v)^{2}\frac{\mathbb{I}_{\alpha
aa}}{2}+(1-(1-v)^{2})\frac{\mathbb{I}}{4}$
$\displaystyle=(1-v)^{2}\frac{\left|aa\right\rangle\\!\left\langle
aa\right|+\left|aa^{\perp}\right\rangle\\!\left\langle
aa^{\perp}\right|}{2}+(1-(1-v)^{2})\frac{\mathbb{I}}{4}=\frac{1}{2}\rho_{aa}+\frac{1}{2}\rho_{aa^{\perp}},$

with $\mathbb{I}_{\alpha aa}$ the projector over the ${\rm
span}(\\{\ket{\alpha},\ket{aa}\\})$. Following the same argumentation, one can
see that for the remaining pairs of preparations $(\beta,bb)$ and $(aa,bb)$,
if we let

$\displaystyle\rho_{\beta^{\perp}}$
$\displaystyle:=(1-v)^{2}\left|\beta^{\perp}\right\rangle\\!\left\langle\beta^{\perp}\right|+(1-(1-v)^{2})\frac{\mathbb{I}}{4},$
$\displaystyle\rho_{bb^{\perp}}$
$\displaystyle:=(1-v)^{2}\left|bb^{\perp}\right\rangle\\!\left\langle
bb^{\perp}\right|+(1-(1-v)^{2})\frac{\mathbb{I}}{4},$

with $\ket{bb^{\perp}},\ket{\beta^{\perp}}\in{\rm
span}(\\{\ket{\beta},\ket{bb}\\})$ and $\\!\left\langle
bb\vphantom{bb^{\perp}}\middle|bb^{\perp}\vphantom{bb}\right\rangle\\!=\\!\left\langle\beta\vphantom{\beta^{\perp}}\middle|\beta^{\perp}\vphantom{\beta}\right\rangle\\!=0$,
the operational equivalence for $(\beta,bb)$ is satisfied:

$\displaystyle\frac{1}{2}\rho_{\beta}+\frac{1}{2}\rho_{\beta^{\perp}}=\frac{1}{2}\rho_{bb}+\frac{1}{2}\rho_{bb^{\perp}}.$

And letting

$\displaystyle\rho^{\prime}_{aa^{\perp}}$
$\displaystyle:=(1-v)^{2}\left|\overline{aa}^{\perp}\right\rangle\\!\left\langle\overline{aa}^{\perp}\right|+(1-(1-v)^{2})\frac{\mathbb{I}}{4},$
$\displaystyle\rho^{\prime}_{bb^{\perp}}$
$\displaystyle:=(1-v)^{2}\left|\overline{bb}^{\perp}\right\rangle\\!\left\langle\overline{bb}^{\perp}\right|+(1-(1-v)^{2})\frac{\mathbb{I}}{4},$

with $\ket{\overline{aa}^{\perp}},\ket{\overline{bb}^{\perp}}\in{\rm
span}(\\{\ket{aa},\ket{bb}\\})$, and $\\!\left\langle
aa\vphantom{\overline{aa}^{\perp}}\middle|\overline{aa}^{\perp}\vphantom{aa}\right\rangle\\!=\\!\left\langle
bb\vphantom{\overline{bb}^{\perp}}\middle|\overline{bb}^{\perp}\vphantom{bb}\right\rangle\\!=0$,
the operational equivalence for $(aa,bb)$ is satisfied:

$\displaystyle\frac{1}{2}\widetilde{\rho_{aa}}+\frac{1}{2}\rho^{\prime}_{aa^{\perp}}=\frac{1}{2}\widetilde{\rho_{bb}}+\frac{1}{2}\rho^{\prime}_{bb^{\perp}}.$

Notice that $\rho_{aa^{\perp}},\rho^{\prime}_{aa^{\perp}}$ and
$\rho_{bb^{\perp}},\rho^{\prime}_{bb^{\perp}}$ are alternative choices of
orthogonal preparations, tailored to each pair of preparations appearing in
the operational equivalences.

###  C.3 Noise parameter and Error term in the NC bound

In this subsection, we find the expression for each of the measurement error
probabilities appearing in the error term in Eq. (11) as a function of the
noise parameter $v$ of the depolarizing channel.

For $x\in\\{a,b\\}$,

$\displaystyle 1-\epsilon_{x}$
$\displaystyle=p(M_{x}|P_{x})=\operatorname{Tr}\left[\left(\left(1-v\right)\left|x\right\rangle\\!\left\langle
x\right|+v\frac{\mathbb{I}_{ab}}{2}\right)\left(\left(1-v\right)\left|x\right\rangle\\!\left\langle
x\right|+v\frac{\mathbb{I}_{ab}}{2}\right)\right]$
$\displaystyle=(1-v)^{2}+2v(1-v)\operatorname{Tr}\left[\frac{\left|x\right\rangle\\!\left\langle
x\right|}{2}\right]+v^{2}\operatorname{Tr}\left[\frac{\mathbb{I}_{ab}}{4}\right]=(1-v)^{2}+v(1-v)+\frac{v^{2}}{2}$
$\displaystyle\implies\epsilon_{x}=v-\frac{v^{2}}{2}.$

For $\chi\in\\{\alpha,aa\\}$,

$\displaystyle 1-\epsilon_{\alpha}$
$\displaystyle=p(M_{\alpha}|P_{\alpha})=\operatorname{Tr}[\rho_{\alpha}M_{\alpha}]=1-\epsilon_{aa}=p(M_{aa}|P_{aa})=\operatorname{Tr}[\rho_{aa}M_{aa}]$
$\displaystyle=\operatorname{Tr}\left[\left(\left(1-v\right)^{2}\left|\chi\right\rangle\\!\left\langle\chi\right|+\left(1-(1-v)^{2}\right)\frac{\mathbb{I}}{4}\right)\left(\left(1-v\right)\left|\chi\right\rangle\\!\left\langle\chi\right|+v\frac{\mathbb{I}}{4}\right)\right]$
$\displaystyle=(1-v)^{3}+\frac{(1-v)^{2}v}{4}+\frac{(1-(1-v)^{2})(1-v)}{4}+\frac{(1-(1-v)^{2})v}{4}$
$\displaystyle=\frac{1}{4}(4-9v+9v^{2}-3v^{3})$
$\displaystyle\implies\epsilon_{aa}=\epsilon_{\alpha}=\frac{3}{4}v(3-3v+v^{2}),$

and analogously for $\chi$ in $\\{\beta,bb\\}$ and $\\{aa,bb\\}$. Hence,

$\displaystyle{\rm Err}$
$\displaystyle=\epsilon_{\alpha}+\epsilon_{\beta}+\epsilon_{a}+\epsilon_{b}+2(\epsilon_{aa}+\epsilon_{bb})=2(v-\frac{v^{2}}{2})+6\cdot\frac{3}{4}v(3-3v+v^{2})=\frac{1}{2}v(31-29v+9v^{2}).$

Following the same arguments, it is easy to see that, for the case of
symmetric confusabilities, the error term in Eq. (11) becomes, as a function
of $v$,

$\displaystyle{\rm Err^{\prime}}=\frac{1}{8}v(31-21v+9v^{2})$

###  C.4 Quantum performance

In this last subsection, we compute the global average fidelity $F_{g}^{\rm
Q}$ in the noisy setting of the optimal quantum cloner for the ideal setting
as a function of the noise channel’s parameter $v$.

$\displaystyle F_{g}^{\rm Q}$
$\displaystyle=\frac{1}{2}\operatorname{Tr}[M_{aa}\rho_{\alpha}]+\frac{1}{2}\operatorname{Tr}[M_{bb}\rho_{\beta}]$
$\displaystyle=\operatorname{Tr}\left[\left(\left(1-v\right)^{2}\left|xx\right\rangle\\!\left\langle
xx\right|+\left(1-(1-v)^{2}\right)\frac{\mathbb{I}}{4}\right)\left(\left(1-v\right)\left|\chi\right\rangle\\!\left\langle\chi\right|+v\frac{\mathbb{I}}{4}\right)\right]$
$\displaystyle=(1-v)^{3}|\\!\left\langle
xx\vphantom{\chi}\middle|\chi\vphantom{xx}\right\rangle\\!|^{2}+\frac{(1-(1-v)^{2})(1-v)}{4}+\frac{(1-v)^{2}v}{4}+\frac{(1-(1-v)^{2})v}{4}$
$\displaystyle=(1-v)^{3}|\\!\left\langle
xx\vphantom{\chi}\middle|\chi\vphantom{xx}\right\rangle\\!|^{2}+\frac{1}{4}v(3-3v+v^{2}).$

Hence,

$F_{g}^{\rm Q,noisy}(v):=(1-v)^{3}F_{g}^{\rm Q,opt}+\frac{1}{4}v(3-3v+v^{2}).$