Contextual advantage for state-dependent cloning
Matteo Lostaglio
1,2
and Gabriel Senno
1
1
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
2
QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands
A number of noncontextual models exist
which reproduce different subsets of quan-
tum 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 quan-
tum state cloning which are indeed nonclassi-
cal 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 ex-
plained by any noncontextual model. We de-
rive 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 noncontex-
tual models, but also provides an experimental
witness of noncontextuality.
An important guiding principle for quantum theo-
rists is the identification of genuine nonclassical effects
certified by rigorous theorems. Given a quantum phe-
nomenon, the relevant question is: Are there classi-
cal models able to reproduce the observed operational
data? Here we investigate this question in the context
of a cloning experiment.
The no-cloning theorem [13] is widely regarded
as a central result in quantum theory. Informally,
the theorem states the impossibility of copying quan-
tum information, and is contrasted with the fact that
classical information, on the other hand, can be per-
fectly copied. More precisely, there is no machine
(formally, a quantum channel) that can take two dis-
tinct and nonorthogonal states {|ψ
1
i, |ψ
2
i} sent at
random as inputs and output the corresponding copies
{|ψ
1
i |ψ
1
i, |ψ
2
i |ψ
2
i} [4].
While no-cloning is often regarded as an intrinsi-
cally 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 op-
erational 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 mea-
surement outcomes. This analysis forces us to look for
any plausible alternative explanation of the empirical
data collected in a quantum experiment before we cer-
tify it as “nonclassical”. But, which ontological models
should be deemed “classical"?
Clearly, the broader the chosen notion of classical-
ity 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 differ-
ent 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 intro-
duced 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 noncon-
textual theories is simple: non-orthogonal quantum
states |ψ
1
i, |ψ
2
i correspond to overlapping probability
distributions µ
1
(λ), µ
2
(λ) over the posited set of phys-
ical states λ and there is no deterministic nor stochas-
tic process mapping {µ
1
, µ
2
} to {µ
1
µ
1
, µ
2
µ
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.
1
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 as-
pect 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
1
Crucially, cloning should be distinguished from the notion
of broadcasting. Broadcasting only requires the creation of a
joint distribution with marginals µ
i
and can be done perfectly
by a generalized CNOT.
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 1
arXiv:1905.08291v3 [quant-ph] 23 Apr 2020
orthogonal pure states cannot be reproduced by any
noncontextual model which complies with the opera-
tional phenomenology featured in a quantum cloning
experiment. Specifically, contextuality provides an
advantage to the maximum copying fidelity. Our re-
sult directly links contextuality to a quantum advan-
tage [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 corre-
sponding to certain sets of operational instructions.
2
We can distinguish three kinds of black-boxes:
1. A preparation black-box P
s
initialises the system;
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. A measurement black-box M
s
0
takes a prepara-
tion P
s
as input and returns an outcome x with
probability p(x|P
s
, M
s
0
).
4. An experiment consists of collecting the statistics
p(x|T (P
s
), M
s
0
) for various choices of the black
boxes P
s
, T and M
s
0
.
The set of P
s
, T, M
s
0
and corresponding observed
statistics p(x|T (P
s
), M
s
0
) are the defining elements of
an operational theory. Noncontextuality is a restric-
tion 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. Makes every preparation P
s
correspond to sam-
pling from a probability distribution µ
s
(λ) over
some set of ontic variables λ. λs are referred to
as ‘hidden variables’ in the context of Bell non-
locality and they form a (measurable) set Λ.
2. Represents transformations by matrices T (λ
0
|λ)
of transition probabilities (T (λ
0
|λ) 0,
R
0
T (λ
0
|λ) = 1 λ) acting on the corresponding
probability density.
3. Represents a measurement M
s
0
by a response
function ξ
s
0
(x|λ) giving the probability of out-
come x given that the hidden variable takes the
value λ (ξ
s
0
(x|λ) 0,
P
x
ξ
s
0
(x|λ) = 1 λ).
2
While empirical data is always to some degree theory-laden,
the word “operational” here signifies that we are striving to-
wards the ideal of the most low-level instructions we can imag-
ine (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”.
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 respec-
tively prepares P
γ
, γ = α, β; a test measurement M
xx
for the
target preparation P
xx
is performed and passed with prob-
ability P (M
aa
|P
α
) (or P (M
bb
|P
β
)). Bottom: ontological
description of the same experiment, where preparing P
x
cor-
responds to sampling λ with probability µ
x
(λ), the cloning
machine maps λ 7→ λ
0
with probability T (λ
0
|λ) and M
xx
gives a ‘pass’ outcome with probability ξ
xx
(1|λ
0
).
An ontological model then defines its predictions as
p(x|T (P
s
), M
s
0
) =
Z
dλdλ
0
µ
s
(λ)T (λ
0
|λ)ξ
s
0
(x|λ
0
).
(1)
Two operational procedures (be them preparations,
measurements or transformations) are said to be op-
erationally equivalent if they cannot be distinguished
by any experiment. Noncontextuality, in the gener-
alized form introduced in [6], is a restriction to on-
tological 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 [6, 15].
In this work we will be concerned with operational
equivalences only at the level of preparations. Two
preparations P
s
and P
s
0
are operationally equivalent
if they cannot be distinguished by any measurements:
p(x|P
s
, M) = p(x|P
s
0
, M), M,
which, for short, we will denote by P
s
' P
s
0
. The
assumption of (preparation) noncontextuality is then
P
s
' P
s
0
µ
s
(λ) = µ
s
(λ
0
). (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 noncontex-
tual ontological models include classical Hamiltonian
mechanics, Hamiltonian mechanics with a resolution
limit on phase space [7] and Spekken’s toy model [8].
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 2
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 assump-
tion 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
) p(M
s
|P ) and ξ
s
(1|λ
0
) ξ
s
(λ
0
).
Let P
a
and P
b
denote the experimental procedures
followed to prepare the states |ai and |bi to be cloned.
As an operational signature of the fact that |ai and
|bi are two pure and, in general, nonorthogonal states,
we consider the ‘test measurements’ M
a
, M
b
, with
outcomes x {0, 1}, giving the operational statis-
tics p(M
a
|P
a
) = p(M
b
|P
b
) = 1. In the quantum
formalism, this statistics is reproduced by perform-
ing the projective measurements {|aiha|, 1 |aiha|}
and {|bihb|, 1 |bihb|} (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 exper-
iment one observes c
ab
= |ha|bi|
2
.
The two preparations P
a
, P
b
go through a cloning
machine T , which outputs new preparations P
α
=
T (P
a
), P
β
= T (P
b
). In quantum theory, the optimal-
state dependent cloning operation is a unitary U
and, hence, the preparations P
α
and P
β
correspond
to pure states |αi := U |a 0i, |βi := U |b 0i respec-
tively, with |0i the initial state of some ancillary
register. Operationally, and similarly to the discus-
sion above, the purity of the outputs implies that we
can perform test measurements M
α
, M
β
satisfying
p(M
α
|P
α
) = 1, p(M
β
|P
β
) = 1 (again, by performing
the measurements described in the quantum formal-
ism as {|αihα|, 1 |αihα|} and {|βihβ|, 1 |βihβ|}).
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 ob-
serves the statistics p(M
aa
|P
aa
) = p(M
bb
|P
bb
) =
1, p(M
bb
|P
aa
) = |haa|bbi|
2
= |ha|bi|
4
. In a quan-
tum experiment this is realized by preparing states
|aai, |bbi and performing the projective measurements
{|aaihaa|, 1 |aaihaa|}, {|bbihbb|, 1 |bbihbb|}.
Then, denoting by c
αaa
:= P (M
aa
|P
α
), c
βbb
:=
P (M
bb
|P
β
), the (global) cloning fidelity is opera-
tionally defined to be
F
g
:=
1
2
c
αaa
+
1
2
c
βbb
,
i.e., the average probability that the imperfect clones
P
α
and P
β
pass the corresponding test measure-
ments for the ideal clones, M
aa
and M
bb
respec-
tively. In quantum theory, the optimal cloning uni-
tary achieves [17]
F
Q,opt
g
:=
1
4
q
(1 + c
ab
)(1 +
c
ab
)
+
q
(1 c
ab
)(1
c
ab
)
2
, (3)
with c
ab
= |ha|bi|
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 opera-
tional 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 |ai, |bi: the existence of preparations P
a
, P
b
sat-
isfying p(M
a
|P
a
) = p(M
b
|P
b
) = 0 and such that the
mixture P
a
/2 + P
a
/2 (tossing a fair coin and follow-
ing either P
a
or P
a
) is operationally equivalent to the
mixture P
b
/2+P
b
/2: P
a
/2+P
a
/2 ' P
b
/2+P
b
/2.
In the idealized quantum experiment one observes
this operational statistics by preparing pure states
|a
i, |b
i in the span of {|ai, |bi} and satisfying
a
a
=
b
b
= 0 as well as
1
2
|aiha|+
1
2
a

a
=
1
2
|bihb| +
1
2
b

b
. The same discussion can be re-
peated for each of the pairs {(a, b), (α, aa), (β, bb)}.
To conclude, here is an operational account (with-
out 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
, M
s
such that
O1 p(M
s
|P
s
) = 1, p(M
s
|P
s
) = 0 for s =
a, b, α, β, aa, bb.
O2
1
2
P
s
+
1
2
P
s
'
1
2
P
s
0
+
1
2
P
s
0
, for all (s, s
0
) in
{(a, b), (α, aa), (β, bb)}.
3 Optimal cloning is contextual - ideal
scenario
In any ontological model, a cloning experiment is de-
scribed as follows (see Fig. 1). A preparation device
randomly prepares either P
a
or P
b
, i.e., it samples
a λ from either the distribution µ
a
(λ) or µ
b
(λ). This
state is sent into the cloning machine that maps λ into
some new λ
0
with probability T(λ
0
|λ). For example, if
λ = (x
1
, p
1
) one could have λ
0
= (x
0
1
, p
0
1
, x
0
2
, p
0
2
). This
λ
0
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 λ
0
, the device gives an outcome x with
probability ξ
aa
(λ
0
) or ξ
bb
(λ
0
).
The assumption of noncontextuality (more pre-
cisely, preparation noncontextuality [6]) and linearity
applied to the operational equivalences in O2 requires
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 3
that any noncontextual ontological model must sat-
isfy (see Eq. (2))
1
2
µ
s
(λ) +
1
2
µ
s
(λ) =
1
2
µ
s
0
(λ) +
1
2
µ
s
0
(λ), (4)
for all (s, s
0
) in {(a, b), (α, aa), (β, bb)} and λ Λ.
Our main result is that no noncontextual ontological
model can reproduce the operational features listed
O1-O2 and match the optimal cloning fidelity pre-
dicted by quantum theory. More precisely:
Theorem 1 (Optimal cloning fidelity in noncontex-
tual models). Let P
α
= T (P
a
), P
β
= 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 non-
contextual model we have that
F
g
F
NC
g
= 1
c
ab
2
+
c
aa,bb
2
. (5)
Proof. The first part of the proof essentially follows
the argument given in Ref. [6] Sec. VIIIA and repro-
duced in Ref. [5] Sec. IVA, slightly adapted to use the
fewer assumptions of the statement. We have that
1 = p(M
k
|P
k
) =
Z
S
k
dλµ
k
(λ)ξ
k
(λ), k = s, s
0
,
where S
k
denotes the support of µ
k
. From this equa-
tion, it follows that ξ
k
(λ) = 1 almost everywhere on
S
k
(that is, modulo sets of measure zero). Further-
more,
0 = p(M
k
|P
k
) =
Z
S
k
µ
k
(λ)ξ
k
(λ), k = s, s
0
,
from which it follows that ξ
k
(λ) = 0 almost every-
where on S
k
. Hence, S
k
S
k
= modulo sets of
zero measure.
The operational equivalence of assumption 1 im-
plies that in a noncontextual model
µ
s
(λ) + µ
s
(λ) = µ
s
0
(λ) + µ
s
0⊥
(λ), λ Λ. (6)
Since S
s
S
s
= S
s
0
S
s
0⊥
= modulo a set of
zero measure, this implies µ
s
(λ) = µ
s
0
(λ) for almost
all λ S
s
S
s
0
. Hence, using the facts above, the `
1
norm distance between µ
s
and µ
s
0
reads (kµ
s
µ
s
0
k :=
R
|µ
s
(λ) µ
s
0
(λ)|).
kµ
s
µ
s
0
k =
Z
Λ\S
s
dλµ
s
0
(λ) +
Z
Λ\S
s
0
dλµ
s
(λ)
= 2 2
Z
S
s
S
s
0
dλµ
s
(λ)
= 2 2
Z
S
s
S
s
0
dλµ
s
(λ)ξ
s
0
(λ).
Note that the last integral can be extended to Λ.
In fact, by contradiction suppose that ξ
s
0
(λ) 6= 0
for some nonzero measure set X S
s
\S
s
0
. Then,
from Eq. (6), it follows that, for almost all λ X,
0 < µ
s
(λ) = µ
s
0⊥
(λ). However, as we discussed
ξ
s
0
(λ) = 0 almost everywhere on S
s
0⊥
, which gives
the desired contradiction. Hence the integral can be
extended to S
s
S
s
0
and, trivially, to all Λ. In con-
clusion,
kµ
s
µ
s
0
k = 22
Z
Λ
dλµ
s
(λ)ξ
s
0
(λ) = 2(1c
ss
0
), (7)
where c
ss
0
= p(M
s
0
|P
s
). Using the triangle inequality,
kµ
aa
µ
bb
k kµ
aa
µ
α
k+ kµ
α
µ
β
k+ kµ
β
µ
bb
k.
By definition, µ
α
(λ) =
R
0
T (λ|λ
0
)µ
a
(λ
0
), for a
stochastic matrix T (λ|λ
0
). Similarly, µ
β
(λ) =
R
0
T (λ|λ
0
)µ
b
(λ
0
), with the same stochastic matrix.
Since
R
dλT (λ|λ
0
) = 1 and T (λ|λ
0
) 0, one can read-
ily verify from the convexity of the absolute value that
kµ
α
µ
β
k kµ
a
µ
b
k (data processing inequality),
which implies
kµ
aa
µ
bb
k kµ
α
µ
aa
k+kµ
a
µ
b
k+kµ
β
µ
bb
k. (8)
We can apply Eq. (7) to each of the couples (s, s
0
) on
the right hand side of Eq. (8), obtaining
kµ
aa
µ
bb
k 2(1c
αaa
)+2(1c
ab
)+2(1c
βbb
). (9)
Let us now show kµ
aa
µ
bb
k 2(1 c
aa,bb
). First,
notice that
kµ
aa
µ
bb
k =
Z
S
aa
\S
bb
dλµ
aa
(λ) +
Z
S
bb
\S
aa
dλµ
bb
(λ) +
Z
R
1
(µ
aa
(λ) µ
bb
(λ)) +
Z
R
2
(µ
bb
(λ) µ
aa
(λ)),
with R
1
:= {λ S
aa
S
bb
: µ
aa
(λ) µ
bb
(λ)} and
R
2
:= (S
aa
S
bb
)\R
1
. Next,
kµ
aa
µ
bb
k = 2
1
Z
R
1
dλµ
bb
(λ)
Z
R
2
dλµ
aa
(λ)
2 2
Z
R
1
R
2
=S
aa
S
bb
dλµ
aa
(λ)
= 2 2
Z
S
aa
S
bb
dλµ
aa
(λ)ξ
bb
(λ)
2(1 c
aa,bb
)
where the first inequality follows from µ
aa
(λ)
µ
bb
(λ) λ R
1
and the second equality follows from
ξ
bb
(λ) = 1 almost everywhere in S
bb
. Finally, substi-
tuting this in Eq. (9) and rearranging the terms gives
1
2
c
αaa
+
1
2
c
βbb
1
c
ab
2
+
c
aa,bb
2
and since F
g
=
1
2
c
αaa
+
1
2
c
βbb
the global cloning
achieved by non-contextual ontological models that
comply with the operational features O1-O2 is upper
bounded as in Eq. (5).
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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. (3)).
In Fig. 2 we compare the optimal quantum cloning
(global) fidelity of Eq. (3) with the maximum non-
contextual cloning fidelity of Eq. (5), taking into ac-
count 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 noncontextu-
ality. Hence, the phenomenology of optimal cloning
cannot be reproduced within noncontextual ontologi-
cal models. Contextuality provides an advantage for
the maximum copying fidelity.
3
Interestingly, the above derivation also gives an al-
ternative, 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
0
= p(M
s
0
|P
s
) and the `
1
distance
between the corresponding probability distributions:
kµ
s
µ
s
0
k = 2(1 c
ss
0
). (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 distinguish-
ing two preparations P
a
and P
b
is at most 1/2+
kµ
a
µ
b
k/4, it immediately follows s
ab
1 c
ab
/2,
which is the optimal state discrimination probability
in noncontextual models, as given in Ref. [5]. Con-
versely, it is not immediately obvious how the tech-
niques of Ref. [5] could be adapted to obtain our re-
sult 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 µ
s
. Con-
sider a model in which µ
ss
= µ
s
µ
s
and ξ
s
(λ) = 1 if
3
Of course, when c
ab
= 0 - as it is for classical, i.e., orthogo-
nal, states - both the quantum and the noncontextual fidelities
are 1.
λ S
s
and zero otherwise. A cloning strategy that
saturates the bound is as follows: if the input λ is
in S
a
\S
b
, output (λ, λ
0
), with λ
0
sampled according
to µ
a
; otherwise, output (λ, λ
0
) with λ
0
sampled ac-
cording to µ
b
. Notice that this sets µ
β
= µ
b
µ
b
and,
hence, c
βbb
= 1 (µ
b
is copied perfectly). On the other
hand, µ
α
(λ, λ
0
) = µ
a
(λ)µ
a
(λ
0
) for λ S
a
\S
b
and
µ
α
(λ, λ
0
) = µ
a
(λ)µ
b
(λ
0
) for λ S
a
S
b
and, hence,
c
αaa
=
Z
dλdλ
0
µ
α
(λ, λ
0
)ξ
aa
(λ, λ
0
)
=
Z
S
a
×S
a
dλdλ
0
µ
α
(λ, λ
0
)
=
Z
(S
a
\S
b
)×S
a
dλdλ
0
µ
a
(λ)µ
a
(λ
0
) +
Z
(S
a
S
b
)×S
a
dλdλ
0
µ
a
(λ)µ
b
(λ
0
)
= (1 c
ab
) + c
ab
· c
ba
= 1 c
ab
+ c
2
ab
,
where, in the last equality, we use the opera-
tional fact that c
ab
= c
ba
. Finally, this gives
F
g
=
1
2
(1 c
ab
+ c
2
ab
) +
1
2
= F
NC
g
. In Appendix A we
complete this strategy with a concrete choice of µ
a
,
µ
b
, µ
aa
, µ
bb
, µ
α
and µ
β
complying with O1 and
satisfying Eq. (4) for all the operational equivalences
in O2.
This optimal strategy seems to suggest the follow-
ing intuition behind the theorem: our assumption of
preparation noncontextuality on the input prepara-
tions P
a
, P
b
imply that the distributions µ
a
(λ) and
µ
b
(λ) overlap “too much” (formally, it implies maxi-
mal ψ-epistemicity, c
ab
=
R
S
b
dλµ
a
(λ) [18]), hence the
cloning performance turns out worse than in quan-
tum mechanics. Furthermore, noncontextuality im-
plies that µ
a
and µ
b
coincide on their overlap, which
implies a direct relation between c
ab
and the `
1
norm
kµ
a
µ
b
k. Crucially the latter cannot be increased
by the cloning machine, since k · k decreases under
post-processing.
However, this mechanism can only be part of the
story. First, the cloning performance is not mono-
tonically 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 neces-
sarily correspond to two independent copies µ
a
µ
a
,
µ
b
µ
b
(this assumption, which we do not make, is
called preparation independence [19]). Nevertheless,
we showed that a no-go theorem results from the ob-
served overlaps c
ab
, c
aa,bb
and noncontextuality as-
sumptions only as a consequence of information pro-
cessing inequalities and the triangle inequality.
We note in passing that our proof technique can be
abstracted and applied to other tasks as follows:
1. First, given a set of observed overlaps {c
ss
0
}, non-
contextuality applied to the operational equiva-
lences
1
2
P
s
+
1
2
P
s
'
1
2
P
s
0
+
1
2
P
s
0
gives the equa-
tions (10).
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 5
2. Second, verify if the equations (10) are compati-
ble with triangle and data processing inequalities
and the performance of quantum protocol un-
der 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 - be-
yond idealizations
Theorem 1 is a no-go result for noncontextual ontolog-
ical models aimed at explaining the phenomenology
of state-dependent quantum cloning. However, the
inequality derived in Eq. (5) is not a proper noncon-
textuality 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 corre-
lations in O1, such as those generated by a cloning ex-
periment 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 experimen-
tally testable statement reduces to the elimitation of
the idealization in O1. Specifically, we want to weaken
it to
O1ni p(M
s
|P
s
) 1
s
, p(M
s
|P
s
)
s
for s =
a, b, α, β, aa, bb,
where ‘ni’ stands for ‘non-ideal’.
Theorem 2 (Optimal cloning fidelity in noncontex-
tual models noise-robust version). With the nota-
tion of Thm. 1, suppose that one observes the opera-
tional features O1ni and O2. Then, for any noncon-
textual model we have that
F
g
F
NC,ni
g
= 1
c
ab
2
+
c
aa,bb
2
+ Err. (11)
where Err =
1
2
(
b
+ 2
bb
+
aa
).
Note that, while we gave an independent and sim-
pler proof of Theorem 1, we can now see it as a corol-
lary of the result above once all error terms are set
of zero. Another interesting case is when all error
terms are equal,
b
=
bb
=
aa
:= , which gives
F
NC,ni
g
= 1
c
ab
2
+
c
aa,bb
2
+ 2. 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
|kµ
s
µ
s
0
k 2(1 c
ss
0
)| 2
s
0
, (12)
and similarly if we exchange s and s
0
. 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 noncon-
textual model and beyond the ideal scenario, between
the `
1
distance of two epistemic states and their opera-
tionally 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 non-
contextual bound on state discrimination of Ref. [5],
s
ab
1
2
+
1
4
kµ
a
µ
b
k 1
c
ab
b
2
.
4.1 An explicit noise model
Having derived a noise-robust version of our noncon-
textual bound, the next step is to investigate whether
quantum mechanics violates it. We consider a stan-
dard noise model in which the ideal quantum prepara-
tions, measurements and unitary transformation are
all thwarted by a depolarizing channel N
v
with noise
level v [0, 1]:
N
v
(ρ) = (1 v) ρ + vI/4.
A direct calculation (see Appendix C) shows that
this sets = 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
Q,noisy
g
(v) := (1 v)
3
F
Q,opt
g
+
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 be-
tween c
ab
and F
g
is not necessarily above the non-
contextual bound. For example, for v = 0.015 a vi-
olation can be observed only for c
ab
[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 cur-
rent experiments. In fact, in terms of the parame-
ter C
s
= 1/2 p(M
s
|P
s
) + 1/2 p(M
s
|P
s
) defined in
Ref. [20] (C
s
= 1 in the ideal scenario), v = 0.015 cor-
responds to C
s
0.9851 for s = a, b and C
s
0.9667
for s = aa, bb, and Ref. [20] experimentally realized
C
s
= 0.9969.
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 6
0.0 0.2 0.4 0.6 0.8 1.0
0.000
0.005
0.010
0.015
0.020
c
ab
v
Figure 3: Noise-resistance of the quantum advantage in
cloning. This plot shows the maximum value of the noise pa-
rameter v of a depolarizing channel (affecting preparations,
measurements and transformation) for which the quantum
value of the cloning fidelity (Eq. (13)) is above the noncon-
textual bound, as a function of the confusability between the
inputs c
ab
.
4.2 Remaining assumptions
As we mentioned, the only remaining idealiza-
tion is the operational verification of O2. Let
us suppose that, in an experiment, after do-
ing tomography,
4
one determines that the actual
experimental realizations of the ideal preparations are
P
(1)
a
, P
(1)
a
, P
(1)
b
, P
(1)
b
, P
(1)
aa
, P
(1)
aa
, P
(1)
bb
, P
(1)
bb
, P
(1)
α
, P
(1)
α
,
P
(1)
β
and P
(1)
β
. 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 experi-
mentally 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 C of the prepara-
tions P
(1)
s
. By the linearity of Eq. (1), one can then
compute the measurement statistics of all the prepa-
rations in C. Therefore, as put forward in Sec. IV of
Ref. [20], to go ahead with the experimental verifica-
tion of Theorem 2 one only needs to find ‘secondary’
preparations P
(2)
s
in C whose measurement statistics
satisfy the operational equivalences in O2; that is, we
only need
O2ni O2 is satisfied for some preparations P
(2)
s
in the
convex hull C of the experimental preparations
P
(1)
s
.
This post-processing hence allows one to apply The-
orem 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
4
We will later discuss the assumption that can one access a
tomographically complete set of measurements.
one pays in this construction is that the correspond-
ing noise parameters
0
s
= p(M
s
|P
(2)
s
) in O1ni will in
general be larger. Note that, even if
0
s
is too large
compared to
s
to see any violation in Theorem 2,
one can get around this issue by adding extra experi-
mental preparations P
(1)
extra
to enlarge C, as explicitly
done in Ref. [20]. To summarize, there are good gen-
eral 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 can-
not be conclusively tested by any experimental means.
In a nonlocality experiment, for example, these in-
clude the assumption that the two sides cannot com-
municate and the ability to choose the measurement
freely, i.e., independently of any other variable rele-
vant to the experiment. In a contextuality experi-
ment 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 measure-
ments (just like a future theory may allow signalling).
Recent work has shown that the problem can be miti-
gated 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 ob-
served in the optimal state-dependent quantum
cloning is incompatible with the predictions of ev-
ery noncontextual ontological model. In particular,
for given overlap, the noncontextual global cloning
fidelity is strictly smaller than the quantum predic-
tion. A similar result continues to hold in more re-
alistic experiments which are unavoidably affected by
noise (while the effect can be ‘washed out’ by exces-
sive experimental imperfections). This identifies con-
textuality as the resource for optimal state-dependent
quantum cloning.
From a foundational point of view, it would be rel-
evant to explore whether the relation between contex-
tuality 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 contex-
tual 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 `
1
norm and confusability developed here to
understand what aspects of other quantum informa-
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 7
tion 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).
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Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 9
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 distribu-
tions µ
aa
, µ
bb
, µ
aa
, µ
bb
and µ
α
satisfying the operational features targeted by Theorem 1. The supports of
all these distributions, which we set to be subsets of [0, 2] ×[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 µ
β
µ
bb
,
it follows that to satisfy the operational equivalence for (µ
β
, µ
bb
) we must have µ
β
µ
bb
. 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
(c) S
bb
(d) S
bb
(e) S
α
Figure 4: Supports of distributions µ
aa
, µ
bb
, µ
aa
, µ
bb
and µ
α
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).
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 10
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
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:
O1ni p(M
s
|P
s
) 1
s
, p(M
s
|P
s
)
s
for s = a, b, α, β, aa, bb,
O2
1
2
P
s
+
1
2
P
s
'
1
2
P
s
0
+
1
2
P
s
0
, for all (s, s
0
) in {(a, b), (α, aa), (β, bb)}.
Then, for any noncontextual model we have that
F
NC
g
1 +
min{
b
c
ab
,
a
c
ba
}
2
+
min{c
aa,bb
+
bb
, c
bb,aa
+
aa
}
2
+
aa
+
bb
2
. (14)
For the proof of Theorem 3, we make use of the following lemma relating the `
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
0
be preparations. Suppose there exists preparations P
s
, P
s
0⊥
and a two outcome mea-
surement M
s
such that
1.
1
2
P
s
+
1
2
P
s
'
1
2
P
s
0
+
1
2
P
s
0⊥
,
2. p(M
k
|P
k
) 1
k
, p(M
k
|P
k
)
k
, k = s, s
0
.
Then, in a noncontextual ontological model,
2 max{1 c
ss
0
s
0
, 1 c
s
0
s
s
} kµ
s
µ
s
0
k 2 min{1 c
ss
0
+
s
0
, 1 c
s
0
s
+
s
}, (15)
Proof. We denote by S
s
the support of µ
s
. Define a partition S
s
S
s
0
= t
4
i=1
R
i
, as summarized in Figure 5:
R
1
= S
s
\(S
s
S
s
0
), R
4
= S
s
0
\(S
s
S
s
0
).
R
2
= {λ S
s
S
s
0
|µ
s
(λ) µ
s
0
(λ)}, R
3
= {λ S
s
S
s
0
|µ
s
(λ) < µ
s
0
(λ)}.
Figure 5: Sketch of the relevant regions in the proof of Lemma 4.
Then,
kµ
s
µ
s
0
k =
Z
|µ
s
(λ) µ
s
0
(λ)|
=
Z
R
1
dλµ
s
(λ) +
Z
R
4
dλµ
s
0
(λ) +
Z
R
2
[µ
s
(λ) µ
s
0
(λ)] +
Z
R
3
[µ
s
0
(λ) µ
s
(λ)]
= 2
Z
R
2
R
3
[µ
s
(λ) + µ
s
0
(λ)] +
Z
R
2
[µ
s
(λ) µ
s
0
(λ)] +
Z
R
3
[µ
s
0
(λ) µ
s
(λ)]
= 2 2
Z
R
3
dλµ
s
(λ) 2
Z
R
2
dλµ
s
0
(λ). (16)
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 11
Consider,
c
ss
0
Z
R
3
dλµ
s
(λ)
Z
R
2
dλµ
s
0
(λ) =
Z
R
1
R
2
R
3
dλµ
s
(λ)ξ
s
0
(λ)
Z
R
3
dλµ
s
(λ)
Z
R
2
dλµ
s
0
(λ)
Z
R
1
R
2
dλµ
s
(λ)ξ
s
0
(λ)
Z
R
2
dλµ
s
0
(λ)
=
Z
R
1
R
2
[µ
s
0
(λ) + µ
s
0
(λ) µ
s
(λ)]ξ
s
0
(λ)
Z
R
2
dλµ
s
0
(λ)
=
Z
R
2
dλµ
s
0
(λ)ξ
s
0
(λ) +
Z
R
1
R
2
[µ
s
0
(λ) µ
s
(λ)]ξ
s
0
(λ)
Z
R
1
R
2
dλµ
s
0
(λ)ξ
s
0
(λ)
Z
dλµ
s
0
(λ)ξ
s
0
(λ) = p(M
s
0
|P
s
0
)
s
0
,
where we used ξ
s
0
1 in the first inequality and assumption 1 and non-contextuality in the second equality.
In the third equality, we used
R
R
1
dλµ
s
0
(λ)ξ
s
0
(λ) = 0 and in the final inequality we used assumption 2. Then,
using Eq. (16),
kµ
s
µ
s
0
k 2(1 c
ss
0
+
s
0
).
Furthermore, recalling that ξ
s
0
= 1 ξ
s
0
,
Z
R
3
dλµ
s
(λ) +
Z
R
2
dλµ
s
0
(λ) c
ss
0
=
Z
R
3
dλµ
s
(λ) +
Z
R
2
dλµ
s
0
(λ)
Z
R
1
R
2
R
3
dλµ
s
(λ)ξ
s
0
(λ)
Z
R
3
dλµ
s
0
(λ)ξ
s
0⊥
(λ) +
Z
R
2
dλµ
s
0
(λ)ξ
s
0⊥
(λ)
Z
R
1
dλµ
s
ξ
s
0
(λ)
Z
R
3
dλµ
s
0
(λ)ξ
s
0⊥
(λ) +
Z
R
2
dλµ
s
0
(λ)ξ
s
0⊥
(λ)
Z
dλµ
s
0
(λ)ξ
s
0
(λ) = p(M
s
0
|P
s
0
)
s
0
.
where in the first inequality we used that µ
s
(λ) µ
s
0
(λ) in R
3
and µ
s
(λ) µ
s
0
(λ) in R
2
. In the final inequality,
we used assumption 2. Hence, we have that
c
ss
0
Z
R
3
dλµ
s
(λ)
Z
R
2
dλµ
s
0
(λ)
s
0
and, using Eq. (16), that
kµ
s
µ
s
0
k 2(1 c
ss
0
+
s
0
). (17)
Finally, noting that kµ
s
µ
s
0
k = kµ
s
0
µ
s
k and that the above derivation is symmetric under the exchange
of s with s
0
we arrive to the desired result
2 max{1 c
ss
0
s
0
, 1 c
s
0
s
s
} kµ
s
µ
s
0
k 2 min{1 c
ss
0
+
s
0
, 1 c
s
0
s
+
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 `
1
norm under stochastic processes (which gives kµ
α
µ
β
k kµ
a
µ
b
k), one can show that
the following equation holds (see Eq. (8)):
kµ
aa
µ
bb
k kµ
α
µ
aa
k + kµ
a
µ
b
k + kµ
β
µ
bb
k. (19)
Using both upper and lower bounds for the `
1
distance derived in Lemma 4, this implies
2 max{1 c
aa,bb
bb
, 1 c
bb,aa
aa
}
2(1 c
αaa
) + 2
aa
+ 2 min{1 c
ab
+
b
, 1 c
ba
+
a
}
+2(1 c
βbb
) + 2
bb
,
which can be rearranged to give the claimed bound on F
NC
g
.
Accepted in Quantum 2020-04-20, click title to verify. Published under CC-BY 4.0. 12
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 N
v
with noise level v [0, 1]:
N
v
(ρ) = (1 v) ρ + v
I
4
.
Therefore, for x {a, b}, the ideal input preparations transform as
|x0i 7→ N
v
(|x0ihx0|) = (1 v) |x0ihx0| + v
I
4
,
so that the actual input preparations become
ρ
x
:= Tr
2
[N
v
(|x0ihx0|)] = (1 v) |xihx| + v
I
ab
2
,
with I
ab
the projector over the span({|ai, |bi}). The ideal cloning transformation U becomes N
v
U = (1
v) U + vD, where D(ρ) = I/4 for all ρ. Hence, the actual outcomes χ {α, β} correspondent to input x {a, b}
become
ρ
χ
:= N
v
U(ρ
x
) = [(1 v)U + vD]
(1 v) |x0ihx0| + v
I
4
= (1 v)
2
|χihχ| + (1 (1 v)
2
)
I
4
.
The actual target copies would become N
v
(|xxihxx|). 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
ρ
xx
:= N
v
N
v
(|xxihxx|) = (1 v)
2
|xxihxx| + (1 (1 v)
2
)
I
4
.
Finally, for the ideal measurements, they transform as (for x {a, b}, χ {α, β}),
{|xihx|, I
ab
|xihx|} 7→ M
x
:=
(1 v) |xihx| + v
I
ab
2
, (1 v)(I
ab
|xihx|) + v
I
ab
2
,
{|xxihxx|, I |xxihxx|} 7→ M
xx
:=
(1 v) |xxihxx| + v
I
4
, (1 v)(I |xxihxx|) + v
I
4
,
{|χihχ|, I |χihχ|} 7→ M
χ
:=
(1 v) |χihχ| + v
I
4
, (1 v)(I |χihχ|) + v
I
4
.
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 {a, b}, let
ρ
x
:= (1 v)
x

x
+ v
I
ab
2
,
with |x
i span({|ai, |bi}) and
x
x
= 0. Note that these are naturally thought as the noisy version of the
perfect orthogonal preparations, ρ
x
= N
v
(
x
0

x
0
). Now, let us check that the operational equivalence is
satisfied,
1
2
ρ
a
+
1
2
ρ
a
= (1 v)
|aiha| +
a

a
2
+ v
I
ab
2
=
I
ab
2
= (1 v)
|bihb| +
b

b
2
+ v
I
ab
2
=
1
2
ρ
b
+
1
2
ρ
b
.
Next, we consider the pair of preparations s (α, aa). Let
ρ
s
:= (1 v)
2
|sihs| + (1 (1 v)
2
)
I
4
,
with |s
i span({|αi, |aai}), and
s
s
= 0. ρ
s
can be seen as the state resulting from preparing |s
i and
letting the noise channel act for two steps, i.e., ρ
s
= N
v
N