Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen-Specker theorem: The Peres-Mermin square

Abstract

When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen-Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp measurements, but also preparations as well, and for which a quantum no-go result still holds. According to this notion, the model need only specify a probability distribution over the outcomes of a measurement in a context-independent way, rather than specifying a particular outcome. It also implies novel constraints of context-independence for the representation of preparations. In this article, we describe a general technique for translating proofs of the Kochen-Specker theorem into inequality constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a noncontextual model. We focus on algebraic state-independent proofs, using the Peres-Mermin square as our illustrative example. Our technique yields the necessary and sufficient conditions for a particular set of correlations (between the preparations and the measurements) to admit a noncontextual model. The inequalities thus derived are demonstrably robust to noise. We specify how experimental data must be processed in order to achieve a test of these inequalities. We also provide a criticism of prior proposals for experimental tests of noncontextuality based on the Peres-Mermin square.

Full text

New J. Phys. 19 (2017)123031 https://doi.org/10.1088/1367-2630/aa9168
PAPER
Deriving robust noncontextuality inequalities from algebraic proofs
of the Kochen–Specker theorem: the Peres–Mermin square
Anirudh Krishna
1,2,3
, Robert W Spekkens
1
and Elie Wolfe
1
1
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
2
Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada
3
Author to whom any correspondence should be addressed.
E-mail: anirudh.krishna@usherbrooke.ca,rspekkens@perimeterinstitute.ca and ewolfe@perimeterinstitute.ca
Keywords: noncontextuality, quantum foundations, Peres–Mermin square
Abstract
When a measurement is compatible with each of two other measurements that are incompatible with
one another, these define distinct contexts for the given measurement. The Kochen–Specker theorem
rules out models of quantum theory that satisfy a particular assumption of context-independence: that
sharp measurements are assigned outcomes both deterministically and independently of their context.
This notion of noncontextuality is not suited to a direct experimental test because realistic measurements
always have some degree of unsharpness due to noise. However, a generalized notion of
noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp
measurements, but also preparations as well, and for which a quantum no-go result stillholds.
According to this notion, the model need only specify a probability distribution over the outcomes of a
measurement in a context-independent way, rather than specifying a particular outcome. It alsoimplies
novel constraints of context-independence for the representation of preparations. In this article, we
describe a general technique for translating proofs of the Kochen–Specker theorem into inequality
constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a
noncontextual model. We focus onalgebraic state-independent proofs, using the Peres–Mermin square
as our illustrative example. Our technique yields the necessary and sufficient conditions for a particular
set of correlations (between the preparations and the measurements)to admit a noncontextual model.
The inequalities thus derivedare demonstrably robust to noise. We specify how experimental data must
be processed in order to achieve a test of these inequalities. We also provide a criticism of prior proposals
for experimental tests of noncontextuality based on the Peres–Mermin square.
1. Introduction
Ontological models of quantum theory are an attempt to explain the statistical predictions of quantum theory.
They take every system to be associated with a space of possible physical states, termed ontic states, every
quantum state to be represented by a statistical distribution over these ontic states, and every measurement to be
represented by a conditional probability distribution for the outcome given the ontic state [1]. Hidden variable
models are examples of ontological models, but so is the physicistʼs orthodox conception of quantum theory,
wherein the ontic states are simply the pure quantum states, not supplemented by any additional variables
4
.
The principle of noncontextuality is an assumption about ontological models that seeks to capture a notion
of classicality. It started its life as an assumption about outcome-deterministic ontological models of quantum
theory, that is, ontological models wherein the outcome of every measurement was fixed deterministically
by the ontic state (in contrast to the orthodox conception). This assumption was famously demonstrated to
be in contradiction with the predictions of quantum theory by Kochen and Specker [2]and Bell [3]. The
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4
The latter is termed the ψ-complete ontological model in [1].
© 2017 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft

Kochen–Specker theorem is one of the strongest constraints on the intepretation of quantum theory.
Furthermore, failing to admit of a noncontextual model appears to be a resource. For instance, in the context of
the state injection model for quantum computation [4,5], the failure of noncontextuality has been shown in
some cases to be necessary for achieving universal quantum computation [6,7].
In [8], a generalized notion of noncontextuality was proposed. For measurements, it constitutes a relaxation
of what was assumed by Kochen–Specker and by Bell. Specifically, it allows the assignment of measurement
outcomes by ontic states to be indeterministic. In this way, it redefined the notion of noncontextuality for
measurements in a way that excised the notion of determinism. This is desirable from a foundational perspective
as it allows one to separate the issue of noncontextuality from that of determinism (recall that Bellʼs notion of
local causality does not presume that the outcomes of measurements are fixed deterministically). Additionally, it
can be shown that the assumption of outcome determinism is unwarranted for any unsharp measurement (i.e., a
measurement for which one cannot find a basis of preparations relative to which it is perfectly predictable), and
every measurement appearing in a real experiment is of this sort [9]. As such, this generalization is important if
one hopes to turn the proven theoretical advantages for computation into practical advantages, because in
practice, sharpness is an idealization that is never strictly satisfied.
Although the revised notion of noncontextuality yields a weaker constraint on the representation of
measurements in the ontological model than did the traditional notion
5
, it naturally applies not only to
measurements but to preparations as well, and thereby implies novel constraints on how quantum states can be
represented by distributions over ontic states in the model
6
. It was argued in [8]that whatever motivations can
be given for assuming noncontextuality for one type of procedure, such as a measurement, this same motivation
can be given for assuming it of any other type of procedure, such as a preparation. Consequently, the only
natural assumption to consider in this approach is that the revised notion of noncontextuality applies to all
procedures. This assumption is termed universal noncontextuality or simply noncontextuality. We will
henceforth refer to the traditional notion of noncontextuality as KS-noncontextuality (for Kochen–Specker)to
avoid any confusion.
In [8], it was shown that quantum theory does not admit of a universally noncontextual ontological model. It
was also demonstrated that if one replaces the assumption of KS-noncontextuality for measurements with the
assumption of universal noncontextuality for all procedures, the relaxation of the constraints on the
representation of measurements is compensated by the strengthening of the constraints on the representation of
preparations in such a way that any proof that quantum theory fails to admit of a KS-noncontextual model can
be translated into a proof that it fails to admit of a universally noncontextual model.
Much of the research on noncontextuality to date has centred on the question of whether quantum theory
admits of a noncontextual model. A more general question, which has been the impetus for much recent work, is
whether one can devise a direct experimental test of the assumption of a noncontextual ontological model, one
that is independent of the validity of quantum theory. Just as a Bell inequality is a constraint on experimental
statistics that follows directly from the assumption of a locally causal ontological model, without any reference to
the quantum formalism, what one wants of a test of noncontextuality is a constraint on experimental statistics
that follows directly from the assumption of a noncontextual ontological model, without any reference to the
quantum formalism. Such constraints will here be termed noncontextuality inequalities. If experimental statistics
are found to violate these inequalities, then one can conclude that not just quantum theory but any operational
theory that can do justice to the experimental statistics—and therefore nature itself—must fail to admit of such a
model, thereby constraining the form of all future physical theories.
The generalized notion of noncontextuality proposed in [8]was defined in such a way as to be applicable to
any operational theory, not just quantum theory, such that if an experiment yields data supporting an
operational theory distinct from quantum theory, the question of whether it admits of a noncontextual model is
still meaningful. The definition asserts that an ontological model of an operational theory is noncontextual if two
experimental procedures that are statistically indistinguishable at the operational level are statistically
indistinguishable at the ontological level. They key point is that the notion of statistical indistinguishability at the
operational level can be assessed in any operational theory
7
.
It has been shown that violations of noncontextuality inequalities defined in terms of this notion can imply
advantages for information processing which are independent of the validity of quantum theory. For example,
they imply an advantage for the cryptographic task of parity-oblivious random access codes [10–12]. Such
5
A consequence of this relaxation is that, by its lights, the ψ-complete ontological model is found to represent measurements
noncontextually. However, it fails to represent preparations noncontextually, as noted in [8].
6
It also implies novel constraints on how quantum channels can be represented by conditional probability distributions from the space of
ontic states to itself.
7
The fact that the notion of statistical indistinguishability is applicable not just pairs of measurements, but to pairs of preparations and
transformations respectively as well, is what allows the generalized notion of noncontextuality to be applicable to these other procedures.
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New J. Phys. 19 (2017)123031 A Krishna et al

inequalities also hold promise for making the results on quantum computational advantages discussed above
robust to noise and for expressing the origin of the advantage in a manner that is independent of the validity of
quantum theory.
Several recent works have considered the question of how to derive noncontextuality inequalities and how to
subject them to experiment test [13–15]. The present work is concerned with a special case of this problem,
namely, how to derive noncontextuality inequalities starting from any given proof of the Kochen–Specker theorem,
that is, from a proof of the failure of KS-noncontextuality in quantum theory. As noted above, [8]showed how,
in general, to convert a proof of the failure of KS-noncontextuality in quantum theory into a proof of the failure
of universal noncontextuality in quantum theory, so the outstanding problem is how to convert a proof of the
failure of universal noncontextuality in quantum theory into an operational noncontextuality inequality.
Note that any test of noncontextuality that is devised from a particular no-go theorem requires an
experimentalist to target a particular set of preparations and a particular set of measurements, each with
specified relations holding among their members (we will say more about the nature of these relations in due
course). A more general version of the problem, however, is to figure out how to infer from any experimental
data—that is, from an experiment that was not designed to target particular preparations or measurements or
any particular relations among them—whether or not it admits of a noncontextual model. Because a test of
noncontextuality is a test of classicality, having the capability to test the assumption of noncontextuality on any
experimental data is clearly of greater utility than merely knowing how to implement a dedicated experiment for
testing the hypothesis of noncontextuality. Pusey [15]identified the conditions that are both necessary and
sufficient for the existence of a noncontextual model for experimental data derived from the simplest
experimental scenario in which such conditions are expected to be nontrivial. Unfortunately, this simplest
scenario does not arise within operational quantum theory
8
. Extending Puseyʼs analysis to more general
scenarios is an important open problem.
Nonetheless, there are also advantages to building sets of noncontextuality inequalities from specific proofs
of the Kochen–Specker theorem, because such proofs have nontrivial structural properties. Different proofs—
and there is now a great diversity of these—capture what is surprising about the failure of noncontextuality in
different ways, and these intuitions are likely to be helpful in identifying the applications thereof.
We here focus on deriving noncontextuality inequalities from state-independent proofs of the Kochen–
Specker theorem.
Reference [14]has already demonstrated how one can derive one such inequality from any state-
independent geometric proof of the Kochen–Specker theorem, that is, any proof expressed in terms of an
uncolourable set of rays. Here, we extend this work in two important ways: (1)we provide a technique for
finding all of the noncontextuality inequalities that apply to a certain set of correlations starting from any state-
independent proof of the Kochen–Specker theorem, and (2)we show how to do so for proofs that are expressed
algebraically rather than geometrically. We expand on each of these points presently, in reverse order.
The distinction between geometric and algebraic proofs of the failure of KS-noncontextuality in quantum
theory is not fundamental because one can convert any algebraic proof into a geometric form and vice-versa.
Nonetheless, each proof style has its advantages. The first known proofs were geometric uncolorability proofs.
Algebraic proofs arose later, but in many respects they have a logic that is easier to grasp. Indeed, the paradigm
example of a proof of the Kochen–Specker theorem is now arguably the algebraic version of the Peres–Mermin
square proof [16,17], which will be the example we focus on here.
Furthermore, the algebraic structure suggests generalizations of these proofs that might not be obvious from
the geometric perspective [18,19]. Although one could derive a noncontextuality inequality for the Peres–
Mermin square by first expressing the latter as a geometric proof (as in [16])and then applying the technique
described in [14], it is more useful to have a technique for deriving noncontextuality inequalities that is native to
the algebraic approach. We here provide such a technique.
In order to turn a proof of the failure of universal noncontextuality in quantum theory into a
noncontextuality inequality, one must operationalize the description of the experiment provided in the no-go
theorem, purging it of any reference of the quantum formalism, and one must robustify the constraints on
experimental data that are derived from noncontextuality, which means that these constraints must provide
quantitative bounds that can be violated in principle even if the experimental operations are noisy. This
progression was achieved in [14], but the resulting inequality provided an upper bound on just a single
operational quantity (an average, over certain preparation-measurement pairs, of the degree of correlation
between them). The technique described in the present article goes much further towards providing a means of
8
Recall that a set of measurements is said to be tomographically complete for a system if the statistics for any measurement on the system can
be computed from the statistics of the measurements in this set. Puseyʼs simplest scenario is one wherein a tomographically complete set of
measurements consists of just two binary-outcome measurements. This scenario does not arise in operational quantum theory, because the
simplest quantum system, a qubit, requires three binary-outcome measurements for tomographic completeness.
3
New J. Phys. 19 (2017)123031 A Krishna et al

deriving all of the noncontextuality inequalities that hold for a given set of preparations and measurements.
Although we focus on a subset of the correlations between preparations and measurements that arise in the
construction, for this restricted set of experimental data, satisfaction of the inequalities that we derive is both
necessary and sufficient for the existence of a noncontextual model.
Finally, we note a difference in the way experiments are described in this article relative to previous
treatments of inequalities for universal noncontextuality [10,13,14]. We here use the notion of a source, that is, a
process which samples a classical variable from a distribution, chooses which preparation procedure to
implement on the system based on the value sampled and outputs both the system and the variable. This choice
ensures that our derived noncontexuality inequalities are easier to compare with Bell inequalities.
The remainder of the paper is structured as follows.
In section 2, we provide an overview of operational theories (2.1)and ontological models (2.2). In particular,
we discuss the concepts of operational equivalence and of compatibility (applied to measurements and sources)
and illustrate the concepts with quantum examples. We provide formal definitions of measurement
noncontextuality and preparation noncontextuality, in particular, a characterization of these assumptions in
terms of expectation values for the outcomes of measurements and sources given the ontic state.
In section 3, we review the well-known proof of the failure of KS-noncontextuality in quantum theory based
on the Peres–Mermin square (3.1), and we show how to translate this no-go theorem into one that demonstrates
the failure of universal noncontextuality in quantum theory (3.2).
Section 4is the heart of the article, describing our technique for turning quantum no-go theorems into
operational noncontextuality inequalities. In the first section (4.1), we operationalize the description of the
quantum measurements and sources that appear in the Peres–Mermin-inspired proof of the failure of universal
noncontextuality, thereby obtaining a notion of a Peres–Mermin experimental scenario that is purged of any
reference to quantum theory. This provides a template for how to achieve this operationalization for any such
construction. The following five sections (4.2–4.6)describe how to derive noncontextuality inequalities from
such an operational construction, using Peres–Mermin as the illustrative example. We also show how the ideal
quantum realization of the measurements and sources in the Peres–Mermin scenario violate these inequalities
(4.6.1), and we demonstrate the robustness of these inequalities to noise (4.6.2), by showing how they can be
violated by partially depolarized versions of the ideal quantum realizations of the measurements and sources.
In section 5, we clarify what must be done experimentally in order to test the noncontextuality inequalities
we have derived, and in section 6we provide our concluding remarks.
Appendix Adiscusses the problem of computationally converting between the vertex and halfspace
representations of a polytope, appendix Bdiscusses the symmetries of our noncontextuality inequalities under
deterministic processings of the experimental procedures, and appendix Cdemonstrates that a certain class of
inequalities on experimental statistics are trivial. Finally, appendix Dreviews a previous proposal for how to
implement an experimental test of noncontextuality based on the Peres–Mermin square, and argues against its
adequacy.
2. Preliminaries
2.1. Operational concepts
2.1.1. Operational theories
The primitive elements of an operational theory are preparations and measurements, each specified as lists of
instructions to be performed in the laboratory.
Asource is a device that implements one of a set of preparation procedures on a system, sampled from some
probability distribution, and has a classical outcome that heralds which preparation has in fact been
implemented. (The use of the term ‘source’to refer to such a device is conventional in both classical and
quantum Shannon theory, where it is the standard way of modelling the input to a communication channel
[20].)We will denote a source by Sand the variable describing its classical outcome by
s
.
A measurement, denoted
M
, accepts as input a system and returns a classical outcome, denoted by the
variable
m
.
An operational theory provides an algorithm for computing the probability distribution for the outcome of
any measurement acting on any preparation, and consequently it allows the computation of the joint probability
distribution over the outcome of any measurement
M
and the outcome of any source S,
msMS
p
r, ,{∣ }
.We
refer to this as simply the joint distribution on the measurement-source pair (M,S).
2.1.2. Operational equivalence
Consider two measurement procedures,
M
1and
M2
, whose outcomes are random variables, denoted
m
1
and
m
2
respectively.
M
1and
M2
are said to be operationally equivalent if they define the same joint distribution for all
4
New J. Phys. 19 (2017)123031 A Krishna et al

possible sources:
S m sM S m sM S:pr , , pr , , . 1
11 2 2
"={∣ } {∣ } ()
Letting

denote an operational equivalence class of measurements, we can express the operational equivalence
of two measurements
M
1and
M2
by specifying that they belong to the same class,
MM,. 2
12
Î()
Similarly, two sources, S1and S2, whose outcomes are random variables
s
1
and
s
2respectively, are said to be
operationally equivalent if they define the same joint distibution for all possible measurements:
MmsMS msMS:pr, , pr, , . 3
11 2 2
"={∣ } {∣ } ()
Letting

denote an operational equivalence class of sources, we can express the operational equivalence of two
sources, S1and S2, by stipulating that they are in the same class,
SS,. 4
12 Î()
Because the joint distribution
smMS
p
r, ,{∣ }
is computable from the operational theory, so too is the
correlation between the outcome
s
of the source Sand the outcome
m
of the measurement
M
,
sm sm s m M Spr , , . 5
MS
sm
,
,
å
áñ = {∣ } ()
Such correlations will be the quantities appearing in our noncontextuality inequalities.
2.1.3. Compatibility
In this section, we briefly review the notion of compatibility. The interested reader is pointed to [21]for a
detailed overview.
Informally, two or more devices are said to be compatible if their output can be obtained by classical post-
processing of the output of a single device. More precisely, one can define the notion of compatibility in terms of
a notion of simulatability.
Consider two measurements,
M
and
M¢
, which accept an input system and output random variables
m
and
m¢
respectively. We say that
M
can simulate
M¢
if there exists a conditional distribution mm
p
r¢{∣}
such that
S m sM S m m m sM S:pr , , pr pr , , . 6
m
å
"¢¢=¢{∣ } {∣}{∣ } ()
Two measurements
M
1and
M2
are said to be compatible if both of them can be simulated by some third
measurement
M
, that is, if there exists mm
p
r1
{∣}
and
mm
p
r
2
{∣}
such that
S m sM S m m m sM S
S m sM S m m m sM S
:pr , , pr pr , , ,
:pr , , pr pr , , . 7
m
m
11 1
22 2
å
å
"=
"=
{∣ } {∣}{∣ }
{∣ } {∣}{∣} ()
Similar definitions hold for sources. We say that a source Swith classical outcome
s
simulates source S
¢
with
classical outcome
s
¢
if there exists a conditional distribution
ss
p
r¢
{∣}
such that
MmsMS ssmsMS:pr, , pr pr, ,. 8
s
å
"¢¢=¢{∣ } {∣}{∣} ()
When it comes to defining a notion of compatibility of sources, there is a nuance relative to the case of
measurements. The definition we adopt will apply only to those pairs of sources, S1and S2, that are operationally
equivalent when one marginalizes over their outcomes, that is, it presumes that S1and S2are such that
MmsMS msMS: pr, , pr, , . 9
ss
11 2 2
12
åå
"={∣ } { ∣ } ()
Every set of sources we consider in this article will have this property. Two such sources, S1and S2, are said to be
compatible if there exists a third source Sthat simulates them both, that is, if there exists ss
p
r1
{∣}
and
ss
p
r
2
{∣}
such that
M msM S ss msMS M msMS ss msMS:pr, , pr pr, ,,<