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arXiv:1710.08434v1 [quant-ph] 23 Oct 2017
All the noncontextuality inequalities for arbitrary prepare-and-measure experiments
with respect to any fixed sets of operational equivalences
David Schmid, Robert W. Spekkens, and Elie Wolfe
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5∗
Within the framework of generalized noncontextuality, we introduce a general technique for
systematically deriving noncontextuality inequalities for any experiment involving finitely many
preparations and finitely many measurements, each of which has a finite number of outcomes. Given
any fixed sets of operational equivalences among the preparations and among the measurements as
input, the algorithm returns a set of noncontextuality inequalities whose satisfaction is necessary
and sufficient for a set of operational data to admit of a noncontextual model. Additionally,
we show that the space of noncontextual data tables always defines a polytope. Finally, we
provide a computationally efficient means for testing whether any set of numerical data admits of a
noncontextual model, with respect to any fixed operational equivalences. Together, these techniques
provide complete methods for characterizing arbitrary noncontextuality scenarios, both in theory
and in practice.
I. INTRODUCTION
Proofs of the failure of noncontextuality place strong
constraints on our understanding of nature. For example,
the argument by Kochen and Specker [1] showed that
quantum theory cannot be explained by an ontological
model in which every projective measurement has its
outcome determined by hidden variables, independently
of what other measurements it is implemented jointly
with (that is, independently of context).
A generalized notion of noncontextuality was defined
in Ref. [2]. Heuristically, it asserts that an ontological
model of some operational theory is noncontextual if
and only if laboratory operations which cannot be
operationally distinguished are represented identically
in the model. It was shown that the operational
predictions of quantum theory are inconsistent with the
existence of such a model. Furthermore, this generalized
notion of noncontextuality (which we henceforth refer
to simply as “noncontextuality”) applies to arbitrary
operational theories—not just quantum theory—so one
can ask whether any given set of data describing possible
statistics for a prepare-and-measure experiment admits
of a noncontextual model. We will refer to such sets
of data as data tables, since one can organize the data
into a table giving the probability for each pairing of a
preparation with a measurement outcome.
For any prepare-and-measure experiment, this article
presents a systematic method for characterizing all
the data tables consistent with the principle of
noncontextuality, given any fixed set of operational
equivalences among the preparations and any fixed set of
operational equivalences among the measurements. The
inputs to our algorithm are these two sets of operational
equivalences, and the output is a set of inequalities on
the elements of the data table. The satisfaction of
these inequalities is necessary and sufficient for the data
∗dschmid@perimeterinstitute.ca
table to admit of a noncontextual model with respect
to the specified operational equivalences. Our method
shows that the space of noncontextual data tables for
any prepare-and-measure scenario defines a polytope,
which we term the generalized-noncontextual polytope, in
analogy with the local polytope of a Bell scenario [3].
In addition, this article provides a systematic and
efficient method for deciding if any given data table,
specified numerically, admits of a noncontextual model,
given any fixed set of operational equivalences.
Foundationally, the failure of noncontextuality plays a
key role as a notion of nonclassicality which has broad
applicability, and which subsumes other notions such
as the negativity of quasi-probability representations [4],
the generation of anomalous weak values [5], and
violations of local causality [2]. As such, our method
for deriving inequalities allows one to quantitatively
identify the boundary between the classical and
the nonclassical in arbitrary prepare-and-measure
scenarios. Furthermore, it allows one to directly
generate noise-robust noncontextuality inequalities from
arbitrary proofs of the failure of noncontextuality in
quantum theory [6], thereby allowing one to find
necessary and sufficient conditions for noncontextuality
in scenarios where only necessary conditions were
previously known [7–11].
Practically, quantifying the boundary between classical
and nonclassical is important for identifying tasks which
admit of a quantum advantage. For example, the failure
of noncontextuality has been shown to be a resource in
various tasks, providing advantages for cryptography [7,
12,13] and computation [14–16]. If a given protocol
for quantum cryptography/communication or a given
quantum circuit can be cast into the form of a
prepare-and-measure experiment, then one can apply
our methods to determine whether the given protocol
or circuit admits of a noncontextual model. By
this sort of study, one might identify new quantum
information-processing tasks for which contextuality is
a resource. Our methods should also prove useful
2
for the benchmarking of real quantum devices, just
as Bell inequality violations allow one to certify
randomness [17] and achieve key distribution [18,19] in
a device-independent fashion.
Because the generalized notion of noncontextuality of
Ref. [2] presumes neither the correctness of quantum
theory nor that the preparations and measurements
are noiseless, the noncontextuality inequalities derived
from it can be applied directly to real experimental
data. One such inequality has been violated with high
confidence in a recent experiment, demonstrating that
nature does not admit of a noncontextual model [8].
To date, such experiments have targeted preparations
and measurements that are believed (based on quantum
intuitions) to yield a violation of one particular
noncontextuality inequality. This article provides a
more general and more powerful approach: given a
numerically-specified data table, we provide an efficient
algorithm to determine whether any noncontextuality
inequality is violated by it ,and if one is, to determine
that inequality and its degree of violation.
Our method for finding the generalized-noncontextual
polytope is comprised of two distinct computational
tasks. The first task is to catalog all extremal solutions
which satisfy some initial set of linear constraints; i.e., it
is an instance of the vertex enumeration problem. That
catalog allows us to formulate a condition for membership
in the generalized-noncontextual polytope in terms of
an existential quantifier. The second task, then, is
quantifier elimination, and requires eliminating variables
for a system of linear inequalities. As elaborated herein,
a variety of standard algorithms are readily available for
efficiently solving both of these tasks. Moreover, vertex
enumeration and quantifier elimination algorithms are
already widely used in quantum foundations1.
II. OPERATIONAL AND ONTOLOGICAL
PRELIMINARIES
An operational theory specifies a set of laboratory
procedures, such as preparations and measurements,
as well as a prescription for finding the probability
distribution p(m|M, P) over outcomes mof any given
measurement Mwhen implemented following any given
preparation P.
1Familiar applications of vertex enumeration and quantifier
elimination—albeit not always referred to by these
names—include deriving standard Bell inequalities [20–23],
deriving entropic inequalities for generalized correlation
scenarios [24–26], and many others [10,27–31]. For instance,
Bell inequalities are defined as the convex hull of all local
strategies; i.e. they precisely characterize the region in
probability-space spanned by deterministic strategies [3]. This
relates to vertex enumeration, because the convex hull problem
(finding inequalities given extreme points) is equivalent to vertex
enumeration (finding extreme points given inequalities) from an
algorithmic perspective (see for example Ref. [28, App. A]).
Two preparations P1and P2are termed operationally
equivalent if they generate the same statistics for all
possible measurements:
∀M:p(m|M, P1) = p(m|M , P2).(1)
We denote this operational equivalence relation by
P1≃P2.
Similarly, two measurement procedures M1and M2are
termed operationally equivalent if they generate the same
probability distribution over their outcomes (denoted m1
and m2) for all possible preparations:
∀P:p(m1|M1, P ) = p(m2|M2, P ).(2)
We denote this operational equivalence relation by
M1≃M2.
We will refer to the event of a measurement Myielding
an outcome mas a measurement effect, denoted [m|M].
If one samples the choice of laboratory procedure Oi
from some probability distribution piand then forgets the
value of i, we introduce the shorthand notation PipiOi
for the effective procedure so defined. The procedures
here could be preparations or measurement effects.
An ontological model attempts to explain the
probabilities p(m|M, P ) in an operational theory via a
set Λ of ontic states. An ontic state λ∈Λ specifies
all the physical properties of the system, and causally
mediates correlations between the preparation and the
measurement. For every laboratory preparation P, the
model specifies a probability distribution µP(λ), where
∀λ:µP(λ)≥0,(3)
ZΛ
dλ µP(λ) = 1.(4)
Whenever preparation Pis implemented, the ontic state
λis sampled from an associated probability distribution
µP(λ). Every measurement Mgenerates an outcome m
as a probabilistic function of the ontic state according to
some fixed response function ξm|M(λ), where
∀λ, m :ξm|M(λ)≥0,(5)
∀λ:X
m
ξm|M(λ) = 1.(6)
For the ontological model to reproduce an operational
theory’s empirical predictions, one requires that
∀m, M, P :p(m|M, P ) = ZΛ
ξm|M(λ)µP(λ)dλ. (7)
Finally, note that an effective laboratory procedure
PipiOiis represented in an ontological model by
the corresponding convex mixture of the ontological
representations of the individual operations Oi(see
Eq. (7) of [4] and the surrounding discussion).
An ontological model which respects the principle of
preparation noncontextuality [2] is one in which two
3
operationally equivalent preparations are represented by
the same distribution over ontic states; that is,
P1≃P2implies that (8)
∀λ:µP1(λ) = µP2(λ).
An ontological model which respects the principle
of measurement noncontextuality [2] is one in which
two operationally equivalent measurement effects are
represented by the same response function; that is,
[m1|M1]≃[m2|M2] implies that (9)
∀λ:ξm1|M1(λ) = ξm2|M2(λ).
In this paper, the term noncontextuality refers
to universal noncontextuality [2], which posits
noncontextuality for all procedures, including
preparations and measurements.
III. PROBLEM SETUP
The scenario we are considering has a set
{P1, P2, ..., Pg}of gpreparations, a set {M1, M2, ..., Ml}
of lmeasurements, a set of doutcomes {1,2, ..., d}for
each measurement, a set of operational equivalences
among the preparations, denoted OEP, and a set of
operational equivalences among the measurements,
denoted OEM. There are no restrictions on any of
these sets, beyond the fact that they must be finite,
as they are in any real experiment2. Furthermore, we
have not presumed that one knows anything about
the preparations and measurements, beyond the fact
that they can be performed repeatedly so as to gather
statistics. Without loss of generality, we treat each
measurement as having exactly doutcomes for some
sufficiently large value of d(since any measurement with
d∗< d outcomes can be redefined to have doutcomes,
d−d∗of which never occur).
The input to our algorithm is a specification of the
operational equivalences OEPand OEM, and the desired
output is a set of inequalities such that a data table
{p(m|Mi, Pj)}i,j,m admits a universally noncontextual
model if and only if all the inequalities are satisfied.
Although it is not obvious from the definition of universal
2In any real experiment with continuous variable systems, one
must coarse-grain the outcomes to a finite set to obtain nonzero
probabilities of any given event. There is also a nuance
concerning experiments with a finite number of preparations,
measurements and outcomes: the full set of operational
equivalences among these might be infinite, but one can always
find a finite generating set of operational equivalences whose
implications for noncontextual data tables are equivalent to
the implications of the full infinite set. A first example
of this procedure can be found in Ref. [32], and a general
method for finding generating sets of operational equivalences
is forthcoming.
noncontextuality, we will find that the final inequalities
will be linear in the probabilities.
Generically, each of the operational equivalences
s∈ OEPis of the form
X
j
αs
PjPj≃X
j′
βs
Pj′Pj′(10)
for some sets of convex weights {αs
Pj}jand {βs
Pj′}j′
(a set of convex weights is a list of nonnegative real
numbers which sum to one). Hence, the principle
of preparation noncontextuality, Eq. (8), implies that
the same functional relationships must hold among the
ontological representations of the preparations; in other
words,
∀λ:X
j
αs
PjµPj(λ) = X
j′
βs
Pj′µPj′(λ).(11)
Similarly, each of the operational equivalences r∈ OEM
is of the form
X
i,m
αr
m|Mi[m|Mi]≃X
i′,m′
βr
m′|Mi′[m′|Mi′] (12)
for some sets of convex weights {αr
m|Mi}i,m and
{βr
m′|Mi′}i′,m′. Hence, the principle of measurement
noncontextuality, Eq. (9), implies that the same
functional relationships must hold among the ontological
representations of the effects; in other words,
∀λ:X
i,m
αr
m|Miξm|Mi(λ) = X
i′,m′
βr
m′|Mi′ξm′|Mi′(λ).(13)
The question of whether a data table admits
a universally noncontextual model, then, may be
compactly summarized as follows:
Formulation F1 of the existence of a universally
noncontextual model: A universally noncontextual
model for a data table {p(m|Mi, Pj)}i,j,m exists (with
respect to the sets of operational equivalences OEPand
OEM) if and only if
∃Λ,∃{µPj(λ)}j,λ,{ξm|Mi(λ)}i,m,λ such that:
∀λ, i, m :ξm|Mi(λ)≥0,(F1a)
∀λ, i :X
m
ξm|Mi(λ) = 1,(F1b)
∀λ, r :X
i,m
(αr
m|Mi−βr
m|Mi)ξm|Mi(λ) = 0,(F1c)
∀λ, j :µPj(λ)≥0,(F1d)
∀j:Zλ
µPj(λ) = 1,(F1e)
∀λ, s :X
j
(αs
Pj−βs
Pj)µPj(λ) = 0,(F1f)
∀i, j, m :ZΛ
ξm|Mi(λ)µPj(λ)dλ =p(m|Mi, Pj).(F1g)
Eqs. (F1a) to (F1g) represent, respectively: positivity
of the response functions (Eq. (3)); normalization of
4
the response functions (Eq. (4)), the consequences of
noncontextuality implied by the operational equivalences
in OEM(Eq. (13)); positivity of the distributions
associated with the preparations (Eq. (5)); normalization
of the distributions associated with the preparations
(Eq. (6)); the consequences of noncontextuality implied
by the operational equivalences in OEP(Eq. (11)); and
the expression for the probabilities in the data table in
terms of the ontological model (Eq. (7)).
There are two key obstacles to deriving constraints
directly on {p(m|Mi, Pj)}i,j,m from the implicit
constraints imposed by Eqs. (F1a)-(F1g). First,
the ontic state space Λ is unknown and possibly of
unbounded cardinality, so that it is not obvious a
priori whether there is an algorithm to solve the
problem. Second, even if the number of ontic states
were known to be finite, so that the problem could in
principle be solved by quantifier elimination methods,
the probabilities in Eq. (F1g) are nonlinear in the
unknown parameters {ξm|Mi(λ)}i,m,λ and {µPj(λ)}j,λ
appearing in the quantifiers; hence, the problem would
be one of nonlinear quantifier elimination, which is
computationally difficult. We overcome both of these
problems by leveraging the convex structure of the
space of response functions: we find the finite set of
convexly-extremal noncontextual assignments to the
measurements, identify the set of ontic states with it, and
then parametrize the distributions corresponding to the
preparations in terms of their probability assignments to
these ontic states. Thus, the unknown parameters form
a finite set, and furthermore the operational probabilities
are linear in these parameters.
Finally, we perform linear quantifier elimination to
obtain constraints on the operational probabilities alone.
IV. CHARACTERIZING THE
GENERALIZED-NONCONTEXTUAL POLYTOPE
A. Enumerating the convexly-extremal
noncontextual measurement assignments
No matter what the form or size of the ontic state
space Λ, a measurement noncontextual assignment of
probabilities to all doutcomes of all lmeasurements, for
a particular ontic state λ∗, is an (ld)-component vector
ξ(λ∗)≡ξ1|M1(λ∗),..., ξd|M1(λ∗), ξ1|M2(λ∗),..., ξd|Ml(λ∗)
subject to the constraints of Eqs. (F1a)-(F1c). We
call such an (ld)-component vector a noncontextual
measurement assignment. The set of all such assignments
defines a polytope:
Characterization P1 of the noncontextual
measurement-assignment polytope: The
(ld)-component vector ξ(λ∗)lies inside the noncontextual
measurement-assignment polytope if and only if
∀i, m :ξm|Mi(λ∗)≥0,(P1a)
∀i:X
m
ξm|Mi(λ∗) = 1 ,(P1b)
∀r:X
i,m
(αr
m|Mi−βr
m|Mi)ξm|Mi(λ∗) = 0 .(P1c)
In what follows, it is critical to characterize
this polytope by its vertices rather than its facets.
The vertices are the convexly-extremal noncontextual
measurement assignments. (Note that if there are no
operational equivalences among the measurements, then
these extremal assignments are deterministic, that is, all
of the elements of the vector have value 0 or 1.) In
general, to find the vertices of a polytope that is given in
terms of its facet inequalities, one must solve the vertex
enumeration problem [33–37]. Many excellent software
packages are freely available for vertex enumeration3.
We introduce the notation κas a discrete variable
ranging over the vertices, and we indicate the
(explicit) noncontextual measurement assignment
of vertex κ∗by the (ld)-component vector
˜
ξ(κ∗)≡˜
ξ1|M1(κ∗),..., ˜
ξd|M1(κ∗),˜
ξ1|M2(κ∗),..., ˜
ξd|Ml(κ∗)
.
Now, since any point in a polytope can be
written as a convex mixture of the vertices, the
noncontextual measurement-assignment polytope can be
defined alternatively but equivalently as the convex hull
of its vertices:
Characterization P2 of the noncontextual
measurement-assignment polytope: The
(ld)-component vector ξ(λ∗)lies inside the noncontextual
measurement-assignment polytope if and only if there
exist some convex weights {w(κ|λ∗)}ksuch that:
∀i, m :ξm|Mi(λ∗) = X
κ
w(κ|λ∗)˜
ξm|Mi(κ),(P2a)
where κranges over the vertices found by performing
vertex enumeration on the linear constraints of
characterization P1.
Below, we presuppose that one has indeed characterized
the noncontextual measurement-assignment polytope by
finding its vertices explicitly.
3Dedicated software for performing vertex enumeration includes
traf from PORTA,skeleton64f from skeleton, and lcdd gmp from
cddlib, the latter notably being readily available on Linux and
MacOS. An especially versatile computational geometry suite for
Linux is polymake.
5
B. Constructing a noncontextual model with
known ontic states and linearly constrained
parameters
Suppose one has a universally noncontextual model of
the experiment in the sense of formulation F1, where the
ontic state space need not be of finite cardinality. The
results of the previous subsection imply that it is always
possible to infer from this model another universally
noncontextual model wherein the ontic state space is of
finite cardinality, as follows.
By substituting Eq. (P2a) into Eq. (7) for some i, j ,
each operational probability can be written in terms of a
finite sum:
p(m|Mi, Pj) = ZΛ
ξm|Mi(λ)µPj(λ)dλ (14a)
=ZΛX
κ
˜
ξm|Mi(κ)w(κ|λ)µPj(λ)dλ (14b)
=X
κ
˜
ξm|Mi(κ)ZΛ
w(κ|λ)µPj(λ)dλ(14c)
=X
κ
˜
ξm|Mi(κ)νPj(κ),(14d)
where we have defined
νPj(κ)≡ZΛ
w(κ|λ)µPj(λ)dλ. (14e)
Because νPj(κ∗) for a given vertex κ∗is a convex
combination of the values of µPj(λ), we can infer
that each νPj(κ) is a valid probability distribution,
and furthermore that the set {νPj(κ)}respects
noncontextuality with respect to the operational
equivalences in OEP.
Thus, if any noncontextual ontological model exists,
then there must also exist a noncontextual model with
an ontic state space of finite cardinality. The latter
model is constructed by identifying one ontic state with
each extremal noncontextual measurement assignment,
and then imagining every preparation as a probability
distribution over those ontic states, as done in Eq. (14e).
In other words,
Formulation F2 of the existence of a
universally noncontextual model: For a data
table {p(m|Mi, Pj)}i,j,m, an ontological model that is
universally noncontextual with respect to the operational
equivalences in OEPand OEMexists if and only if
∃{νPj(κ)}j,κ such that:
∀κ, j :νPj(κ)≥0,(F2a)
∀j:X
κ
νPj(κ) = 1 ,(F2b)
∀κ, s :X
j
(αs
Pj−βs
Pj)νPj(κ) = 0 ,(F2c)
∀i, j, m :X
κ
˜
ξm|Mi(κ)νPj(κ) = p(m|Mi, Pj),(F2d)
where κranges over the discrete set of vertices of the
polytope defined by Eqs. (P1)or (P2).
In this formulation, each operational probability
p(m|Mi, Pj) is given as a linear function of a finite
set of unknown parameters. This is because the
only unknown parameters on the right-hand side of
Eq. (F2d) are {νPj(κ)}κ, while the {˜
ξm|Mi(κ)}κare
specified numerically—they are the solution of the vertex
enumeration problem described in the previous section.
Achieving linearity in all the constraints is a critical
intermediate step towards finding a final quantifier-free
formulation, as we do in the next section, and is critical
for the numerical methods we introduce in Section V.
C. The inequalities formulation of the
generalized-noncontextual polytope
To obtain constraints that refer only to operational
probabilities, we eliminate the unobserved {νPj(κ)}j,κ
from the system of equations (F2), obtaining a system
of linear inequalities over the {p(m|Mi, Pj)}i,j,m alone.
The linearity of the final inequalities follows from the
linearity of the inequalities and equalities in Eqs. (F2).
This establishes that the space of noncontextual data
tables defines a polytope.
The standard method for solving this problem of
linear quantifier elimination is the Chernikov-refined
Fourier-Motzkin algorithm [38–42], which is implemented
in a variety of software packages4.
4Dedicated software for eliminating variables from a set of
linear inequalities via the Fourier-Motzkin algorithm includes
fmel from PORTA,fme from qskeleton, and fourier from
lrs. From a geometric perspective, each variable in a linear
system is an axis of some high-dimensional coordinate system;
consequently, eliminating a variable is equivalent to projecting
the polytope onto a hyperplane orthogonal to that particular
axis. As such, polytope projection is the titular topic of most
of the relevant literature on linear quantifier elimination [38–42].
Furthermore, polytope projection and vertex enumeration are
intimately related: One can define the vertex enumeration
6
Denoting the quantifier-free list of linear facet
inequalities of the generalized-noncontextual polytope
by {h1, h2, ..., hn} ≡ H (for ‘halfspaces’) and letting
the coefficients of a specific facet inequality hbe given
by γh
i,j,m while γh
0indicates the constant term in that
inequality, we find that:
Formulation F3 of the existence of a
universally noncontextual model: For a data
table {p(m|Mi, Pj)}i,j,m, an ontological model that is
universally noncontextual with respect to the operational
equivalences in OEPand OEMexists if and only if
∀h∈ H :X
i,j,m
γh
i,j,mp(m|Mi, Pj) + γh
0≥0,(F3a)
where His the set of ninequalities resulting from
eliminating all free parameters {νPj(κ)}j,κ in the
formulation of Eq. (F2).
V. DOES A GIVEN NUMERICAL DATA TABLE
ADMIT OF A NONCONTEXTUAL MODEL?
To date, experimental tests of generalized
noncontextuality [7,8] have targeted the specific
preparations, measurements, and operational
equivalences of some particular quantum no-go
theorem [7–10,43]. By abstracting away the
quantum-specific elements of the proof and describing
the experiment in entirely operational terms, one can
identify operational features of a set of preparations
and measurements, such that any theory exhibiting
those features fails to admit of a noncontextual
ontological model. To test these operational features,
previous experiments have used post-processed data
to enforce specific operational equivalences appearing
in the quantum no-go argument. (See the “secondary
procedures” technique described in Ref. [8].)
We here introduce a much more general analysis
technique, in which one need not target any
specific preparations, measurements, and operational
equivalences. Rather, arbitrary numerical data tables
can be directly analyzed. With respect to whatever
operational equivalences happen to be manifest in the
data5, one can use the methods we present below to
problem as a task of linear quantifier elimination, and therefore
any polytope projection algorithm can be used to perform
vertex enumeration, albeit less efficiently than specialized
algorithms [33–37]. Conversely, a brute-force technique for
polytope pro jection is to first enumerate the polytope’s vertices,
manually discard the to-be-eliminated coordinates from each
each vertex, and then reconvert back to inequalities using a
convex hull algorithm. This roundabout method of performing
polytope projection is generally suboptimal, but can be used in
practice.
5A detailed analysis of how these operational equivalences can be
computed from data is forthcoming in a separate article.
efficiently test whether the numerical data table admits
of a noncontextual model or not. Because answering
this yes-no question does not require deriving the full set
of noncontextuality inequalities for the scenario under
study, it is computationally very efficient6. Furthermore,
analyzing data in this manner always allows for larger
inequality violations, since the post-processing required
in the secondary procedures technique of Ref. [8] always
introduces additional noise.
To test whether a numerically specified data table
{p∗(m|Mi, Pj)}i,j,m admits of a noncontextual model, we
leverage the formulation in Eq. (F2). All the equality
constraints of Eqs. (F2b), (F2c), and (F2d) can be
encoded in a single matrix equality constraint,
M·x=b∗,(15)
where Mcontains the parameters αs
Pj−βs
Pjand the
quantities {˜
ξm|Mi(κ)}i,m,κ,xcontains the unknown
parameters {νPj(κ)}j,κ, and b∗contains the probabilities
{p∗(m|Mi, Pj)}i,j,m, as well as zeroes and ones
corresponding to the right-hand sides of Eqs. (F2b) and
(F2c). Eq. (F2a) becomes simply x≥0. Hence,
for a numerically specified {p∗(m|Mi, Pj)}i,j,m , the
formulation of Eq. (F2) defines a linear program (LP)7.
The primal LP is the search for a solution to a linear
system of equations, namely:
∃xsuch that
M·x=b∗,
and x≥0.
(16)
Because no objective function to maximize or minimize is
specified in the LP defined by Eq. (16), this means the LP
is just checking for the existence of an xwhich satisfies
the constraints and hence guarantees the existence of a
noncontextual model, via Eq. (F2).
Whenever the primal LP is infeasible – that is, no
solution can be found – one can obtain a certificate of
primal infeasibility, also known as the Farkas dual [47,
48]. The certificate of primal infeasibility is obtained by
6An analagous pair of problems with widely differing
computational difficulties has long been appreciated it the
study of Bell nonlocality. Obtaining al l the Bell inequalities
which characterize some nonlocality scenario can be quite
difficult, but ascertaining if a particular correlation admits
a local model or not can be resolved with the application of
a single linear program [44,45]. The same (efficient) linear
program can be used to return a single Bell inequality which
certifies the nonlocality of the given correlation [46].
7Linear programming is used across many fields; specialized LP
software packages include Mosek,Gurobi, and CPLEX .
7
solving the complementary8linear system
min
y
y·b∗such that
1≥y·M≥0.
(17)
Farkas’ lemma states that either the primal LP is feasible,
or else the certificate yresulting from Eq. (17) satisfies
the strict inequality y·b∗<0.
Farkas’s lemma is easily proven: Plainly, if there exists
such a y(i.e, not only y·M≥0but also y·b∗<0),
then there cannot exist an xwhich satisfies the primal
LP of Eq. (16); since the inequalities
x≥0,(18a)
and y·M≥0,(18b)
and y·b∗=y·M·x<0,(18c)
can not all be satisfied simultaneously.
Of relevance to this work is that we may interpret the
certificate yresulting from Eq. (17) as a noncontextuality
inequality, since Farkas’ lemma ensures that y·b≥0
for every bfor which the primal LP is feasible. The
extent to which y·b∗is negative is identically the amount
by which the corresponding noncontextuality inequality
is violated by the (contextual) {p∗(m|Mi, Pj)}i,j,m.
When interpreting certificates of primal infeasibility as
noncontextuality inequalities, one deduces the constant
term from those elements of b∗which do not depend on
{p∗(m|Mi, Pj)}i,j,m. In practice, therefore, the constant
term is the sum of those elements in ywhich correspond
to the normalization conditions of Eq. (F2b).
This technique allows an experimenter to optimally
certify the contextuality of a numerical data table
{p∗(m|Mi, Pj)}i,j,m without first performing the
computationally expensive task of finding all the
noncontextuality inequalities, i.e., without doing any
work to transform formulation F2 into formulation F3.
Instead, by seeking a certificate of primal infeasibility
– a single query to a linear program – one obtains the
noncontextuality inequality which best witnesses the
contextuality of the data table.
VI. APPLICATIONS
A. Generalized-noncontextual polytope in the
simplest nontrivial case
As argued in Ref. [32], the simplest possible scenario in
which the principle of noncontextuality implies nontrivial
8The complementary LP defined in Eq. (17) is meant to explain
how infeasibility certificates are generated in practice. Note,
however, that the Farkas dual of an LP is not the same as the
LP’s dual formulation, although the concepts are related. See
Refs. [47,48], as well as Ref. [30, Theorem 1 and supplementary
materials]).
constraints on operational probabilities involves four
preparations and two binary-outcome measurements9.
We imagine for simplicity that the preparations satisfy
the operational equivalence
1
2P1+1
2P2≃1
2P3+1
2P4(19)
and that there are no operational equivalences among the
measurements.
We denote the operational probability p(0|Mi, Pj)
by pij. (By normalization, probability p(1|Mi, Pj) is
then 1 −pij .) Vertex enumeration finds 4 vertices
for the noncontextual measurement-assignment polytope,
corresponding to the four deterministic assignments
(ξ0|M1(λ), ξ0|M2(λ)) ∈ {(0,0),(0,1),(1,0),(1,1)}. Each of
the 4 preparations defines a probability distribution over
these 4 ontic states, so there are 16 free parameters to
be eliminated. Linear quantifier elimination finds the
polytope of data tables consistent with the principle
of noncontextuality and the operational equivalence of
Eq. (19) to be:
∀i, j : 0 ≤pij ≤1,(20a)
p12 +p22 −p23 −p14 ≤1,(20b)
p12 +p22 −p13 −p24 ≤1,(20c)
p22 +p13 −p12 −p24 ≤1,(20d)
p12 +p23 −p22 −p14 ≤1,(20e)
p22 +p14 −p12 −p23 ≤1,(20f)
p23 +p14 −p21 −p22 ≤1,(20g)
p12 +p24 −p22 −p13 ≤1,(20h)
p13 +p24 −p12 −p22 ≤1.(20i)
Note that the two probabilities which do not appear in
Eqs. (20b)-(20i), p11 and p21, are fixed by the operational
equivalence relation, Eq. (19):
p11 =p13 +p14 −p12 ,(21a)
p21 =p23 +p24 −p22 .(21b)
Ineqs. (20) and Eqs. (21) tightly define the
generalized-noncontextual polytope for this scenario.
Furthermore, all of Ineqs. (20) are equivalent under
relabeling. That is, any one of the inequalities can
generate all 8 by applying relabelings which respect the
operational equivalences: M1↔M2,P1↔P2, and
(P1, P2)↔(P3, P4). Similarly, each of Eqs. (21) is
equivalent to the other under the same relabelings.
As an illustration of how a noncontextuality inequality
can be derived from a numerically specified (contextual)
9Ref. [32] also assumes that these two measurements are
tomographically complete, but we do not make this assumption
here. See Refs. [8,32] for details on the issue of tomographic
completeness.
8
data table, consider the following example
p11 = 1, p12 = 0, p13 = 1, p14 = 0,
p21 = 1, p22 = 0, p23 = 0, p24 = 1,(22)
which respects the operational equivalence relation of
Eq. (19), but maximally violates Ineq. (20i), since it
has p13 +p24 −p12 −p22 = 2 6≤ 1. Indeed, when we
construct the primal linear program per Eq. (16), we
find it to be infeasible, and we find that the certificate of
infeasibility returned by our numerical solver corresponds
to Ineq. (20i).
1. Relevance to parity-oblivious multiplexing
In the communication task of “parity-oblivious
multiplexing”, an agent Alice wishes to communicate
two bits to an agent Bob, in such a way that Bob can
extract information about either of the two bits but
cannot extract any information about their parity [7].
This task involves four preparations (associated to the
four possibilities for the values of the two bits) and two
measurements (corresponding to which bit Bob wishes
to learn about), and the parity-obliviousness condition
implies an operational equivalence relation among the
preparations, namely, that of Eq. (19). Consequently,
this task fits precisely the operational scenario considered
in this section.
In Ref. [7], it was shown that contextuality provides an
advantage for the task of parity-oblivious multiplexing:
the maximum probability of succeeding at this task in a
noncontextual model is 3
/4, so that any higher probability
of success requires contextuality. In a quantum world, for
instance, one can succeed with probability 1
4(2 + √2) ≈
0.85.
If one identifies our preparations P1,P2,P3, and
P4with Ref. [7]’s preparations P00 ,P11,P01 , and P10 ,
respectively, then the inequality in Ref. [7] is our facet
Ineq. (20i). This is the same inequality which witnesses
the failure of noncontextuality for the data table defined
in Eq. (22), which is to be expected, as this particular
data table describes a set of probabilities which achieve
the maximum logically possible probability of success at
parity-oblivious multiplexing.
Additionally, one could apply our algorithm to
generalized types of parity-oblivious multiplexing. For
instance, Ref. [7] derived a bound on the probability of
success in a noncontextual model of n-bit parity-oblivious
multiplexing, in which Alice wishes to communicate n
bits to Bob under the constraint that Bob can learn no
information about the parity between any two of the bits.
These bounds are tight, and can be saturated by a na¨ıve
classical strategy. However, one could further use our
techniques to learn whether or not they constitute facet
inequalities of the generalized-noncontextual polytope,
as well as to find the full generalized-noncontextual
polytope.
For the still more general case where the different
n-bit strings which Alice wishes to send do not have
equal a priori probabilities, our method can also find the
generalized-noncontextual polytope, from which one can
immediately infer the maximum success probability for
the task.
B. Generalized-noncontextual polytopes for
scenarios relevant to state discrimination
One way in which one can generalize the simplest
operational scenario described above is to increase
the number of binary-outcome measurements from two
to three while still not assuming any operational
equivalences among them, so that the only operational
equivalence relation remains the one between the
preparations (Eq. (19)). This operational scenario can
also be related to an information-theoretic task, namely,
the task of minimum error state discrimination, as noted
by two of the present authors in Ref. [49].
In the quantum version of this task, an agent wishes
to guess which of two pure quantum states a system was
prepared in given a single sample of the system, where the
identity of the two quantum states is known. Quantum
theory prescribes a particular trade-off relation between
the probability of success and the non-orthogonality of
the two quantum states, and Ref. [49] showed that
this trade-off contradicts the principle of generalized
noncontextuality. The ideal quantum realization of
minimum error state discrimination fits the operational
scenario described above: the two pure quantum
states define two of the preparation procedures, while
their orthogonal complements in the 2d subspace that
they span define the other two. The fact that the
equal mixture of any two orthogonal pure states in a
2d subspace is independent of the basis implies the
operational equivalence of Eq. (19). Finally, the degree
of nonorthogonality has an operational interpretation
as the probability of one state passing a test for
the other (termed the confusability). Therefore, the
measurements of each of the two bases, together with
the discriminating measurement, provide the three
binary-outcome measurements in the scenario.
The facet noncontextuality inequalities for this
operational scenario are given in Appendix D of
Ref. [49]10, and these are seen to imply a nontrivial upper
bound on the probability of successful discrimination
for a given confusability. Hence, contextuality provides
an advantage for minimum error state discrimination.
The quantum probability of successful discrimination
for a given confusability is higher than that allowed
10 Actually, the polytope given therein is the intersection of
the generalized-noncontextual polytope with two additional
inequalities, which are implied by making sensible labeling
choices.
9
in a noncontextual model, and hence partakes in this
contextual advantage.
Using our technique, one can also immediately derive
the generalized-noncontextual polytope for more general
minimum error state discrimination scenarios, such as
those in which the quantum states (preparations) being
discriminated are sampled with unequal probabilities,
or in which there are more than two quantum states
(preparations).
Because state discrimination in various
forms is a primitive for many other quantum
information-processing tasks, such analyses should
be valuable for identifying the circumstances in which
contextuality constitutes a resource.
C. Generalized-noncontextual polytopes for a
scenario involving both preparation and
measurement noncontextuality
So far, our examples have involved operational
equivalences only among preparations. In this section, we
revisit the scenario considered in the recent experimental
test of noncontextuality in Ref. [8], which involves
operational equivalences among the preparations and
also among the measurements. Specifically, we imagine
a set of six preparations and three binary-outcome
measurements, where the preparations satisfy the
operational equivalences
1
2P1+1
2P2≃1
2P3+1
2P4≃1
2P5+1
2P6,(23)
and the measurement effects satisfy the operational
equivalence
1
3[0|M1] + 1
3[0|M2] + 1
3[0|M3]
≃1
3[1|M1] + 1
3[1|M2] + 1
3[1|M3].
(24)
(See Ref. [8] for a discussion of the significance of these
operational equivalences.)
We denote the operational probability p(0|Mi, Pj) by
pij and p(1|Mi, Pj) by pij . Vertex enumeration finds 6
vertices for the noncontextual measurement-assignment
polytope, corresponding to the four indeterministic
assignments defined by (ξ0|M1(λ), ξ0|M2(λ), ξ0|M3(λ)) ∈
{(0,1
2,1),(1
2,0,1),(1,0,1
2),(1,1
2,0),(0,1,1
2),(1
2,1,0)}. Each
of the 6 preparations defines a probability distribution
over these 6 ontic states, so there are 36 free parameters
to be eliminated. Linear quantifier elimination finds the
polytope of data tables consistent with the principle of
noncontextuality and with the operational equivalences
of Eq. (23) and Eq. (24). We find that this polytope has
1596 facet inequalities.
Plainly, 1596 inequalities is far too many to list
explicitly. However, by considering the physical
symmetries of this scenario, we can significantly simplify
our description of these facets. Since the scenario
is invariant under various relabelings of measurements
[Eqs. (25a-25b)], outcomes [Eq. (25c)], and preparations
[Eqs. (25d-25f)]—i.e. those relabelings which respect
the operational equivalences—we know a priori that
the generalized-noncontextual polytope will possess
significant internal symmetry. The symmetry group
which leaves our polytope invariant is generated by the
six relabelings
M1↔M2(25a)
M1↔M3(25b)
([0|M1],[0|M2],[0|M3]) ↔([1|M1],[1|M2],1|M3]) (25c)
P1↔P2(25d)
(P1, P2)↔(P3, P4) (25e)
(P1, P2)↔(P5, P6).(25f)
We use parentheses to indicate a coherent relabelling:
e.g., the outcomes of three measurements can be flipped
per Eq. (25c), but only if all three measurements have
their outcomes relabeled simultaneously. This is in
contrast to an exchange like P1↔P2per Eq. (25d),
which can be performed in isolation. The total order
of this symmetry group is 576.
Under this group, we find that the 1596 facet
inequalities admit classification into seven symmetry
classes. We therefore explicitly list a single representative
inequality from each class:
Inequality Terms Upper
Bound
Orbit
Size
p11 ≤1 35
p11 +p23 +p35 ≤2.5 48
p11 +p22 +p35 ≤2.572
p11 −p14 −2p15 −2p22 +2 p23 +2 p35 ≤3 576
2p11 −p22 +2 p23 ≤3144
p11 −p15 +p22 +p23 +2 p35 ≤4 576
p11 −p15 +2 p22 +2 p35 ≤4144
(26)
The number of inequalities in each symmetry class
is given by the “Orbit Size” in Ineqs. (26). The
generalized-noncontextual polytope for this scenario is
defined by the 1596 facet inequalities, as well as by
equalities which hold for any data table (contextual
or noncontextual) admitting the operational equivalence
relations per Eqs. (23) and (24). These equalities fall into
three distinct symmetry classes, represented by
p11 +p14 =p12 +p15 (27a)
p11 +p21 +p31 = 3/2 (27b)
p11 +p11 = 1.(27c)
The first two equalities are enforced by the operational
equivalence relations, Eqs. (23) and (24), respectively,
while the third equality is guaranteed by normalization
of measurements, Eq. (6).
To test if a given data table lies inside this
generalized-noncontextual polytope, one could
reconstruct all 1596 inequalities from the seven given
10
in Ineqs. (26), but it is likely much easier to instead
artificially generate equivalent-up-to-symmetries data
tables from one’s actual data table, and then to test each
of those against the seven canonical inequalities11. One
only needs to consider at most 576 data table variants
(per the group order), although in practice there will be
fewer variants to consider if the data table one wishes to
investigate possesses any internal symmetry of its own.
Noting that12
p11 +p23 +p35 =p12 +p24 +p36 ,(28)
the single inequality derived (and experimentally
violated) in Ref. [8, Eq. (6)] is recognized as a facet
of the generalized-noncontextual polytope, namely it is
precisely the second inequality in Ineqs. (26):
2 (p11 +p23 +p35)≤5.(29)
One can also derive Ineq. (29) directly (and efficiently!)
by using the linear program presented in Section V.
Namely, it is the inequality corresponding to the
certificate of infeasibility returned by our numerical
solver, when we construct the primal linear program
using Eq. (16) together with the ideal (contextual)
quantum data table.
VII. CONCLUSIONS
For arbitrary prepare-and-measure experiments, we
have presented a method for finding necessary and
sufficient conditions for a data table to admit of a
noncontextual model, subject to any fixed sets of
operational equivalences among preparations and among
measurements. We have also presented an efficient
method for determining whether a numerical data table
is noncontextual, in this same setting.
We have provided worked examples of each of
these methods, in the process deriving necessary
and sufficient conditions for operational scenarios
in which only necessary conditions were previously
known. Equivalently, we have derived the full
generalized-noncontextual polytopes for scenarios in
which only a single facet inequality was previously known.
The operational scenario studied in Section VI A is of
relevance to parity-oblivious multiplexing [7], while the
operational scenario studied in Section VI C originates in
a recent experimental test of contextuality [8].
A precursor to the current work can be found in [27]. A
distinct method, introduced in Ref. [43], also allows one
to derive all of the facet inequalities for many operational
scenarios. However, the method therein is not fully
general, as it applies only to scenarios in which one
special equivalence class of preparations is singled out
(see Section III. B. of Ref. [43] for details). It would be
interesting to compare the two approaches, e.g., in terms
of computational efficiency, and to modify the approach
in Ref. [43] to make it as general as the approach
described in this article.
ACKNOWLEDGMENTS
D.S. thanks Ravi Kunjwal for useful discussions. This
research was supported by a Discovery grant of the
Natural Sciences and Engineering Research Council
of Canada and by Perimeter Institute for Theoretical
Physics. Research at Perimeter Institute is supported by
the Government of Canada through the Department of
Innovation, Science and Economic Development Canada
and by the Province of Ontario through the Ministry of
Research, Innovation and Science.
———————————————————————–
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