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arXiv:1704.01153 · New Journal of Physics

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

Anirudh Krishna, Robert W. Spekkens, Elie Wolfe

Published Oct 5, 2017• 31 citations
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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.

DOI
10.1088/1367-2630/aa9168

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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.
The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors-the context-are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.
When it isn't possible to tell two distinct experimental procedures apart purely from their input/output statistics, then it seems a plausible hypothesis that the two procedures must be physically identical. We call such a hypothesis "noncontextuality", an instance of Leibniz's principle of the identity of indiscernibles. Read in the contrapositive, this hypothesis entails that any physical distinctions between two experimental procedures must necessarily lead to a difference in their operational statistics. The results I present in this thesis concern the failure of this hypothesis -- a failure dubbed "contextuality" -- when one tries to embed an operational theory (such as quantum theory) in the ontological models framework. The Kochen-Specker theorem demonstrates the failure of noncontextuality for deterministic ontological models of quantum theory, i.e., those ontological models where the ontic/physical state of the system fixes the outcome of any projective measurement on the system in a deterministic manner. This thesis goes beyond the Kochen-Specker (KS) theorem by asking what operational facts must be verified in experiments to conclude that Nature does not admit noncontextual ontological models, not even indeterministic ones. This leads to noncontextuality inequalities that are robust to noise in the preparations and measurements. In the particular case of quantum theory, these inequalities are meaningful even when unsharp measurements (or POVMs) are allowed, a feature not shared by the traditional approach to KS-noncontextuality where unsharp measurements are excluded by fiat: allowing them renders even trivial POVMs (proportional to identity) maximally KS-contextual. The sense in which trivial POVMs are indeed "trivial" (or "noncontextual") is clear in our approach: they are simply too noisy to lead to violation of our noncontextuality inequalities.
A characterization of noncontextual models which fall within the ambit of Fine's theorem is provided. In particular, the equivalence between the existence of three notions is made explicit: a joint probability distribution over the outcomes of all the measurements considered, a measurement-noncontextual and outcome-deterministic (or KS-noncontextual, where KS stands for Kochen-Specker) model for these measurements, and a measurement-noncontextual and factorizable model for them. A KS inequality, therefore, is implied by each of these three notions. Following this characterization of noncontextual models that fall within the ambit of Fine's theorem, nonfactorizable noncontextual models which lie outside the domain of Fine's theorem are considered. While outcome determinism for projective (sharp) measurements in quantum theory can be shown to follow from the assumption of preparation noncontextuality, such a justification is not available for nonprojective (unsharp) measurements which ought to admit outcome-indeterministic response functions. The Liang-Spekkens-Wiseman (LSW) inequality is cited as an example of a noncontextuality inequality that should hold in any noncontextual model of quantum theory without assuming factorizability. Three other noncontextuality inequalities, which turn out to be equivalent to the LSW inequality under relabelings of measurement outcomes, are derived for Specker's scenario. The polytope of correlations admissible in this scenario, given the no-disturbance condition, is characterized.
The testability of the Kochen-Specker theorem is a subject of ongoing controversy. A central issue is that experimental implementations relying on sequential measurements cannot achieve perfect compatibility between the measurements and that therefore the notion of noncontextuality does not apply. We demonstrate by an explicit model that such compatibility violations may yield a violation of noncontextuality inequalities, even if we assume that the incompatibilities merely originate from context-independent noise. We show, however, that this problem can be circumvented by combining the ideas behind Leggett-Garg inequalities with those of the Kochen-Specker theorem.
Generalized noncontextuality as defined by Spekkens is an attempt to make precise the distinction between classical and quantum theories. It is a restriction on models used to reproduce the predictions of quantum mechanics. There has been considerable progress recently in deriving experimentally robust noncontextuality inequalities. The violation of these inequalities is a sufficient condition for an experiment to not admit a generalized noncontextual model. \nIn this thesis, we present an algorithm to automate the derivation of noncontextuality inequalities. At the heart of this algorithm is a technique called Fourier Motzkin elimination (abbrev. FM elimination). \nAfter a brief overview of the generalized notion of contextuality and FM elimination, we proceed to demon- strate how this algorithm works by using it to derive noncontextuality inequalities for a number of examples. \nBelinfante’s construction, presented first for the sake of pedagogy, is based on a simple proof by Belinfante to demonstrate the invalidity of von Neumann’s notion of noncontextuality. We provide a quantum realization on a single qubit that can violate this inequality. \nWe then go on to discuss paradigmatic proofs of noncontextuality such as the Peres-Mermin square, Mermin’s Star and the 18 vector construction and use them to derive robust noncontextuality inequalities using the algorithm. \nWe also show how one can generalize the noncontextuality inequalities obtained for the Peres-Mermin square to systems with continuous variables.
Contextuality provides one of the fundamental characterizations of quantum phenomena, and can be used as a resource in lots of quantum information processes. In this letter, we derive a class of gauge-like equivalent noncontextual inequalities from different noncontextual models of the proofs for Kochen-Specker theorem based on Greenberger-Horne-Zeilinger states. These noncontextual inequalities are equivalent up to some correlation items holding both for noncontextual hidden variable theories and quantum mechanics, and can be converted to an extreme simplest Mermin-like inequality. Then, using single-photon hyperentangled Greenberger-Horne-Zeilinger states encoded by spin, path and orbital angular momentum, we experimentally test all these state-dependent noncontextual models of the proofs for the Kochen-Specker theorem only by checking the extreme simplest Mermin-like inequality. The experiment results show that the Mermin-like inequality can be obviously violated and quantum mechanics is exclusively contextual.