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Limited preparation contextuality in quantum theory leads to Cirel'son bound

Manik Banik, Some Sankar Bhattacharya, Amit Mukherjee, Arup Roy, Andris Ambainis, Ashutosh Rai

Published Jun 17, 2015• 0 citations
OpenAlex

Abstract

Kochen-Specker (KS) theorem lies at the heart of the foundations of quantum mechanics. It establishes impossibility of explaining predictions of quantum theory by any noncontextual ontological model. The notion of KS contextuality has been generalized for arbitrary experimental procedures (preparation, measurement, and transformation procedure). Interestingly, it has been shown that preparation contextuality powers parity-oblivious multiplexing, a two party information theoretic game. Thus, using resources of a given operational theory, the maximum success probability achievable in such a game suffices as a \emph{bona-fide} measure of preparation contextuality for the underlying theory. In this work we show that preparation contextuality in quantum theory is more restricted compared to a general operational theory known as \emph{box world}. Moreover, we find that this limitation of quantum theory implies the quantitative bound on quantum nonlocality as depicted by the Cirel'son bound.

Related papers

Limited preparation contextuality in quantum theory and its relation to the Cirel'son bound

Manik Banik, Some Sankar Bhattacharya, Amit Mukherjee +3 more

The Kochen--Specker (KS) theorem lies at the heart of the foundations of quantum mechanics. It establishes the impossibility of explaining predictions of quantum theory by any noncontextual ontological model. Spekkens generalized the notion of KS contextuality in [Phys. Rev. A 71, 052108 (2005)] for arbitrary experimental procedures (preparation, measurement, and transformation procedures). Interestingly, later on it was shown that preparation contextuality powers parity-oblivious multiplexing [Phys. Rev. Lett. 102, 010401 (2009)], a two-party information theoretic game. Thus, using resources of a given operational theory, the maximum success probability achievable in such a game suffices as a bona fide measure of preparation contextuality for the underlying theory. In this work we show that preparation contextuality in quantum theory is more restricted compared to a general operational theory known as box world. Moreover, we find that this limitation of quantum theory implies the quantitative bound on quantum nonlocality as depicted by the Cirel'son bound.

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Contextuality for preparations, transformations, and unsharp measurements

Robert W. Spekkens

The Bell-Kochen-Specker theorem establishes the impossibility of a noncontextual hidden variable model of quantum theory, or equivalently, that quantum theory is contextual. In this paper, an operational definition of contextuality is introduced which generalizes the standard notion in three ways: (i) it applies to arbitrary operational theories rather than just quantum theory, (ii) it applies to arbitrary experimental procedures rather than just sharp measurements, and (iii) it applies to a broad class of ontological models of quantum theory rather than just deterministic hidden variable models. We derive three no-go theorems for ontological models, each based on an assumption of noncontextuality for a different sort of experimental procedure; one for preparation procedures, another for unsharp measurement procedures (that is, measurement procedures associated with positive-operator valued measures), and a third for transformation procedures. All three proofs apply to two-dimensional Hilbert spaces, and are therefore stronger than traditional proofs of contextuality.

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Robert W. Spekkens, D. H. Buzacott, A. J. Keehn +2 more

In a noncontextual hidden variable model of quantum theory, hidden variables determine the outcomes of every measurement in a manner that is independent of how the measurement is implemented. Using a generalization of this notion to arbitrary operational theories and to preparation procedures, we demonstrate that a particular two-party information-processing task, "parity-oblivious multiplexing," is powered by contextuality in the sense that there is a limit to how well any theory described by a noncontextual hidden variable model can perform. This bound constitutes a "noncontextuality inequality" that is violated by quantum theory. We report an experimental violation of this inequality in good agreement with the quantum predictions. The experimental results also provide the first demonstration of 2-to-1 and 3-to-1 quantum random access codes.

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Ravi Kunjwal

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.

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An ontological model of an operational theory is considered to be universally noncontextual if both preparation and measurement noncontextuality assumptions are satisfied in that model. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely linked to suitable Bell expression whose local bound is greater than universal non-contextual bound. Thus, for a given state, even if quantum theory does not exhibit non-locality, it may still reveal non-classicality by violating the universal non-contextual bound. Further, we consider a different communication game to demonstrate that for a given choices of observables in quantum theory, even if there is no logical proof of preparation and measurement contextuality exist, the universal quantum contextuality can be revealed through that communication game. Such a game thus test a weaker form of universal non-contextuality with minimal assumption.

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Maximally Epistemic Interpretations of the Quantum State and Contextuality

Matthew Leifer, O. J. E. Maroney

We examine the relationship between quantum contextuality (in both the standard Kochen-Specker sense and in the generalized sense proposed by Spekkens) and models of quantum theory in which the quantum state is maximally epistemic. We find that preparation noncontextual models must be maximally epistemic, and these in turn must be Kochen-Specker noncontextual. This implies that the Kochen-Specker theorem is sufficient to establish both the impossibility of maximally epistemic models and the impossibility of preparation noncontextual models. The implication from preparation noncontextual to maximally epistemic then also yields a proof of Bell's theorem from an Einstein-Podolsky-Rosen-like argument.

arXiv:1208.5132•Physical Review Letters•Mar 20, 2013•92 citations
Kochen-Specker (KS) theorem lies at the heart of the foundations of quantum mechanics. It establishes impossibility of explaining predictions of quantum theory by any noncontextual ontological model. The notion of KS contextuality has been generalized for arbitrary experimental procedures (preparation, measurement, and transformation procedure). Interestingly, it has been shown that preparation contextuality powers parity-oblivious multiplexing, a two party information theoretic game. Thus, using resources of a given operational theory, the maximum success probability achievable in such a game suffices as a \emph{bona-fide} measure of preparation contextuality for the underlying theory. In this work we show that preparation contextuality in quantum theory is more restricted compared to a general operational theory known as \emph{box world}. Moreover, we find that this limitation of quantum theory implies the quantitative bound on quantum nonlocality as depicted by the Cirel'son bound.
The Kochen--Specker (KS) theorem lies at the heart of the foundations of quantum mechanics. It establishes the impossibility of explaining predictions of quantum theory by any noncontextual ontological model. Spekkens generalized the notion of KS contextuality in [Phys. Rev. A 71, 052108 (2005)] for arbitrary experimental procedures (preparation, measurement, and transformation procedures). Interestingly, later on it was shown that preparation contextuality powers parity-oblivious multiplexing [Phys. Rev. Lett. 102, 010401 (2009)], a two-party information theoretic game. Thus, using resources of a given operational theory, the maximum success probability achievable in such a game suffices as a bona fide measure of preparation contextuality for the underlying theory. In this work we show that preparation contextuality in quantum theory is more restricted compared to a general operational theory known as box world. Moreover, we find that this limitation of quantum theory implies the quantitative bound on quantum nonlocality as depicted by the Cirel'son bound.
The Bell-Kochen-Specker theorem establishes the impossibility of a noncontextual hidden variable model of quantum theory, or equivalently, that quantum theory is contextual. In this paper, an operational definition of contextuality is introduced which generalizes the standard notion in three ways: (i) it applies to arbitrary operational theories rather than just quantum theory, (ii) it applies to arbitrary experimental procedures rather than just sharp measurements, and (iii) it applies to a broad class of ontological models of quantum theory rather than just deterministic hidden variable models. We derive three no-go theorems for ontological models, each based on an assumption of noncontextuality for a different sort of experimental procedure; one for preparation procedures, another for unsharp measurement procedures (that is, measurement procedures associated with positive-operator valued measures), and a third for transformation procedures. All three proofs apply to two-dimensional Hilbert spaces, and are therefore stronger than traditional proofs of contextuality.
In a noncontextual hidden variable model of quantum theory, hidden variables determine the outcomes of every measurement in a manner that is independent of how the measurement is implemented. Using a generalization of this notion to arbitrary operational theories and to preparation procedures, we demonstrate that a particular two-party information-processing task, "parity-oblivious multiplexing," is powered by contextuality in the sense that there is a limit to how well any theory described by a noncontextual hidden variable model can perform. This bound constitutes a "noncontextuality inequality" that is violated by quantum theory. We report an experimental violation of this inequality in good agreement with the quantum predictions. The experimental results also provide the first demonstration of 2-to-1 and 3-to-1 quantum random access codes.
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.
An ontological model of an operational theory is considered to be universally noncontextual if both preparation and measurement noncontextuality assumptions are satisfied in that model. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely linked to suitable Bell expression whose local bound is greater than universal non-contextual bound. Thus, for a given state, even if quantum theory does not exhibit non-locality, it may still reveal non-classicality by violating the universal non-contextual bound. Further, we consider a different communication game to demonstrate that for a given choices of observables in quantum theory, even if there is no logical proof of preparation and measurement contextuality exist, the universal quantum contextuality can be revealed through that communication game. Such a game thus test a weaker form of universal non-contextuality with minimal assumption.
We examine the relationship between quantum contextuality (in both the standard Kochen-Specker sense and in the generalized sense proposed by Spekkens) and models of quantum theory in which the quantum state is maximally epistemic. We find that preparation noncontextual models must be maximally epistemic, and these in turn must be Kochen-Specker noncontextual. This implies that the Kochen-Specker theorem is sufficient to establish both the impossibility of maximally epistemic models and the impossibility of preparation noncontextual models. The implication from preparation noncontextual to maximally epistemic then also yields a proof of Bell's theorem from an Einstein-Podolsky-Rosen-like argument.