Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1201.4421 · Physical Review A

Quantum discord and noncontextual hidden-variable models

Kazuo Fujikawa

Published Jan 31, 2012• 6 citations
arXivHTMLPDF

Abstract

It is shown that theoretically viable noncontextual hidden-variable models in $d=2$ lead to conflicting dispersion-free expressions in the analysis of the conditional measurement of two non-orthogonal projectors. No satisfactory criterion of the quantum discord, which relies on the analysis of conditional measurement, is formulated in the $d=2$ hidden-variable space due to a lack of uniqueness of the dispersion-free representation. We also make a speculative comment on a ``many-worlds interpretation'' of hidden-variable models to account for the conditional measurement.

DOI
10.1103/physreva.85.012114

Related papers

Quantum Discord, CHSH Inequality and Hidden Variables -- Critical reassessment of hidden-variables models

Kazuo Fujikawa

Hidden-variables models are critically reassessed. It is first examined if the quantum discord is classically described by the hidden-variable model of Bell in the Hilbert space with $d=2$. The criterion of vanishing quantum discord is related to the notion of reduction and, surprisingly, the hidden-variable model in $d=2$, which has been believed to be consistent so far, is in fact inconsistent and excluded by the analysis of conditional measurement and reduction. The description of the full contents of quantum discord by the deterministic hidden-variables models is not possible. We also re-examine CHSH inequality. It is shown that the well-known prediction of CHSH inequality $|B|\leq 2$ for the CHSH operator $B$ introduced by Cirel'son is not unique. This non-uniqueness arises from the failure of linearity condition in the non-contextual hidden-variables model in $d=4$ used by Bell and CHSH, in agreement with Gleason's theorem which excludes $d=4$ non-contextual hidden-variables models. If one imposes the linearity condition, their model is converted to a factored product of two $d=2$ models which describes quantum mechanical separable states. The CHSH inequality thus does not test the hidden-variables model in $d=4$. This observation is consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to dispersion-free representations. As for hidden-variables models, there exist no viable local non-contextual models in any dimensions.

arXiv:1302.0916•Feb 5, 2013•0 citations

Conditional Measurement in Noncontextual Hidden-Variable Models

Kazuo Fujikawa

The noncontextual hidden-variable models in d = 2, such as the ones constructed by Bell and by Kochen and Specker, have difficulties in accounting for the conditional measurement of two non-orthogonal projectors. An idea of branching in the hidden-variable space, which provides a means to realize the notion of reduction effectively and describe the state preparation, is suggested as a way to resolve the difficulties associated with the conditional measurement.

arXiv:1204.3229•Progress of Theoretical Physics•Jun 1, 2012•1 citation

Quantum violation of stochastic noncontextual hidden-variable theories

Shasanka Mohan Roy, Virendra Singh

We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state \ensuremath{\lambda} specifies not definite values, but expectation values A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}), . . . of observables A, . . . and noncontextuality means that A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (${2}^{\mathit{n}}$-1)/2, quantum violation of classical noncontextuality grows exponentially with n.

Physical Review A•Oct 1, 1993•21 citations

From the Kochen-Specker Theorem to Noncontextuality Inequalities without Assuming Determinism

Ravi Kunjwal, Robert W. Spekkens

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.

arXiv:1506.04150•Physical Review Letters•Sep 9, 2015•87 citations

Three noncontextual hidden variable models for the Peres-Mermin square

Gábor Hofer‐Szabó

I will argue that the Peres-Mermin square does not necessarily rule out a value-definite (deterministic) noncontextual hidden variable model if the operators are not given a physical interpretation satisfying the following two requirements: (i) each operator is uniquely realized by a single physical measurement; (ii) commuting operators are realized by simultaneous measurements. To underpin this claim, I will construct three hidden variable models for three different physical realizations of the Peres-Mermin square: one violating (i), another violating (ii), and a third one violating both (i) and (ii).

arXiv:2011.06508•Nov 1, 2020•8 citations

Quantum Measurements Are Noncontextual

Robert B. Griffiths

Quantum measurements are noncontextual, with outcomes independent of which other commuting observables are measured at the same time, when consistently analyzed using principles of Hilbert space quantum mechanics rather than classical hidden variables.

arXiv:1302.5052•Feb 20, 2013•1 citation
It is shown that theoretically viable noncontextual hidden-variable models in d=2d=2d=2 lead to conflicting dispersion-free expressions in the analysis of the conditional measurement of two non-orthogonal projectors. No satisfactory criterion of the quantum discord, which relies on the analysis of conditional measurement, is formulated in the d=2d=2d=2 hidden-variable space due to a lack of uniqueness of the dispersion-free representation. We also make a speculative comment on a ``many-worlds interpretation'' of hidden-variable models to account for the conditional measurement.
Hidden-variables models are critically reassessed. It is first examined if the quantum discord is classically described by the hidden-variable model of Bell in the Hilbert space with d=2d=2d=2. The criterion of vanishing quantum discord is related to the notion of reduction and, surprisingly, the hidden-variable model in d=2d=2d=2, which has been believed to be consistent so far, is in fact inconsistent and excluded by the analysis of conditional measurement and reduction. The description of the full contents of quantum discord by the deterministic hidden-variables models is not possible. We also re-examine CHSH inequality. It is shown that the well-known prediction of CHSH inequality ∣B∣≤2|B|\leq 2∣B∣≤2 for the CHSH operator BBB introduced by Cirel'son is not unique. This non-uniqueness arises from the failure of linearity condition in the non-contextual hidden-variables model in d=4d=4d=4 used by Bell and CHSH, in agreement with Gleason's theorem which excludes d=4d=4d=4 non-contextual hidden-variables models. If one imposes the linearity condition, their model is converted to a factored product of two d=2d=2d=2 models which describes quantum mechanical separable states. The CHSH inequality thus does not test the hidden-variables model in d=4d=4d=4. This observation is consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to dispersion-free representations. As for hidden-variables models, there exist no viable local non-contextual models in any dimensions.
We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state \ensuremath{\lambda} specifies not definite values, but expectation values A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}), . . . of observables A, . . . and noncontextuality means that A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (2n{2}^{\mathit{n}}2n-1)/2, quantum violation of classical noncontextuality grows exponentially with n.
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.
I will argue that the Peres-Mermin square does not necessarily rule out a value-definite (deterministic) noncontextual hidden variable model if the operators are not given a physical interpretation satisfying the following two requirements: (i) each operator is uniquely realized by a single physical measurement; (ii) commuting operators are realized by simultaneous measurements. To underpin this claim, I will construct three hidden variable models for three different physical realizations of the Peres-Mermin square: one violating (i), another violating (ii), and a third one violating both (i) and (ii).