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arXiv:1204.3229 · Progress of Theoretical Physics

Conditional Measurement in Noncontextual Hidden-Variable Models

Kazuo Fujikawa

Published Jun 1, 2012• 1 citation
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Abstract

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.

DOI
10.1143/ptp.127.989

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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.
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.
It will be shown that the Peres-Mermin square admits value-definite noncontextual hidden-variable models if the observables associated with the operators can be measured only sequentially but not simultaneously. Namely, sequential measurements allow for noncontextual models in which hidden states update between consecutive measurements. Two recent experiments realizing the Peres-Mermin square by sequential measurements will also be analyzed along with other hidden-variable models accounting for these experiments.
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.
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.
Meyer [Phys. Rev. Lett. 83, 3751 (1999)] recently showed that the Kochen-Specker theorem, which demonstrates the impossibility of a deterministic hidden variable description of ideal spin measurements on a spin 1 particle, can be effectively nullified if only finite precision measurements are considered. We generalize this result: one can ascribe consistent outcomes to a dense subset of the set of projection valued measurements, or to a dense subset of the set of positive operator valued measurements, on any finite dimensional system. Hence no Kochen-Specker-like contradiction can rule out hidden variable theories indistinguishable from quantum theory by finite precision measurements in either class.
Since the enlightening proofs of quantum contextuality first established by Kochen and Specker, and also by Bell, various simplified proofs have been constructed to exclude the noncontextual hidden variable theory of our nature at the microscopic scale. The conflict between the noncontextual hidden variable theory and quantum mechanics is commonly revealed by Kochen-Specker sets of yes-no tests, represented by projectors (or rays), via either logical contradictions or noncontextuality inequalities in a state-(in)dependent manner. Here we propose a systematic and programmable construction of a state-independent proof from a given set of nonspecific rays in C3{\mathbb{C}}^{3}C3 according to their Gram matrix. This approach brings us a greater convenience in the experimental arrangements. Besides, our proofs in C3{\mathbb{C}}^{3}C3 can also be generalized to any higher-dimensional systems by a recursive method.