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arXiv:quant-ph/0408039 · Physical Review A

Comment on “All quantum observables in a hidden-variable model must commute simultaneously”

Koji Nagata

Published Jun 14, 2006• 4 citations
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Abstract

Malley discussed [Phys. Rev. A 69, 022118 (2004)] that all quantum observables in a hidden-variable model for quantum events must commute simultaneously. In this comment, we discuss that Malley's theorem is indeed valid for the hidden-variable theoretical assumptions, which were introduced by Kochen and Specker. However, we give an example that the local hidden-variable (LHV) model for quantum events preserves noncommutativity of quantum observables. It turns out that Malley's theorem is not related to the LHV model for quantum events, in general.

DOI
10.1103/physreva.73.066101

Related papers

Reply to “Comment on ‘All quantum observables in a hidden-variable model must commute simultaneously’” [Phys. Rev. A73, 066101 (2006)]

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Nagata [Phys. Rev. A 73, 066101 (2006)] questions whether a general no-go theorem of Malley [Phys. Rev. A 69, 022118 (2004)] applies to local hidden variables and outlines a ``counterexample.'' In fact this is not a counterexample at all, but in seeing why it fails we clarify the significance of Malley's result and its relation to other no-go theorems.

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All quantum observables in a hidden-variable model must commute simultaneously

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Under a standard set of assumptions for a hidden-variable model for quantum events we show that all observables must commute simultaneously. This seems to be an ultimate statement about the inapplicability of the usual hidden-variable model for quantum events. And, despite Bell's complaint that a key condition of von Neumann's was quite unrealistic, we show that these conditions, under which von Neumann produced the first no-go proof, are entirely equivalent to those introduced by Bell and Kochen and Specker. As these conditions are also equivalent to those under which the Bell-Clauster-Horne inequalities are derived, we see that the experimental violations of the inequalities demonstrate only that quantum observables do not commute.

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arXiv:quant-ph/0505223•Physical Review A•Jul 20, 2005•2 citations
Malley discussed [Phys. Rev. A 69, 022118 (2004)] that all quantum observables in a hidden-variable model for quantum events must commute simultaneously. In this comment, we discuss that Malley's theorem is indeed valid for the hidden-variable theoretical assumptions, which were introduced by Kochen and Specker. However, we give an example that the local hidden-variable (LHV) model for quantum events preserves noncommutativity of quantum observables. It turns out that Malley's theorem is not related to the LHV model for quantum events, in general.
Malley and Fine (Phys. Rev. A 2006 73 066102) discussed that all quantum observables would commute simultaneously if we accept a realistic theory of the Bell type (a local realistic theory) for quantum events, provided that all quantum events, including every quantum state and every observable (including every projector), are reproduced by a realistic theory of the Bell type for quantum events. We point out that their claim leads us to a contradiction. We study the relation between a local realistic theory and commutativity of quantum observables in a finite-dimensional space. We show that a realistic theory of the Bell type for quantum events does not impose the commutative algebraic structure into the set of all quantum observables if all quantum events are reproduced by a local realistic theory. We discuss that a violation of Bell locality is derived within a realistic theory of the Kochen?Specker type within quantum events.
Bell nonlocality is an intriguing property of quantum mechanics with far reaching consequences for information processing, philosophy and our fundamental understanding of nature. However, nonlocality is a statement about static correlations only. It does not take into account dynamics, i.e. time evolution of those correlations. Consider a dynamic situation where the correlations remain local for all times. Then at each moment in time there exists a local hidden-variable (LHV) model reproducing the momentary correlations. Can the time evolution of the correlations then be captured by evolving the hidden variables? In this light, we define dynamical LHV models and motivate and discuss potential additional physical and mathematical assumptions. Based on a simple counter example we conjecture that such LHV dynamics does not always exist. This is further substantiated by a rigorous no-go theorem. Our results suggest a new type of nonlocality that can be deduced from the observed time evolution of measurement statistics and which generically occurs in interacting quantum systems.
Under a standard set of assumptions for a hidden-variable model for quantum events we show that all observables must commute simultaneously. This seems to be an ultimate statement about the inapplicability of the usual hidden-variable model for quantum events. And, despite Bell's complaint that a key condition of von Neumann's was quite unrealistic, we show that these conditions, under which von Neumann produced the first no-go proof, are entirely equivalent to those introduced by Bell and Kochen and Specker. As these conditions are also equivalent to those under which the Bell-Clauster-Horne inequalities are derived, we see that the experimental violations of the inequalities demonstrate only that quantum observables do not commute.
We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.
A formula for the commutator of tensor product matrices is used to show that, for qubits, compatibility of quantum multiparty observables almost never implies local compatibility at each site, and to predict when this happens/does not happen in a concise manner. In particular, it is shown that two ``fully nontrivial'' nnn-qubit observables are compatible locally and globally if and only if they are equal up to sign. In addition, the formula gives insight into the construction of paradoxes of the type of the Kochen-Specker theorem, which can then be easily rephrased into proposals for no-hidden-variable experiments of the type of the ``Bell theorem without inequalities.''