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arXiv:quant-ph/0505223 · Physical Review A

Commuting multiparty quantum observables and local compatibility

Claudio Altafini

Published Jul 20, 2005• 2 citations
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Abstract

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'' $n$-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.''

DOI
10.1103/physreva.72.012112

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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'' $n$-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.''
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.
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.
We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the NNN-qubit Pauli group for $N\ensuremath{\ge}4$, thus adding to the proofs that have been presented earlier for the 2- and 3-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has OOO observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2O{}^{O}O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the NNN-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members include all numbers of qubits from 2 up, and show how it can be used to generate projectors-based KS proofs involving only 9 bases (or experimental contexts) in any dimension of the form 2N{2}^{N}2N for $N\ensuremath{\ge}2$. Some implications of our results are discussed.
Quantum mechanics is a nonlocal theory, but not as nonlocal as the no-signalling principle allows. However, there exist quantum correlations that exhibit maximal nonlocality: they are as nonlocal as any nonsignalling correlation and thus have a local content, quantified by the fraction pL{p}_{L}pL​ of events admitting a local description, equal to zero. We exploit the known link between the Kochen-Specker and Bell theorems to derive a maximal violation of a Bell inequality from every Kochen-Specker proof. We then show that these Bell inequalities lead to experimental bounds on the local content of quantum correlations that are significantly better than those based on other constructions. We perform the experimental demonstration of a Bell test originating from the Peres-Mermin Kochen-Specker proof, providing an upper bound on the local content ${p}_{L}\ensuremath{\lesssim}0.22$.