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Bell-like inequalities from symmetrized products of noncommuting observables on qubits

Omar Gamel, Graham R. Fleming

Published Sep 28, 2015• 0 citations
OpenAlex

Abstract

It is well known that noncommuting observables cannot be simultaneously measured in quantum theory. However, under hidden variable models, such observables must simultaneously hold premeasurement values. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fr\'echet bounds, finding that the joint probabilities may be negative. We then use the symmetrization heuristic to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct generalized Bell inequalities with a quantum to classical violation of \frac{3}{2}. The result demonstrates the importance of noncommutativity in the nonclassicality of quantum mechanics. However, the additional symmetrization assumption makes its conclusions weaker than traditional Bell inequalities.

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It is well known that noncommuting observables cannot be simultaneously measured in quantum theory. However, under hidden variable models, such observables must simultaneously hold premeasurement values. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fr'echet bounds, finding that the joint probabilities may be negative. We then use the symmetrization heuristic to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct generalized Bell inequalities with a quantum to classical violation of \frac{3}{2}. The result demonstrates the importance of noncommutativity in the nonclassicality of quantum mechanics. However, the additional symmetrization assumption makes its conclusions weaker than traditional Bell inequalities.
Noncommuting observables cannot be simultaneously measured; however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fréchet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct Quasi-Bell inequalities, Bell inequalities containing products of noncommuting observables, on two qubits. These inequalities place limits on the local hidden variable models that define joint probabilities for noncommuting observables. We find that the Quasi-Bell inequalities have a quantum to classical violation as high as 32 on two qubit, higher than conventional Bell inequalities. The result demonstrates the theoretical importance of noncommutativity in the nonlocality of quantum mechanics and provides an insightful generalization of Bell inequalities.
We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of noncommuting projectors. We present local and nonlocal examples of violation of generalized Bell inequalities in phase space, which sense the negativity of the Wigner function.
A Bell Inequality with Quantum to Classical Violation 32\frac{3}{2}23​
We introduce a Bell inequality in a bipartite spin-12\frac{1}{2}21​ system with a quantum to classical violation ratio of 32\frac{3}{2}23​, exceeding the 2\sqrt{2}2​ violation of the Clauser-Horne-Shimony-Holt inequality. The increase is allowed by the use of products of non-commuting observables symmetrized over all possible orderings in the quantization. This shows that the non-commutativity of operations intrinsic to quantum mechanics contributes to its violation of local realism. It is shown that generalizing the inequality to high order does not increase the violation. This, along with a hidden variable model found by Toner et. al., leads us to conjecture that 32\frac{3}{2}23​ is the maximal possible violation.
This article takes up a suggestion that the reason we cannot find certain hidden variable theories for quantum mechanics, as in Bell’s theorem, is that we require them to assign joint probability distributions on incompatible observables. These joint distributions are problematic because they are empirically meaningless on one standard interpretation of quantum mechanics. Some have proposed getting around this problem by using generalized probability spaces. I present a theorem to show a sense in which generalized probability spaces can’t serve as hidden variable theories for quantum mechanics, so the proposal for getting around Bell’s theorem fails. 1 Introduction2 Bell’s Theorem and Classical Probability Spaces 2.1 Bell’s derivation of the Bell inequalities 2.2 Pitowsky’s derivation of the Bell inequalities3 Incompatible Observables4 Generalized Probability Spaces5 A ‘No-Go’ Theorem6 Conclusions
Joint quantum measurements of noncommuting observables are possible, if one accepts an increase in the measured variances. A necessary condition for a joint measurement to be possible is that a joint probability distribution exists for the measurement. This fact suggests that there may be a link with Bell inequalities, as these will be satisfied if and only if a joint probability distribution for all involved observables exists. We investigate the connections between Bell inequalities and conditions for joint quantum measurements to be possible. Mermin's inequality for the three-particle Greenberger-Horne-Zeilinger state turns out to be equivalent to the condition for a joint measurement on two out of the three quantum systems to exist. Gisin's Bell inequality for three coplanar measurement directions, meanwhile, is shown to be less strict than the condition for the corresponding joint measurement.
Bell inequalities were meant to test quantum mechanics vs local hidden variable models, but can also be used to verify entanglement. For entanglement verification purposes one assumes the validity of quantum mechanics as well as quantum descriptions of one's measurements. With the help of these assumptions it is possible to derive a strengthened Bell inequality whose violation implies entanglement. We generalize known examples of such inequalities by relating the expectation value of the Bell operator to a particular quantitative measure of entanglement, namely, the negativity. Moreover, we obtain statistics illustrating the fact that violating a given (strengthened or not) Bell inequality is a much more rare feat for a quantum state of two qubits than it is to be entangled.