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Mermin's Bell inequality and operators' noncommutativity

Jose Luis Cereceda

Published Jul 3, 2000• 0 citations
OpenAlex

Abstract

The relationship between the noncommutativity of operators and the violation of the Bell inequality is exhibited in the light of the n-particle Bell-type inequality discovered by Mermin [PRL 65, 1838 (1990)]. It is shown that the maximal amount of violation of Mermin's inequality predicted by quantum mechanics decreases exponentially by a factor of 2^{-m/2} whenever any m among the n single-particle commutators happen to vanish.

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The relationship between the noncommutativity of operators and the violation of the Bell inequality is exhibited in the light of the n-particle Bell-type inequality discovered by Mermin [PRL 65, 1838 (1990)]. It is shown that the maximal amount of violation of Mermin's inequality predicted by quantum mechanics decreases exponentially by a factor of 2^{-m/2} whenever any m among the n single-particle commutators happen to vanish.
One of the essential features of quantum theory is the noncommutativity of observables. Moreover, under the non-contextual realistic interpretations of quantum theory, all quantum observables must commute {[J. D. Malley, Phys. Rev. A {\\bf 69}, 022118 (2004)]}. Experimental realizations of violations of Bell inequality say the noncommutativity of observables in an entangled state. Here we show that commutativity can be tested experimentally using uncorrelated states, with no dependence on the algebraic structure of the observables under study. The test is derived from the bounds of an inequality valid for all commuting observables. For nnn spin-1/2 systems we show that an nnn-partite uncorrelated state violates the inequality by an amount that grows exponentially with nnn.
Relations between Bell's inequality and noncommutativity of operators are discussed via the four operators involved in the Clauser et al inequality. The case of all operators commuting (i.e. the six commutators vanish) and the case of three out of the four operators mutually commuting (i.e. five commutators vanish) is shown to abide by the inequality. In the latter case a novel insight is unravelled. The Bell quantum bound is obeyed for the case when four commutators vanish. The probabilistic upper limit of the inequality is reviewed and shown to be 4. In any theory based on Hilbert space, the upper limit is .
In this paper, it is proved that the maximal violation of Mermin's inequalities of nnn qubits occurs only for GHZ's states and the states obtained from them by local unitary transformations. The key point of our argument involved here is by using the certain algebraic properties that Pauli's matrices satisfy, which is based on the determination of local spin observables of the associated Bell-Mermin operators.
In this paper, we introduce the motivation for Bell inequalities and give some background on two different types: CHSH and Mermin's inequalities. We present a proof for each and then show that certain quantum states can violate both of these inequalities. We introduce a new result stating that for four given measurement directions of spin, two respectively from Alice and two from Bob, which are able to produce a violation of the Bell inequality for an arbitrary shared quantum state will also violate the Bell inequality for a maximally entangled state. Then we provide another new result that characterizes all of the two-qubit states that violate Mermin's inequality.
The Mermin-Klyshko inequality for n spin-12\frac{1}{2}21​ particles and two dichotomic observables is generalized to n spin-sss particles and two maximal observables. It is shown that some multiparty multilevel Greenberger-Horne-Zeilinger states [A. Cabello, Phys. Rev. A 63, 022104 (2001)] maximally violate this inequality for any s.s.s. For a fixed n,n,n, the magnitude of the violation is constant for any s,s,s, which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-sss particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed s,s,s, the violation grows exponentially with n,n,n, which provides a generalization to any s of Mermin's conclusion for n $\mathrm{spin}\ensuremath{-} \frac{1}{2}$ particles [Phys. Rev. Lett. 65, 1838 (1990)].
We investigate the violation factor of the Bell-Mermin inequality. Until now, we use an assumption that the results of measurement are ±1\pm 1±1. In this case, the maximum violation factor is 2(n−1)/22^{(n-1)/2}2(n−1)/2. The quantum predictions by nnn-partite Greenberger-Horne-Zeilinger (GHZ) state violate the Bell-Mermin inequality by an amount that grows exponentially with nnn. Recently, a new measurement theory based on the truth values is proposed. The values of measurement outcome are either +1+1+1 or 0. Here we use the new measurement theory. We consider multipartite GHZ state. It turns out that the Bell-Mermin inequality is violated by the amount of 2(n−1)/22^{(n-1)/2}2(n−1)/2. The measurement theory based on the truth values provides the maximum violation of the Bell-Mermin inequality.