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Class of bipartite quantum states satisfying any Bell-type inequality

Elena R. Loubenets

Published Feb 14, 2005• 0 citations
OpenAlex

Abstract

We introduce a bipartite state property sufficient for the validity of a CHSH-form inequality and the perfect correlation form of the original Bell inequality for any bounded quantum observables. A nonseparable state with this property does not necessarily admit a local hidden variable model. 1

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In a general setting, we introduce a new bipartite state property sufficient for the validity of the perfect correlation form of the original Bell inequality for any three bounded quantum observables. A bipartite quantum state with this property does not necessarily exhibit perfect correlations. The class of bipartite states specified by this property includes both separable and nonseparable states. We prove analytically that, for any dimension d ⩾ 3, every Werner state, separable or nonseparable, belongs to this class.
For a nonseparable bipartite quantum state violating the Clauser-Horne-Shimony-Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both ”sides”. 1
For a nonseparable bipartite quantum state violating the Clauser–Horne–Shimony–Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both 'sides'.
Bell inequalities reveal when quantum correlations cannot be explained by any local hidden-variable model, and the CHSH inequality is the most fundamental test of this phenomenon. While entangled two-qubit states can violate CHSH inequalities with suitable local measurements, the situation is more subtle for higher-dimensional systems when the allowed observables are physically restricted. In this work, we study bipartite qutrit systems in which each party is restricted to spin-1 observables S(u)=uxSx+uySy+uzSzS(u)=u_xS_x+u_yS_y+u_zS_zS(u)=ux​Sx​+uy​Sy​+uz​Sz​, with u∈R3u\in\mathbb R^3u∈R3 a unit vector. Hanotel and Loubenets conjectured that nonseparable pure states of two qutrits do not violate the CHSH inequality under this measurement restriction. We prove the stronger result that every bipartite state on C3⊗C3\mathbb C^3\otimes\mathbb C^3C3⊗C3, pure or mixed, satisfies the CHSH inequality for all choices of local spin-1 observables. Thus the absence of CHSH violation is not a property of special states, but a structural limitation of the spin-1 measurement family. Our result clarifies a distinction between two-qutrit entanglement and CHSH nonlocality under natural spin-component measurements.
We introduce a general condition sufficient for the validity of the original Bell inequality (1964) in a local hidden variable (LHV) frame. This condition can be checked experimentally and incorporates only as a particular case the assumption on perfect correlations or anticorrelations usually argued for this inequality in the literature. Specifying this general condition for a quantum bipartite case, we introduce the whole class of bipartite quantum states, separable and nonseparable, that (i) admit an LHV description under any bipartite measurements with two settings per site; (ii) do not necessarily exhibit perfect correlations and may even have a negative correlation function if the same quantum observable is measured at both sites, but (iii) satisfy the ‘perfect correlation’ version of the original Bell inequality for any three bounded quantum observables A 1 , A 2 = B 1 , B 2 at sites ‘ A ’ and ‘ B ’, respectively. Analysing the validity of this general LHV condition under classical and quantum correlation scenarios with the same physical context, we stress that, unlike the Clauser–Horne–Shimony–Holt inequality, the original Bell inequality distinguishes between classicality and quantum separability.
We study the nonlocality of arbitrary dimensional bipartite quantum states. By computing the maximal violation of a set of multi-setting Bell inequalities, an analytical and computable lower bound has been derived for general two-qubit states. This bound gives the necessary condition that a two-qubit state admits no local hidden variable models. The lower bound is shown to be better than that from the CHSH inequality in judging the nonlocality of some quantum states. The results are generalized to the case of high dimensional quantum states, and a sufficient condition for detecting the non-locality has been presented.