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Physical Review Letters

Maximal violation of Bell inequalities for mixed states

Samuel L. Braunstein, A. Mann, M. Revzen

Published Jun 1, 1992• 456 citations
Publisher

Abstract

We show that mixed states can produce maximal violations in the Bell inequality due to Clauser, Horne, Shimony, and Holt (CHSH). This follows from the degeneracy of the operator which is naturally associated with the Bell inequality (here called the Bell operator). We have calculated the form of all the eigenvalues for the generic CHSH Bell operator. Finally, we consider several examples which demonstrate the utility of studying the eigenvalues and eigenvectors of the Bell operator.

DOI
10.1103/physrevlett.68.3259

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We show that mixed states can produce maximal violations in the Bell inequality due to Clauser, Horne, Shimony, and Holt (CHSH). This follows from the degeneracy of the operator which is naturally associated with the Bell inequality (here called the Bell operator). We have calculated the form of all the eigenvalues for the generic CHSH Bell operator. Finally, we consider several examples which demonstrate the utility of studying the eigenvalues and eigenvectors of the Bell operator.
Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.
Bell‐type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson’s bound constitutes a global upper limit for quantum violations of the Clauser‐Horne‐Shimony‐Holt (CHSH) and the Clauser‐Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell‐type inequalities by solving the eigenvalue problem for the operator associated with each Bell‐type inequality. Thereby, we use the min‐max principle to calculate the norm of these self‐adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell‐type inequality is maximally violated.
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We study the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for two-spin systems, of arbitrary spins j1{j}_{1}j1​ and j2{j}_{2}j2​, prepared in an entanglement of spin-coherent states of each of the spins. We show that the Bell-CHSH inequality is quite robustly violated for a wide range of values of the parameters that specify the spin-coherent states, and, for a particular choice of these parameters, maximal violations are obtained. That is, the violations can reach the Tsirelson bound, 222\sqrt{2}22​, for any choices of the spins.