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arXiv:quant-ph/0403175 · Physical Review Letters

Generalizing Tsirelson's Bound on Bell Inequalities Using a Min-Max Principle

Stefan Filipp, Karl Svozil

Published Sep 23, 2004• 37 citations
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Abstract

Bounds on the norm of quantum operators associated with classical Bell-type inequalities can be derived from their maximal eigenvalues. This quantitative method enables detailed predictions of the maximal violations of Bell-type inequalities.
DOI
10.1103/physrevlett.93.130407

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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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The first part of this paper contains an introduction to Bell inequalities and Tsirelson's theorem for the non-specialist. The next part gives an explicit optimum construction for the “hard” part of Tsirelson's theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson's theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.
We consider a subclass of bipartite CHSH-type Bell inequalities. We investigate operations which leave their Tsirelson bound invariant, but change their classical bound. The optimal observables are unaffected except for a relative rotation of the two laboratories. We illustrate the utility of these operations by giving explicit examples. We prove that, for a fixed quantum state and fixed measurement setup except for a relative rotation of the two laboratories, there is a Bell inequality that is maximally violated for this rotation, and we optimize some Bell inequalities with respect to the maximal violation. Finally, we optimize the qutrit to qubit ratio of some dimension witnessing Bell inequalities. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.