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arXiv:1501.01406 · The frontiers collection

A Quantum Mechanical Bound for CHSH-Type Bell Inequalities

Michael Epping, Hermann Kampermann, Dagmar Bruß

Published Nov 15, 2016• 1 citation
arXivHTMLPDF
DOI
10.1007/978-3-319-38987-5_13

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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.
It is well-known that no local model-in theory-can simulate the outcome statistics of a Bell-type experiment as long as the detection efficiency is higher than a threshold value. For the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality this theoretical threshold value is [Formula: see text]. On the other hand, Phys. Rev. Lett. 107, 170404 (2011) outlined an explicit practical model that can fake the CHSH inequality for a detection efficiency of up to 0.5. In this work, we close this gap. More specifically, we propose a method to emulate a Bell inequality at the threshold detection efficiency using existing optical detector control techniques. For a Clauser-Horne-Shimony-Holt inequality, it emulates the CHSH violation predicted by quantum mechanics up to [Formula: see text]. For the Garg-Mermin inequality-re-calibrated by incorporating non-detection events-our method emulates its exact local bound at any efficiency above the threshold. This confirms that attacks on secure quantum communication protocols based on Bell violation is a real threat if the detection efficiency loophole is not closed.