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The European Physical Journal D

Bell inequalities for a sensible family of local hidden variable theories testable at low detection efficiency

Emilio Santos

Published Mar 22, 2007• 12 citations
Publisher
DOI
10.1140/epjd/e2007-00133-6

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We address the problem of closing the detection efficiency loophole in Bell experiments, which is crucial for real-world applications. Every Bell inequality has a critical detection efficiency η that must be surpassed to avoid the detection loophole. Here, we propose a general method for reducing the critical detection efficiency of any Bell inequality to arbitrary low values. This is accomplished by entangling two particles in N orthogonal subspaces (e.g., N degrees of freedom) and conducting N Bell tests in parallel. Furthermore, the proposed method is based on the introduction of penalized N-product (PNP) Bell inequalities, for which the so-called simultaneous measurement loophole is closed, and the maximum value for local hidden-variable theories is simply the Nth power of the one of the Bell inequality initially considered. We show that, for the PNP Bell inequalities, the critical detection efficiency decays exponentially with N. The strength of our method is illustrated with a detailed study of the PNP Bell inequalities resulting from the Clauser-Horne-Shimony-Holt inequality.
The problem to compute the local hidden variable (LHV) value of a Bell inequality plays a central role in quantum nonlocality. In particular, this problem is the first step to characterize the LHV polytope in a given scenario. In this work, we establish a relation between the LHV value of bipartite Bell inequalities and the mathematical notion of excess of a matrix. Inspired by the strongly developed theory of excess, we derive several results having a direct impact in the field of quantum nonlocality. As consequence, we provide the LHV value for infinite families of bipartite Bell inequalities, with an unbounded number of measurement settings, without requiring to solve any optimization problem. We also find tight Bell inequalities for a large number of measurement settings. Furthermore, we show that the entire set of optimal LHV strategies of Bell inequalities, induced by certain Hadamard matrices, equals the entire set of vectors mutually unbiased to certain pair of bases.