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arXiv:1012.5640 · Journal of Physics A Mathematical and Theoretical

Joint measurements and Svetlichny's inequality

Yang Xiang, Wei Ren

Published Jul 20, 2011• 0 citations
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Abstract

We prove that Svetlichny's inequality can be derived from the existence of joint measurements and the principle of no-signaling. Then we show that, on the basis of the assumption of quantum measurement, it would imply the breach of causality if the magnitude of violation of Svetlichny's inequality exceeds the quantum bound.
DOI
10.1088/1751-8113/44/32/325305

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No signalling condition by itself does not answer the question why quantum-mechanics violates Bell's inequality by not more than 2sqrt22\\sqrt{2}2sqrt2. Recently Buhrman and Massar \\cite{massar} have given the answer by using unitarity and linearity of quantum-mechanics. We provide a simple answer to the same with the help of realistic joint measurement in quantum mechanics and Bell's inequality which has been derived under the assumption of existence of joint measurement and no signalling condition.
Quantum mechanics is not the unique no-signaling theory which is endowed with stronger-than-classical correlations, and there exists a broad class of no-signaling theories allowing even stronger-than-quantum correlations. The principle of information causality has been suggested to distinguish quantum theory from these nonphysical theories, together with an elegant information-theoretic proof of the quantum bound of two-particle correlations. In this work, we extend this to genuine NNN-particle correlations that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. We first express Svetlichny's inequality in terms of multipartite no-signaling boxes, then prove that the strongest genuine multipartite correlations lead to the maximal violation of information causality. The maximal genuine multipartite correlations under the constraint of information causality is found to be equal to the quantum mechanical bound. This result consolidates information causality as a physical principle defining the possible correlations allowed by nature, and provides intriguing insights into the limits of genuine multipartite correlations in quantum theory.
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We first present, by using exclusivity principle, a brief proof of the complementarity principle: the sum of squared expectation values of dichotomic (±1) mutually complementary observables can not be greater than 1. Then we prove that the complementarity principle yields tight quantum bounds of violations of N-qubit Svetlichny's inequalities. This result not only demonstrates that exclusivity principle can give tight quantum bound for certain type of genuine multipartite correlations, but also illustrates the subtle relationship between quantum complementarity and quantum genuine multipartite correlations.
Svetlichny's inequality (SI) is a Bell-like inequality, the violation of which can be used to confirm the existence of genuine multipartite correlations. However, poor detector efficiency can possibly cause a so-called detection loophole in actual Svetlichny experiments. We derive an alternative SI to deal with this loophole. If the experimental data can violate this SI, it must result in loophole-free violation of the original SI. We show that the minimum detection efficiency needed for a loophole-free violation of the tripartite SI is about 0.970.970.97. For the general case of nnn particles, we give the analytic expression of the needed detection efficiency, and find that its value monotonically and rapidly approaches 1 as the number of particles increases.