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arXiv:1602.02619

An analytical condition for the violation of Mermin's inequality by any three qubit state

Satyabrata Adhikari, A. S. Majumdar

Published Feb 8, 2016• 4 citations
arXivHTMLPDF

Abstract

Mermin's inequality is the generalization of the Bell-CHSH inequality for three qubit states. The violation of the Mermin inequality guarantees the fact that there exists quantum non-locality either between two or three qubits in a three qubit system. In the absence of an analytical result to this effect, in order to check for the violation of Mermin's inequality one has to perform a numerical optimization procedure for even three qubit pure states. Here we derive an analytical formula for the maximum value of the expectation of the Mermin operator in terms of eigenvalues of symmetric matrices, that gives the maximal violation of the Mermin inequality for all three qubit pure and mixed states.

DOI
10.48550/arxiv.1602.02619

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Mermin's inequality is the generalization of the Bell-CHSH inequality for three qubit states. The violation of the Mermin inequality guarantees the fact that there exists quantum non-locality either between two or three qubits in a three qubit system. In the absence of an analytical result to this effect, in order to check for the violation of Mermin's inequality one has to perform a numerical optimization procedure for even three qubit pure states. Here we derive an analytical formula for the maximum value of the expectation of the Mermin operator in terms of eigenvalues of symmetric matrices, that gives the maximal violation of the Mermin inequality for all three qubit pure and mixed states.
The violation of the Mermin inequality (MI) for multipartite quantum states guarantees the existence of nonlocality between either few or all parties. The detection of optimal MI violation is fundamentally important, but current methods only involve numerical optimization, thus hard to find even for three qubit states. In this paper, we provide a simple and elegant analytical method to achieve the tighter bound of Mermin operator for arbitrary three qubit states. Also the necessary and sufficient conditions for the tightness of the bound for some class of states has been stated.
Abstract The necessary and sufficient criteria for violating the Mermin and Svetlichny inequalities by arbitrary three-qubit states are presented. Several attempts have been made, earlier, to find such criteria, however, those extant criteria are neither tight for most of the instances, nor fully general. We generalize the existing criteria for Mermin and Svetlichny inequalities which are valid for the local projective measurement observables as well as for the arbitrary ones. We obtain the maximal achievable bounds of the Mermin and Svetlichny operators with unbiased measurement observables for arbitrary three-qubit states and with arbitrary observables for three-qubit states having maximally mixed marginals. We find that for certain ranges of measurement strengths, it is possible to violate Mermin and Svetlichny inequalities only by biased measurement observables. The necessary and sufficient criteria of violating any one of the six possible Mermin and Svetlichny inequalities are also derived.
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