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arXiv:quant-ph/0407029

Maximal violation of Mermin's inequalities

Zeqian Chen

Published Jul 5, 2004• 0 citations
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Abstract

In this paper, it is proved that the maximal violation of Mermin's inequalities of $n$ qubits occurs only for GHZ's states and the states obtained from them by local unitary transformations. The key point of our argument involved here is by using the certain algebraic properties that Pauli's matrices satisfy, which is based on the determination of local spin observables of the associated Bell-Mermin operators.

DOI
10.48550/arxiv.quant-ph/0407029

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In this paper, it is proved that the maximal violation of Mermin's inequalities of nnn qubits occurs only for GHZ's states and the states obtained from them by local unitary transformations. The key point of our argument involved here is by using the certain algebraic properties that Pauli's matrices satisfy, which is based on the determination of local spin observables of the associated Bell-Mermin operators.
In this paper, we characterize the maximal violation of Ardehali's inequality of nnn qubits by showing that GHZ's states and the states obtained from them by local unitary transformations are the unique states that maximally violate the Ardehali's inequalities. This concludes that Ardehali's inequalities can be used to characterize maximally entangled states of nnn qubits, as the same as Mermin's and Bell-Klyshko's inequalities.
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.
This Letter presents the first rigorous proof of the conjecture raised by Gisin and Bechmann-Pasquinucci [Phys. Lett. A 246, 1 (1998)]], that the Greenberger-Horne-Zeilinger states of n qubits and the states obtained from them by local unitary transformations are the unique states that maximally violate the Bell-Klyshko inequalities. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy and some subtle mathematical techniques. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give a characterization of maximally entangled states of n qubits in terms of the Bell-type inequality.
In this paper, we introduce the motivation for Bell inequalities and give some background on two different types: CHSH and Mermin's inequalities. We present a proof for each and then show that certain quantum states can violate both of these inequalities. We introduce a new result stating that for four given measurement directions of spin, two respectively from Alice and two from Bob, which are able to produce a violation of the Bell inequality for an arbitrary shared quantum state will also violate the Bell inequality for a maximally entangled state. Then we provide another new result that characterizes all of the two-qubit states that violate Mermin's inequality.