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arXiv:quant-ph/0103068 · Journal of Physics A Mathematical and General

Spectral decomposition of Bell's operators for qubits

Valerio Scarani, Nicolas Gisin

Published Jul 20, 2001• 147 citations
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Abstract

The spectral decomposition is given for the N -qubit Bell operators with two observables per qubit. It is found that the eigenstates (when non-degenerate) are N -qubit GHZ states even for those operators that do not allow the maximal violation of the corresponding inequality. We present two applications of this analysis. In particular, we discuss the existence of pure entangled states that do not violate the Mermin-Klyshko inequality for N ⩾3.

DOI
10.1088/0305-4470/34/30/314

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The spectral decomposition is given for the N -qubit Bell operators with two observables per qubit. It is found that the eigenstates (when non-degenerate) are N -qubit GHZ states even for those operators that do not allow the maximal violation of the corresponding inequality. We present two applications of this analysis. In particular, we discuss the existence of pure entangled states that do not violate the Mermin-Klyshko inequality for N ⩾3.
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.
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.
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.
In the case of bipartite two-qubit systems, we derive an analytical expression of bound Bell operator for any given pure state. Our result not only manifests some properties of Bell inequality, for example, which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also gives the explicit values of maximal violation for any pure state. Finally we point out that any mixed states which can produce maximal violation of Bell inequality must have a maximal concurrence value.