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Gisin's theorem cannot be generalized to all multi-particle states

Marek Żukowski, Časlav Brukner, Wiesław Laskowski, Marcin Wieśniak

Published Oct 15, 2001• 1 citation
OpenAlex

Abstract

Gisin's theorem states that any pure entangled state of two particles violates Clauser-Horne-Shimony-Holt inequality which is a bound on correlations between measurements on two particles obtained within local realistic picture. We find that this theorem cannot be generalized to all multi-particle states for the case where measurements on each particle can be chosen between two arbitrary dichotomic observables. In particular we find a family of pure entangled states of N qubits which do not violate any Bell inequality for correlations between such measurements. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description of correlations.

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Gisin's theorem states that any pure entangled state of two particles violates Clauser-Horne-Shimony-Holt inequality which is a bound on correlations between measurements on two particles obtained within local realistic picture. We find that this theorem cannot be generalized to all multi-particle states for the case where measurements on each particle can be chosen between two arbitrary dichotomic observables. In particular we find a family of pure entangled states of N qubits which do not violate any Bell inequality for correlations between such measurements. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description of correlations.
We derive a single general Bell inequality which is a sufficient and necessary condition for the correlation function for N particles to be describable in a local and realistic picture, for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables. We also derive a necessary and sufficient condition for an arbitrary N-qubit mixed state to violate this inequality. This condition is a generalization and reformulation of the Horodecki family condition for two qubits.
Abstract We present a single general Bell inequality that summarizes all possible Bell type constrains on the correlation function for the N‐particle system (i.e., it is a necessary and sufficient condition for the existence of local realistic models for the correlation function). It is applicable in the case when measurements on each particle can be chosen between two arbitrary dichotomic observables. We also obtain a necessary and sufficient condition for an arbitrary N‐qubit mixed state to violate this general Bell inequality.
In this Rapid Communication, we show analytically that all pure entangled states of two ddd-dimensional systems (qudits) violate the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality. This property was pointed out by Gisin in the qubit case and then generalized by Gisin-Peres and Popescu-Rohrlich to the qudit case based on the Clauser-Horne-Shimony-Holt (CHSH) inequality. We report the Gisin's theorem for two qudits by making use of the CGLMP inequality.
We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to nnn qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.