Gisin's Theorem for Three Qubits

Abstract

We present a Theorem that all generalized Greenberger-Horne-Zeilinger states of a three-qubit system violate a Bell inequality in terms of probabilities. All pure entangled states of a three-qubit system are shown to violate a Bell inequality for probabilities; thus, one has Gisin's theorem for three qubits.PACS numbers: 03.65. Ud, 03.67.Mn, Quantum mechanics violates Bell type inequalities that hold for any local-realistic theory [1,2,3,4,5]. In 1991, Gisin presented a theorem, which states that any pure entangled state of two particles violates a Bell inequality for two-particle correlation functions [6,7]. Bell's inequalities for systems of more than two qubits are the object of renewed interest, motivated by the fact that entanglement between more than two quantum systems is becoming experimentally feasible. Recent investigations show the surprising result that there exists a family of pure entangled N > 2 qubit states that do not violate any Bell inequality for N -particle correlations for the case of a standard Bell experiment on N qubits [8]. By a standard Bell experiment we mean the one in which each local observer is given a choice between two dichotomic observables [9,10,11,12]. This family is the generalized Greenberger-Horne-Zeilinger (GHZ) states given by( 1) with 0 ≤ ξ ≤ π/4. The GHZ states [3] are for ξ = π/4. In 2001, Scarani and Gisin noticed that for sin 2ξ ≤ 1/ √ 2 N −1 the states (1) do not violate the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequalities. Based on which, Scarani and Gisin wrote that "this analysis suggests that MK [in Ref. [9], MABK] inequalities, and more generally the family of Bell's inequalities with two observables per qubit, may not be the 'natural' generalizations of the CHSH inequality to more than two qubits" [8], where CHSH stands for Clauser-Horne-Shimony-Holt. In Ref.[10]Żukowski and Brukner (ŻB) have derived a general Bell inequality for correlation functions for N qubits. TheŻB inequalities include MABK inequalities as special cases. Ref. [9] shows that (a) For N = even, although the generalized GHZ state (1) does not violate MABK inequalities, it violates theŻB inequality and (b) For sin 2ξ ≤ 1/ √ 2 N −1 and N = odd, the correlations between measurements on qubits in the generalized GHZ state (1) satisfy all Bell inequalities for correlation functions, which involve two dichotomic observables per local measurement station.In this Letter, we focus on a three-qubit system, whose corresponding generalized GHZ state reads |ψ GHZ = cos ξ|000 + sin ξ|111 . Up to now, there is no Bell inequality violated by this pure entangled state for the region ξ ∈ (0, π/12] based on the standard Bell experiment. Can Gisin's theorem be generalized to three-qubit pure entangled states? Can one find a Bell inequality that violates |ψ GHZ for the whole region? In the following, we first present a theorem that all generalized GHZ states of a three-qubit system violate a Bell inequality in terms of probabilities; second, we will provide a universal Bell inequality for probabilities tha...

Identification of three-qubit entanglement

“…An important approach to characterize entanglement is Bell inequality [3][4][5][6][7][8]. For instance, N. Gisin has proved that all two-qubit pure entangled states violate the CHSH inequality [4].…”

On Bell-Like Inequalities and Pseudo-Hermitian Operators

“…Soon after, Gisin and Peres provided an elegant proof of this theorem for the case of pure two-qudit systems [24]. Chen et al showed that all pure entangled three-qubit states violate a Bell inequality [25]. Nevertheless, it still remains open whether Gisin's theorem can be generalized to the multi-qudit case.…”

Bell inequality for qubits based on the Cauchy-Schwarz inequality

“…For the three-qubit case, such a difficulty has been overcome in Ref. [11], where a probabilistic Bell inequality was proposed and consequently Gisin's theorem for three qubits naturally returned. Recent development also indicates that Bell inequality is not unique when one studies Gisin's theorem for three qubits, in Ref.…”