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Communications in Theoretical Physics

Entanglement Criterion of N -Qubit State

Zhang Hua, Wu Xiao-Hua

Published Apr 1, 2006• 0 citations
Publisher

Abstract

In this paper, we shall develop a generic scheme to construct the criterion that an N-qubit state has true N -particle entanglement. For the N = 3 case, we show that violation of the Mermin's inequality | E LHV ( x ⊗ y ⊗ y + y ⊗ x ⊗ y + y ⊗ y ⊗ x − x ⊗ x ⊗ x )|⩽2, is sufficient to confirm three-particle entanglement.

DOI
10.1088/0253-6102/45/4/006

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In this paper, we shall develop a generic scheme to construct the criterion that an N-qubit state has true N -particle entanglement. For the N = 3 case, we show that violation of the Mermin's inequality | E LHV ( x ⊗ y ⊗ y + y ⊗ x ⊗ y + y ⊗ y ⊗ x − x ⊗ x ⊗ x )|⩽2, is sufficient to confirm three-particle entanglement.
A recent paper [] presented a bound for the three-qubit Mermin inequality such that the violation of this bound indicates genuine three-qubit entanglement. We show that this bound can be improved for a specific choice of observables. In particular, if spin observables corresponding to orthogonal directions are measured at the qubits (e.g., XXX and YYY spin coordinates), then the bound is the same as the bound for states with a local hidden variable model. As a consequence, it can straightforwardly be shown that in the experiment described by ], genuine three-qubit entanglement was detected.
It is presented a criterion of efficiently detecting all pure entangled states of n qubits by observing fluctuation of Bell's operators in place of expectation in Bell's inequalities. The entanglement witnesses are some of MK Bell operators corresponding to Mermin-Klyshko inequalities, which are easily constructed by resolving a maximum problem and using a computable method, respectively. We illustrate generalized GHZ states, generalized W and pure three-qubit states, respectively, for seeing how good our criterion is to detect the entanglement of nnn qubits. Therefore, we eventually resolve the problem of efficiently detecting pure entangled states of n qubits.
The notions of three-particle entanglement and three-particle nonlocality are discussed in the light of Svetlichny's inequality [Phys. Rev. D 35, 3066 (1987)]. It is shown that there exist sets of measurements, which can be used to prove three-particle entanglement, but which are nevertheless useless at proving three-particle nonlocality. In particular, it is shown that the quantum predictions giving a maximal violation of Mermin's three-particle Bell inequality [Phys. Rev. Lett. 65, 1838 (1990)] can be reproduced by a hybrid hidden variables model in which nonlocal correlations are present only between two of the particles. It should be possible, however, to test the existence of both three-particle entanglement and three-particle nonlocality for any given quantum state via Svetlichny's inequality.
We propose a sufficient entanglement criterion for the multipartite systems based on quantum Fisher information. It complements the well-established positive partial transpose criterion in the sense that it is also able to detect bound entangled states. Moreover, it improves the well-known upper bound on the quantum Fisher information of separable multiqudit states introduced by Toth et al. [G. Toth and T. Vertesi, Phys. Rev. Lett. 120, 020506 (2018)] and hence detects a greater number of metrologically useful entangled states. This criterion is formulated as a simple inequality which must be fulfilled by the quantum Fisher information of separable states of multipartite systems. Violation of this inequality by a state reveals its entanglement and accordingly confirms its usefulness for quantum metrology. The efficiency of this criterion for detection of entanglement is illustrated through some examples.
A possibility to reveal the entanglement in generalized n-qubit two-parameter GHZ states, as well as in any n-qubit states, with the help of the Mermin and Ardehali inequalities from the collection generally called the Mermin–Ardehali–Belinskii–Klyshko inequalities has been studied. Formulas for the calculation of the Mermin and Ardehali correlation functions in any quantum n-qubit states are derived, and criteria of the violation of corresponding inequalities by specific states are obtained. A set of states that are absolutely insensitive to the Mermin and Ardehali operators is revealed. Modified Mermin and Ardehali operators are proposed, the set of which makes it possible to extend the class of n-qubit states, in which quantum correlations can be revealed.