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arXiv:1205.4958 · Journal of Physics A Mathematical and Theoretical

Degrees of entanglement for multipartite systems

A. I. Solomon, C.-L. Ho, Gérard Duchamp

Published Jan 29, 2013• 0 citations

Abstract

We propose a unified mathematical scheme, based on a classical tensor isomorphism, for characterizing entanglement that works for pure states of multipartite systems of any number of particles. The degree of entanglement is indicated by a set of absolute values of the determinants for each subspace of the multipartite systems. Our scheme provides a characterization of the degrees of entanglement when the qubits are measured or lost successively, and leads naturally to necessary and sufficient conditions for multipartite pure states to be separable. For systems with a large number of particles, a rougher indication of the degree of entanglement is provided by the set of mean values of the determinantal values for each subspace of the multipartite systems.

DOI: 10.1088/1751-8113/46/7/075302

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We propose a unified mathematical scheme, based on a classical tensor isomorphism, for characterizing entanglement that works for pure states of multipartite systems of any number of particles. The degree of entanglement is indicated by a set of absolute values of the determinants for each subspace of the multipartite systems. Our scheme provides a characterization of the degrees of entanglement when the qubits are measured or lost successively, and leads naturally to necessary and sufficient conditions for multipartite pure states to be separable. For systems with a large number of particles, a rougher indication of the degree of entanglement is provided by the set of mean values of the determinantal values for each subspace of the multipartite systems.

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The purpose of this paper is to obtain a sufficient and necessary condition as a criteria to test whether an arbitrary multipartite state is entangled or not. Based on the tensor expression of a multipartite pure state, the paper shows that a state is separable iff $| \mathbf{C}(ρ)| $ =0 for pure states and iff

C(ρ)

vanishes for mixed states.

We propose an entanglement tensor to compute the entanglement of a general pure multipartite quantum state. We compare the ensuing tensor with the concurrence for bipartite state and apply the tensor measure to some interesting examples of entangled three-qubit and four-qubit states. It is shown that in defining the degree of entanglement of a multi-partite state, one needs to make assumptions about the willingness of the parties to cooperate. We also discuss the degree of entanglement of the multi-qubit

\ket{W_{M}}

-states.