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Russian Microelectronics

The computable measure of quantum entanglement of multiqubit states

Andrey Yu. Chernyavskiy

Published May 1, 2009• 5 citations
Publisher

Abstract

The search for adequate measures of the entanglement of pure quantum states for an arbitrary number of subsystems is one of the priority problems of quantum information science. In this paper, I present the computable measure of entanglement of an arbitrary quantum state. The measure is the generalization of the von Neumann entropy to an arbitrary number of subsystems.

DOI
10.1134/s106373970903007x

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The search for adequate measures of the entanglement of pure quantum states for an arbitrary number of subsystems is one of the priority problems of quantum information science. In this paper, I present the computable measure of entanglement of an arbitrary quantum state. The measure is the generalization of the von Neumann entropy to an arbitrary number of subsystems.
The quantification and classification of quantum entanglement is a very important and still open question of quantum information theory. In this paper, we describe an entanglement measure for multipartite pure states (the minimum of Shannon's entropy of orthogonal measurements). This measure is additive, monotone under LOCC, and coincides with the reduced von Neumann entropy on bipartite states. A method for numerical calculation of this measure by genetic algorithms is also presented. Moreover, the minimization of entropy technique is extended to fermionic states.
Consider a system consisting of n d-dimensional quantum particles and an arbitrary pure state $|\ensuremath{\Psi}〉$ of the whole system. Suppose we simultaneously perform complete von Neumann measurements on each particle. The Shannon entropy of the outcomes' joint probability distribution is a functional of the state $|\ensuremath{\Psi}〉$ and of n measurements chosen for each particle. Denote $S[\ensuremath{\Psi}]$ the minimum of this entropy over all choices of the measurements. We show that $S[\ensuremath{\Psi}]$ coincides with the entropy of entanglement for bipartite states. We compute $S[\ensuremath{\Psi}]$ for some special multipartite states: the hexacode state ∣H〉|H〉∣H〉 (n=6,(n=6,(n=6, d=2)d=2)d=2) and the determinant states ∣Detn〉|{\mathrm{Det}}_{n}〉∣Detn​〉 (d=n).(d=n).(d=n). The computation yields S[H]=4log2S[H]=4\mathrm{log}2S[H]=4log2 and S[Detn]=log(n!).S[{\mathrm{Det}}_{n}]=\mathrm{log}(n!).S[Detn​]=log(n!). Counterparts of the determinant state defined for d<nd<nd<n are also considered.
We propose a measure of entanglement between two subsystems of arbitrary (but finite) number of levels. This measure is invariant under unitary transformations of the subsystems and varies between 0 (product states) and 1 (maximum entangled states). The case of two two-level systems is worked out explicitly, where the known complementarity between one- and two-particle interference results in a very natural way from a sum rule. Generalizations to more than two subsystems are discussed. Here we show that for three particles there cannot exist a pure state that is completely characterized by three-particle entanglement alone. For the example of the Greenberger-Horne-Zeilinger (GHZ) state, an experimental setup is proposed, in which its corresponding two-particle entanglement would show up.