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Journal of Liaoning Normal University

A Comparison of the Entanglement Measures for Multipartite Pure States

Cao Wan-Cang, Dan Liu, Bai Hong-bo

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

In this paper,a concise and alternative average entropy is introduced,which can be used to analyze entanglement of multipartite pure states.By comparing with the average von Neumann entropy established previously,the results show that the entanglement of multipartite pure states can also be measured in this new scheme.

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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.
The paper is an elaborate comparative assessment of the most relevant entropy-based quantities to be applied in quantifying quantum entanglement in pure, mixed, bipartite, and multipartite quantum systems. The study fills the existing gap in the literature in which most of the previous literature examines individual measures, but a lot of knowledge is lacking on what these measures are regarding their strengths, weaknesses, and the operational usefulness against each other. The study can prove the behaviour of each measure under noise, decoherence, dimensional scaling, and variation of state-type by analysing von Neumann entropy, the Tsallis entropy, entropy of entanglement of formation, concurrence and logarithmic negativity. Numerical studies with pure two-qubit states, Werner states, GHZ states, W states and mixed random states demonstrate specific sensitivity properties and computational limitations which ensure that no single measure can be a universal quantifier. The paper goes on to say that entropy-only measures do not work in mixed regimes because of a dominance in classical correlations. The results have several important implications to quantum communication, quantum simulation, and quantum computing in the NISQ-era, where the choice of measurement has a direct impact on benchmarking, error-correction scheme, and protocol performance based on entanglement. The paper ends with a recommendation of contextual, purpose-based scheme of selecting entanglement measures and suggests a further study on scalable benchmarking, hybrid quantifiers, and operational performance-based entanglement measurements. These observations reinforce the theoretical premise of entanglement studies and enable the advancement of more effective quantum technologies in the field of practice.
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.
The entanglement of a pure bipartite state is uniquely measured by the von-Neumann entropy of its reduced density matrices. Though it cannot specify all the non-local characteristics of pure entangled states. It was proven that for every possible value of entanglement of a bipartite system, there exists an infinite number of equally entangled pure states, not comparable(satisfies Nielsen’s criteria) to each other. In this work, we investigate other correlation measures of pure bipartite states that are able to differentiate the quantum correlations of the states with entropy of entanglement. In Schmidt rank 3, we consider the whole set of states having same entanglement and compare how minutely such states can be distinguished by other correlation measures. Then for different values of entanglement we compare the sets of states belonging to the same entanglement and also investigate the graphs of different correlation measures. We extend our search to Schmidt rank 4 and 5 also.