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arXiv:quant-ph/0105104 · Journal of Mathematical Physics

A uniqueness theorem for entanglement measures

Oliver Rudolph

Published Jun 1, 2001• 141 citations
arXivHTMLPDF

Abstract

We obtain a mathematically simple characterization of all functionals coinciding with the von Neumann reduced entropy on pure states based on the Khinchin–Faddeev axiomatization of Shannon entropy and give a physical interpretation of the axioms in terms of entanglement.
DOI
10.1063/1.1370954

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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.
Abstract We introduce new entanglement measures on the set of density operators on tensor product Hilbert spaces. These measures are based on the greatest cross norm on the tensor product of the sets of trace class operators on Hilbert space. We show that they satisfy all the basic requirements on entanglement measures discussed in the literature, including convexity, invariance under local unitary operations and non-increase under local quantum operations and classical communication. In the second part of this paper we discuss the uniqueness theorem for entanglement measures. We obtain a mathematically simple characterization of all functionals coinciding with the von Neumann reduced entropy on pure states based on the Khinchin-Faddeev axiomatization of Shannon entropy and give a physical interpretation of the axioms in terms of entanglement. We also discuss the uniqueness theorem in the operational approach and argue that the standard argument in favor of the uniqueness theorem for entanglement measures is not fully conclusive and that further axioms have to be added to the list of postulates. 1