The uniqueness theorem for entanglement measures

By M. Donald, Michal Horodecki, Oliver Rudolph. 2001

237 citations recorded by the source corpus.

We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyze properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entangle ment.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/5bd98ab4084f205a69b20cfc25155960e1336ead

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In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Large Violation of Bell Inequalities with Low Entanglement (2010, background): “y. Indeed, for bipartite pure states there exists a universal measure of entanglement, the so called entropy of entanglement: E(j i) = S((j ih j) A); where Sdenotes the usual von Neumann entropy (see [25]). It is easy to see that E(j i) 0 for every state j iand that the maximally entangled state in”
Entanglement of purification through holographic duality (2017, background): “When ρAB is pure, we have ED(ρAB) = EC(ρAB) = S(ρA), and then there is the essentially unique measure (namely the entanglement entropy) which satisfies desirable properties [36].”
Aspects of Generic Entanglement (2004, background): “Even in the bipartite, asymptotic case, there are many different mixedstate “entanglement measures,” most of which are poorly understood, both individually and in relation to each other; see [3, 6, 12, 21, 26, 50] and references therein.”
Quantum resource theories (2018, background): “In other words, Φ has a Kraus operator decomposition in which each Kraus operator is a tensor product (Bennett et al., 1999; Donald et al., 2002).”
Faithful Squashed Entanglement (2010, background): “1 n [14] ? y [26] y y y [27] n LOCC monotonicitya y y y y y y y y [28] asymptotic continuity y [29] ? ? ? y y [30] y [9] n[9] convexity y ? ? ? y y y [31] n strong superadditivity y y y ? n [32, 33] n [34] ? ? subadditivity y ? ? y y y y y monogamy y [11] ? ? n [10] n [10] n [10] n [10] ?”
Geometry of Quantum States: Index (2007, methodology): “The backward implication follows from an explicit conversion protocol proposed by Nielsen, or alternative versions presented in [274, 404, 502].”