On properties of the space of quantum states and their application to construction of entanglement monotones

Abstract

We consider two properties of the set of quantum states as a convex topological space and some their implications concerning the notions of a convex hull and of a convex roof of a function defined on a subset of quantum states. By using these results we analyze two infinite-dimensional versions (discrete and continuous) of the convex roof construction of entanglement monotones, which is widely used in finite dimensions. It is shown that the discrete version may be 'false' in the sense that the resulting functions may not possess the main property of entanglement monotones while the continuous version can be considered as a 'true' generalized convex roof construction. We give several examples of entanglement monotones produced by this construction. In particular, we consider an infinite-dimensional generalization of the notion of Entanglement of Formation and study its properties.

34 pages, the minor corrections have been made

References in corpus (3)

• Entropic characteristics of subsets of states
• Continuity of the von Neumann entropy
• Generalized compactness in linear spaces and its applications

Cited by in corpus (12)

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• Squashed entanglement in infinite dimensions
• On the classical capacity of quantum Gaussian measurement
• Continuity of characteristics of composite quantum systems
• The information capacity of entanglement-assisted continuous variable measurement
• On quantum channels and operations preserving finiteness of the von Neumann entropy
• Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique
• Generalizations of Fano's Inequality for Conditional Information Measures via Majorization Theory
• Entanglement cost for infinite-dimensional physical systems
• Conclusive discrimination by $N$ sequential receivers between $r\ge 2$ arbitrary quantum states