Squashed entanglement in infinite dimensions

Abstract

We analyse two possible definitions of the squashed entanglement in an infinite-dimensional bipartite system: direct translation of the finite-dimensional definition and its universal extension. It is shown that the both definitions produce the same lower semicontinuous entanglement measure possessing all basis properties of the squashed entanglement on the set of states having at least one finite marginal entropy. Is also shown that the second definition gives an adequate extension of this measure to the set of all states of infinite-dimensional bipartite system. A general condition relating continuity of the squashed entanglement to continuity of the quantum mutual information is proved and its corollaries are considered. Continuity bound for the squashed entanglement under the energy constraint on one subsystem is obtained by using the tight continuity bound for conditional mutual information (proved in the Appendix by using Winter's technique). It is shown that the same continuity bound is valid for the entanglement of formation. As a result the asymptotic continuity of the both entanglement measures under the energy constraint on one subsystem is proved.

39 pages, some questions are unsolved, any comments are welcome, in v.2 continuity bounds for the squashed entanglement and for the entanglement of formation under the energy constraint on one subsystem are added, in v.3 remark about asymptotic continuity is added, in v.4 condition relating continuity of the squashed entanglement to continuity of the quantum mutual information is added

References in corpus (3)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography

Cited by in corpus (18)

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• Energy-constrained private and quantum capacities of quantum channels
• Entanglement cost and quantum channel simulation
• Measures of quantum correlations in infinite-dimensional systems
• Energy-constrained two-way assisted private and quantum capacities of quantum channels
• Asymptotic state transformations of continuous variable resources
• Maximum and minimum entropy states yielding local continuity bounds
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
• Continuity of characteristics of composite quantum systems
• Uniform continuity bounds for characteristics of multipartite quantum systems
• On lower semicontinuity of the quantum conditional mutual information and its corollaries
• Entanglement cost for infinite-dimensional physical systems
• Computable lower bounds on the entanglement cost of quantum channels
• Entropic witness for quantum memory in open system dynamics
• On quantum states with a finite-dimensional approximation property
• Features of preparable entangled states in Gaussian quantum networks