Squashed entanglement in infinite dimensions

TL;DR

This paper investigates two definitions of squashed entanglement in infinite-dimensional systems, demonstrating their equivalence on certain states, and establishes continuity properties and bounds under energy constraints.

Contribution

It introduces and compares two definitions of infinite-dimensional squashed entanglement, proving their equivalence and extending the measure to all states with continuity bounds.

Findings

Both definitions produce the same lower semicontinuous entanglement measure.

The second definition adequately extends the measure to all states.

Continuity bounds for squashed entanglement under energy constraints are established.

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.