Rank two bipartite bound entangled states do not exist

Journal article (2003)

Authors

Pawel Horodecki Politechnika Gdanska
John A. Smolin IBM Research
B.M. Terhal IBM Research
Ashish V. Thapliyal University of California
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Published Date

2003

Language

English

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Abstract

We explore the relation between the rank of a bipartite density matrix and the existence of bound entanglement. We show a relation between the rank, marginal ranks, and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n × n Hilbert space. A direct consequence of this result is that there are no bipartite bound entangled states of rank two. We also show that a separability condition in terms of a quantum entropy inequality is associated with the above results. We explore the idea of how many pure states are needed in a mixture to cancel the distillable entanglement of a Schmidt rank n pure state and provide a lower bound of n - 1. We also prove that a mixture of a non-zero amount of any pure entangled state with a pure product state is distillable.