Characterizing operations preserving separability measures via linear preserver problems

Abstract

We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied in quantum information theory. We see, in particular, that for k ≥ 2 the operator norms induced by states with Schmidt rank k are invariant only under local unitaries, the swap operator and the transpose map. However, in the k = 1 case there is an additional isometry: the partial transpose map.

Keywords:

• entanglement
• linear preserver problem
• isometry groups
• separability
• quantum information theory

AMS Subject Classifications:

• 15A86
• 15A60
• 81P40

Acknowledgements

The author is grateful to an anonymous referee who provided a significantly simplified proof of Theorem 4.3. Thanks are also extended to Christian Gogolin, Rajesh Pereira and Andreas Winter for asking helpful questions that led to this work. Thanks to David Kribs for constant support. The author was supported by an NSERC Canada Graduate Scholarship and the University of Guelph Brock Scholarship.