Characterizing Operations Preserving Separability Measures via Linear Preserver Problems

TL;DR

This paper characterizes linear operators that preserve certain quantum state properties related to Schmidt rank and separability, extending classical results and identifying key isometries in quantum information theory.

Contribution

It extends linear preserver problem results to operators preserving Schmidt rank constraints and characterizes isometries for specific quantum norms.

Findings

Operators preserving Schmidt rank k are invariant under local unitaries, swap, and transpose.

Partial transpose is an additional isometry when k=1.

New proofs for multipartite state preservation results.

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 at least 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.