Physics – Quantum Physics
Scientific paper
2010-08-21
Linear Multilinear Algebra 59, 1171-1187 (2011)
Physics
Quantum Physics
16 pages, typos corrected, references added, proof of Theorem 4.3 simplified and clarified
Scientific paper
10.1080/03081087.2011.596540
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.
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