## 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 prove an analogous statement in the multipartite setting. We use this result to develop a bipartite version of a classical result about the structure of maps that preserve rank-1operators 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.

**Authors:**

- Nathaniel Johnston

** ****Download:**

- Official publication in Linear and Multilinear Algebra
- Preprint from arXiv:1008.3633 [quant-ph]
- Local preprint [pdf]

**Related talks:**

- Isometries of Locally Unitarily Invariant Norms – PDF, TeX
- Linear Preserver Problems in Quantum Information Theory – PDF, TeX

** ****Cite as:**

- N. Johnston. Characterizing Operations Preserving Separability Measures via Linear Preserver Problems.
*Linear and Multilinear Algebra*, 59(10):1171–1187, 2011.

**Supplementary Material:**

- An Introduction to Linear Preserver Problems (blog post)

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