zbMATH — the first resource for mathematics

Isometries and relative entropy preserving maps on density operators. (English) Zbl 1241.47034
Let \(H\) be a finite-dimensional complex Hilbert space. A density operator on \(H\) is a positive operator whose trace is \(1\). Let \(S(H)\) denote the set of all density operators on \(H\) and let \(M(H)\) be the subset of all invertible density operators. The first result of the paper asserts that, if \(\phi: S(H) \to S(H)\) is an isometry with respect to the metric which comes from the von Neumann-Schatten \(p\)-norm, then there exists either a unitary or an antiunitary operator \(U\) such that \(\phi(A)=UAU^\ast\), for all \(A\in S(H)\). There are two more results related to preservers in the paper. The transformations on \(S(H)\) and \(M(H)\) are studied which preserve different relative entropies. These results are continuation of an earlier work of the first author.

47B49 Transformers, preservers (linear operators on spaces of linear operators)
47N50 Applications of operator theory in the physical sciences
Full Text: DOI
[1] Aizenman M, Advanced Series in Mathematical Physics 20 (1994)
[2] DOI: 10.1017/CBO9780511535048 · doi:10.1017/CBO9780511535048
[3] DOI: 10.1103/PhysRevA.79.052311 · doi:10.1103/PhysRevA.79.052311
[4] DOI: 10.1023/A:1009822315406 · Zbl 0931.47060 · doi:10.1023/A:1009822315406
[5] DOI: 10.1007/BF02100287 · Zbl 0756.46043 · doi:10.1007/BF02100287
[6] DOI: 10.1103/PhysRevA.72.052310 · doi:10.1103/PhysRevA.72.052310
[7] DOI: 10.1007/BF02771613 · Zbl 0156.37902 · doi:10.1007/BF02771613
[8] Molnár L, Lecture Notes in Mathematics 1895 pp 236– (2007)
[9] DOI: 10.1063/1.2898693 · Zbl 1153.81407 · doi:10.1063/1.2898693
[10] DOI: 10.1016/j.laa.2010.12.007 · Zbl 1220.47051 · doi:10.1016/j.laa.2010.12.007
[11] DOI: 10.1016/j.laa.2010.01.025 · Zbl 1187.47030 · doi:10.1016/j.laa.2010.01.025
[12] DOI: 10.1088/0305-4470/36/1/318 · Zbl 1047.81017 · doi:10.1088/0305-4470/36/1/318
[13] DOI: 10.1088/1751-8113/42/1/015301 · Zbl 1156.81349 · doi:10.1088/1751-8113/42/1/015301
[14] Nielsen M, Quantum Computation and Quantum Information (2000)
[15] DOI: 10.1007/978-3-642-57997-4 · doi:10.1007/978-3-642-57997-4
[16] Petz D, Quantum Information Theory and Quantum Statistics (2008)
[17] Ringrose JR, Compact Non-self-adjoint Operators (1971) · Zbl 0223.47012
[18] DOI: 10.1063/1.3511335 · Zbl 1314.81124 · doi:10.1063/1.3511335
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.