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arXiv:math-ph/0601063 · Journal of Mathematical Physics

Contractivity of positive and trace-preserving maps under Lp norms

David Pérez-Garcı́a, Michael M. Wolf, Dėnes Petz, Mary Beth Ruskai

Published Aug 1, 2006• 135 citations
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Abstract

We provide a complete picture of contractivity of trace preserving positive maps with respect to p-norms. We show that for p>1 contractivity holds in general if and only if the map is unital. When the domain is restricted to the traceless subspace of Hermitian matrices, then contractivity is shown to hold in the case of qubits for arbitrary p⩾1 and in the case of qutrits if and only if p=1, ∞. In all noncontractive cases best possible bounds on the p-norms are derived.

DOI
10.1063/1.2218675

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Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten ppp -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗C^{\ast }C∗ -algebra.