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arXiv:1601.07808 · Journal of Physics A Mathematical and Theoretical

Characterizing the dynamical semigroups that do not decrease a quantum entropy

Paolo Aniello, Dariusz Chruściński

Published Jul 14, 2016• 14 citations
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Abstract

Abstract In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Rényi entropies, as well as a family of functions of density operators strictly related to the Schatten norms. A few remarkable consequences—in particular, a description of the associated infinitesimal generators—are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are also considered.

DOI
10.1088/1751-8113/49/34/345301

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Abstract In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Rényi entropies, as well as a family of functions of density operators strictly related to the Schatten norms. A few remarkable consequences—in particular, a description of the associated infinitesimal generators—are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are also considered.
Entropy plays a fundamental role in several branches of physics. In the quantum setting, one usually considers the von Neumann entropy, but other useful quantities have been proposed in the literature; e.g., the Rényi and the Tsallis entropies. The evolution of an open quantum system, described by a semigroup of dynamical maps (in short, a dynamical semigroup), may decrease a quantum entropy, for some initial condition. We will discuss various characterizations of those dynamical semigroups that, for every initial condition, do not decrease a general class of quantum entropies, which is defined using the notion of Schur concavity of a function. We will not assume that such a dynamical semigroup be completely positive, the physical justification of this condition being controversial. Therefore, we will consider semigroups of trace-preserving, positive — but not necessarily completely positive — linear maps. We will next focus on a special class of (completely positive) dynamical semigroups, the twirling semigroups , having applications in quantum information science. We will argue that the whole class of dynamical semigroups that do not decrease a quantum entropy can be obtained as a suitable generalization of the twirling semigroups.
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.
In this article we consider certain kinds of metrics and relative entropies on sets of density operators acting on a finite-dimensional complex Hilbert space. The metrics are the ones induced by the von Neumann–Schatten p-norms. We describe the general form of ‘a priori’ nonsurjective isometries of the space of all density operators with respect to those distances. In the second part of this article we study mappings preserving entropic quantities. We determine the structure of ‘a priori’ nonsurjective maps on the set of all density operators which leave a certain measure of relative entropy invariant and also characterize the surjective maps on the set of all invertible density operators with similar invariance properties.
We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in continuous time, proving the existence of a quantum extension for Lévy processes. In the second part of this article, we focus on the case where the commutative algebra is isomorphic to $\Acal=l^\infty(1,...,N)$ with NNN either finite or infinite. We propose a classification of the CP maps leaving $\Acal$ stable, producing physical examples of each classes.
We analyze semigroups of decomposable maps on C*-algebras in context of the algebraic structure of associated infinitesimal generators. Case of von Neumann algebras, including B(H)B(\mathcal{H})B(H) for H\mathcal{H}H a Hilbert space, is also addressed. We then elaborate on D-divisible (decomposably divisible) dynamical maps on the Banach space of trace class operators. Our analysis extends earlier results on decomposable dynamical maps on matrix algebras (J. Phys. A: Math. Theor. 56 485202) and provides a partial generalization of the seminal work of Lindblad (Commun. Math. Phys. 48 119-130) on completely positive semigroups.