Canadian Mathematical Bulletin

Maps on Quantum States in -algebras Preserving von Neumann Entropy or Schatten -norm of Convex Combinations

Marcell Gaál

Published Mar 22, 2018• 3 citations

Abstract

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 $p$ -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 $C^{\ast }$ -algebra.

DOI: 10.4153/cmb-2018-011-3

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Canadian Mathematical Bulletin

Maps on Quantum States in -algebras Preserving von Neumann Entropy or Schatten -norm of Convex Combinations

Marcell Gaál

Published Mar 22, 2018• 3 citations

Publisher

Abstract

p

-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^{\ast }

-algebra.

DOI: 10.4153/cmb-2018-011-3

Related papers

TRANSFORMATIONS ON DENSITY OPERATORS PRESERVING GENERALISED ENTROPY OF A CONVEX COMBINATION

Marcell Gaál, Gergő Nagy

Bulletin of the Australian Mathematical SocietyMay 3, 20181 citation

Characterizing the dynamical semigroups that do not decrease a quantum entropy

Paolo Aniello, Dariusz Chruściński

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.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.

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Properties of Segal’s Entropy for Quantum Systems

Andrzej Łuczak, Hanna Podsȩdkowska

International Journal of Theoretical PhysicsFeb 22, 20175 citations

p

C^{\ast }

-algebra.

Let

(\mathcal{M},τ)

and

(\mathcal{N},τ^{\prime})

be tracial von-Neumann algebras and let

φ:\mathcal{M}\to\mathcal{N}

be a strictly completely positive, trace preserving map. Given a positive, invertible

B\in\mathcal{M}

with

τ(B)=1

, a state on

\mathcal{M}

given by a positive

A\in L^1(\mathcal{M}, τ)

is said to be recoverable if

\mathcal{R}(φ(A))=A

where

\mathcal{R}

is the Petz recovery map corresponding to

B

and

φ

. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of

\mathcal{R}\circφ

. We show that there exists a completely positive, trace preserving map

ψ:\mathcal{M}\to\mathcal{M}

such that

ψ(A)

is recoverable for all

A

and

(\mathcal{R}\circφ)^n\toψ

in norm as operators on

L^p(\mathcal{M},τ)

for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in

L^1

. Finally, we prove an interesting decomposition theorem for normal states on

\mathcal{M}