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Annales Academiae Scientiarum Fennicae Mathematica

Mappings preserving Segal's entropy in von Neumann algebras

Andrzej Łuczak, Hanna Podsȩdkowska

Published Jun 1, 2019• 2 citations
Publisher

Abstract

We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change the Segal entropy of the density of a normal, not necessarily normalised, state.Two cases are dealt with: a) no restriction on the map is imposed, b) the map represents a repeatable instrument in measurement theory which means that it is idempotent.

DOI
10.5186/aasfm.2019.4439

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We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change the Segal entropy of the density of a normal, not necessarily normalised, state.Two cases are dealt with: a) no restriction on the map is imposed, b) the map represents a repeatable instrument in measurement theory which means that it is idempotent.
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.
Let (M,τ)(\mathcal{M},τ)(M,τ) and (N,τ′)(\mathcal{N},τ^{\prime})(N,τ′) be tracial von-Neumann algebras and let φ:M→Nφ:\mathcal{M}\to\mathcal{N}φ:M→N be a strictly completely positive, trace preserving map. Given a positive, invertible B∈MB\in\mathcal{M}B∈M with τ(B)=1τ(B)=1τ(B)=1, a state on M\mathcal{M}M given by a positive A∈L1(M,τ)A\in L^1(\mathcal{M}, τ)A∈L1(M,τ) is said to be recoverable if R(φ(A))=A\mathcal{R}(φ(A))=AR(φ(A))=A where R\mathcal{R}R is the Petz recovery map corresponding to BBB and φφφ. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of R∘φ\mathcal{R}\circφR∘φ. We show that there exists a completely positive, trace preserving map ψ:M→Mψ:\mathcal{M}\to\mathcal{M}ψ:M→M such that ψ(A)ψ(A)ψ(A) is recoverable for all AAA and (R∘φ)n→ψ(\mathcal{R}\circφ)^n\toψ(R∘φ)n→ψ in norm as operators on Lp(M,τ)L^p(\mathcal{M},τ)Lp(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 L1L^1L1. Finally, we prove an interesting decomposition theorem for normal states on M\mathcal{M}M.
Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.