Sufficiency and Petz recovery for positive maps

TL;DR

This paper investigates the structure of positive trace-preserving maps in quantum information, establishing conditions for state interconversion and recovery, and extending mathematical frameworks to infinite-dimensional algebras.

Contribution

It introduces the use of minimal sufficient Jordan algebras for understanding positive maps and extends Frenkel's formula to infinite-dimensional von Neumann algebras.

Findings

Equality in quantum data-processing implies existence of recovery maps.

Two quantum dichotomies are interconvertible by PTP maps iff by decomposable maps.

Frenkel's formula is extended to approximately finite-dimensional von Neumann algebras.

Abstract

We study the interconversion of families of quantum states ("statistical experiments") via positive, trace-preserving (PTP) maps and clarify its mathematical structure in terms of minimal sufficient Jordan algebras, which can be seen to generalize the Koashi-Imoto decomposition to the PTP setting. In particular, we show that Neyman-Pearson tests generate the minimal sufficient Jordan algebra, and hence also the minimal sufficient *-algebra corresponding to the Koashi-Imoto decomposition. As applications, we show that a) equality in the data-processing inequality for the relative entropy or the $\alpha$-$z$ quantum R\'enyi divergence implies the existence of a recovery map also in the PTP case and b) that two dichotomies can be interconverted by PTP maps if and only if they can be interconverted by decomposable, trace-preserving maps. We thoroughly review the necessary mathematical background on Jordan algebras. As a step beyond the finite-dimensional case, we prove Frenkel's formula for approximately finite-dimensional von Neumann algebras.