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arXiv · 2604.08380

Sufficiency and Petz recovery for positive maps

Lauritz van Luijk·Henrik Wilming
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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.

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Lauritz van Luijk, Henrik Wilming. 2026-04-09. Sufficiency and Petz recovery for positive maps. https://arxiv.org/abs/2604.08380

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