Sufficiency and Petz recovery for positive maps

• URL: http://arxiv.org/abs/2604.08380v1
• Date: Thu, 09 Apr 2026 15:42:35 GMT
• Title: Sufficiency and Petz recovery for positive maps
• Abstract summary: We study the interconversion of families of quantum states ("statistical experiments") via positive, trace-preserving maps.<n>We show that Neyman-Pearson tests generate the minimal sufficient Jordan algebra.<n>We also prove Frenkel's formula for approximately finite-dimensional von Neumann algebras.
• Score: 2.8647133890966994
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $α$-$z$ quantum Rényi 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 also prove Frenkel's formula for approximately finite-dimensional von Neumann algebras.

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