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arXiv:2604.08380

Sufficiency and Petz recovery for positive maps

Lauritz van Luijk, Henrik Wilming

quant-phmath-phmath.OASubmitted Apr 9, 2026• Announced Apr 10, 2026• 0 citations• Revised May 14, 2026 (v2)
v1 · Apr 9, 2026v2 · May 14, 2026
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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 $α$-$z$ 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 prove Frenkel's formula for approximately finite-dimensional von Neumann algebras.

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58 pages total; Comments welcome!

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