Sufficiency and Petz recovery for positive maps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909042781192192 |
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| author | van Luijk, Lauritz Wilming, Henrik |
| author_facet | van Luijk, Lauritz Wilming, Henrik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08380 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sufficiency and Petz recovery for positive maps van Luijk, Lauritz Wilming, Henrik Quantum Physics Mathematical Physics Operator Algebras 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. |
| title | Sufficiency and Petz recovery for positive maps |
| topic | Quantum Physics Mathematical Physics Operator Algebras |
| url | https://arxiv.org/abs/2604.08380 |