B can steer A relative to a pure state exactly when a state-preserving conditional expectation maps B' onto A.
Sufficiency and Petz recovery for positive maps
6 Pith papers cite this work. Polarity classification is still indexing.
6
Pith papers citing it
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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2026 6roles
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Proves the optimal constant c=1 in the trace-norm bound ||[A, log B]||_1 <= h_2(p) for positive operators A <= B with Tr A = p, Tr B = 1, settling Bravyi's small incremental mixing conjecture.
The f_0-divergence defined via Jordan decomposition integrals coincides with Araki's relative entropy on arbitrary von Neumann algebras, extending Frenkel's finite-dimensional formula.
The authors extend recovery theorems for α-z Rényi divergences from 2-positive maps to merely positive maps between general von Neumann algebras using Jordan algebraic techniques.
Quantum hockey stick f-divergences are extended to general von Neumann algebras, with regularized Rényi versions shown to coincide with standard Petz and sandwiched Rényi divergences.
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Quantum steering is equivalent to state-preserving conditional expectations
B can steer A relative to a pure state exactly when a state-preserving conditional expectation maps B' onto A.
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Sharp Quantum Entropy Mixing Rates and the Operator Layer Cake Theorem
Proves the optimal constant c=1 in the trace-norm bound ||[A, log B]||_1 <= h_2(p) for positive operators A <= B with Tr A = p, Tr B = 1, settling Bravyi's small incremental mixing conjecture.
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Integral representations of $f$-divergences for general von Neumann algebras
The f_0-divergence defined via Jordan decomposition integrals coincides with Araki's relative entropy on arbitrary von Neumann algebras, extending Frenkel's finite-dimensional formula.
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Sufficient positive maps between von Neumann algebras: R\'enyi divergences
The authors extend recovery theorems for α-z Rényi divergences from 2-positive maps to merely positive maps between general von Neumann algebras using Jordan algebraic techniques.
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Hockey stick $f$-divergences
Quantum hockey stick f-divergences are extended to general von Neumann algebras, with regularized Rényi versions shown to coincide with standard Petz and sandwiched Rényi divergences.
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