The paper introduces sufficient real positive maps and shows that minimal sufficient real Jordan algebras are generated by the likelihood-ratio set together with the projected reference state.
Operations that do not disturb quantum states
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Minimal sufficient Jordan algebras characterize sufficiency for positive trace-preserving maps on quantum states, with Neyman-Pearson tests generating them and equality in data-processing inequalities implying Petz recovery.
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Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras
The paper introduces sufficient real positive maps and shows that minimal sufficient real Jordan algebras are generated by the likelihood-ratio set together with the projected reference state.
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Sufficiency and Petz recovery for positive maps
Minimal sufficient Jordan algebras characterize sufficiency for positive trace-preserving maps on quantum states, with Neyman-Pearson tests generating them and equality in data-processing inequalities implying Petz recovery.