Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras

Abstract:We develop a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras for models consisting of general self-adjoint operators, including derivatives of states. Square-root likelihood ratios and symmetric logarithmic derivatives serve as self-adjoint likelihood-type objects, allowing ordinary quantum statistical models and local models to be treated within a unified framework that admits degenerate reference states. We introduce sufficient real unital positive maps and show that their Hermitian fixed points form a real Jordan algebra admitting a sufficient Jordan conditional expectation. The real and complex *-subalgebras generated by this algebra admit sufficient real and complex completely positive conditional expectations, respectively. We characterize the minimal sufficient real *-subalgebra by the likelihood-ratio set and $\rho$-modular invariance. The minimal sufficient real Jordan algebra is generated by the likelihood-ratio set together with the operator obtained by normalizing every irreducible diagonal block of the reference state, relative to the minimal sufficient real *-subalgebra, to trace one. We also obtain Koashi-Imoto type decompositions for real *-subalgebras and real Jordan algebras admitting sufficient conditional expectations, explicitly accounting for inequivalent irreducible Jordan representations. These results separate the likelihood-ratio aspect of sufficiency from its noncommutative modular aspect and identify real Jordan structure as a natural framework for quantum statistical sufficiency.

arXiv:2604.23292 [quant-ph]

(or arXiv:2604.23292v4 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2604.23292 Focus to learn more arXiv-issued DOI via DataCite

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