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

• URL: http://arxiv.org/abs/2604.23292v1
• Date: Sat, 25 Apr 2026 13:13:15 GMT
• Title: Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras
• Abstract summary: We develop a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras.<n>Within this framework, square-root likelihood ratios and symmetric logarithmic derivatives arise naturally as fundamental self-adjoint likelihood-type objects.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras. In contrast to the conventional formulation, which is based on families of states, complex completely positive coarse-grainings, and Radon-Nikodym cocycles associated with faithful reference states, our framework allows models consisting of general self-adjoint operators, including derivatives of states. Within this framework, square-root likelihood ratios and symmetric logarithmic derivatives arise naturally as fundamental self-adjoint likelihood-type objects. This makes it possible to treat ordinary quantum statistical models and local quantum statistical structures within a unified setting. We introduce sufficient real positive maps and show that sufficient complex *-subalgebras, sufficient real *-subalgebras, and sufficient real Jordan algebras correspond respectively to complex completely positive maps, real completely positive maps, and real positive maps. We characterize minimal sufficient real *-subalgebras by the likelihood-ratio set together with rho-modular invariance, and show that the real Jordan algebra generated by the likelihood-ratio set and the projected reference state is the minimal sufficient real Jordan algebra. We also obtain Koashi-Imoto type decompositions for sufficient real *-subalgebras and sufficient real Jordan algebras. Our formulation admits degenerate reference states and separates the likelihood-ratio aspect of sufficiency from its genuinely quantum modular aspect. These results suggest that real Jordan structure provides a natural framework for the statistical aspect of quantum theory beyond the conventional complex *-algebraic setting.

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