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arXiv:2604.23292

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

Koichi Yamagata

quant-phSubmitted Apr 25, 2026• Announced Apr 28, 2026• 0 citations• Revised Sep 22, 2026 (v4)
v1 · Apr 25, 2026v2 · May 9, 2026v3 · May 18, 2026v4 · Sep 22, 2026
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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 $ρ$-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.

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Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real ∗*∗-Subalgebras and Real Jordan Algebras
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 ρρρ-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.
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Local asymptotic normality is a cornerstone of classical asymptotic statistics. Extending it to quantum statistical models entails formidable conceptual and technical challenges arising from their noncommutative nature, above all, the definition of a quantum likelihood ratio, whose very choice dictates the subsequent theoretical development. Employing quantum information geometry based on the symmetric logarithmic derivative, we formulate a quantum Lebesgue decomposition and the square-root likelihood ratio. These constructions provide a natural framework for a quantum analogue of Le Cam’s third lemma, the formulation of quantum local asymptotic normality, and the asymptotic representation results developed in our earlier work linking regular quantum statistical models to quantum Gaussian shift models. In this way, quantum information geometry emerges as a natural and unifying foundation for asymptotic quantum statistics.
We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a dagger-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setting. In combination with previous work this characterizes EJAs as the most general systems allowed in a generalized probabilistic theory that is simultaneously a dagger-effectus. Using the dagger structure we get a notion of dagger-positive maps of the form f = g*g. We give a complete characterization of the pure dagger-positive maps and show that these correspond precisely to the Jordan algebraic version of the sequential product that maps (a,b) to sqrt(a) b sqrt(a). The notion of dagger-positivity therefore characterizes the sequential product.
We prove recovery theorems for ααα-zzz Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and LpL^pLp-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.
Abstract Some features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.
We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities. These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit. Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents. © 2020 FahrenHouse. All rights reserved.