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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
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. 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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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. 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.
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 ααα-zzz quantum Rényi 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.
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.
A statistical experiment on a von Neumann algebra is a parametrized family of normal states on the algebra. This paper introduces the concept of minimal sufficiency for statistical experiments in such operator algebraic situations. We define equivalence relations of statistical experiments indexed by a common parameter set by completely positive or Schwarz coarse-graining and show that any statistical experiment is equivalent to a minimal sufficient statistical experiment unique up to normal isomorphism of outcome algebras. We also establish the relationship between the minimal sufficiency condition for a statistical experiment in this paper and those for subalgebra. These concepts and results are applied to the concatenation relation for completely positive channels with general input and outcome von Neumann algebras. In the case of the quantum-classical channel corresponding to the positive-operator valued measure (POVM), we prove the equivalence of the minimal sufficient condition previously proposed by the author and that in this paper. We also give a characterization of the discreteness of a POVM up to postprocessing equivalence in terms of the corresponding quantum-classical channel.
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 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 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.