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

Koichi Yamagata

arxiv: 2604.23292 · v3 · pith:YCR6NZ7Wnew · submitted 2026-04-25 · 🪐 quant-ph

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

Koichi Yamagata This is my paper

Pith reviewed 2026-05-21 00:23 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum sufficiencyreal Jordan algebrasself-adjoint operatorslikelihood ratiosreal positive mapsminimal sufficiencyKoashi-Imoto decomposition

0 comments

The pith

Real Jordan algebras provide a natural framework for quantum sufficiency in self-adjoint statistical models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras for models made of general self-adjoint operators. This setting replaces the usual reliance on complex completely positive maps and faithful reference states with real positive maps that act directly on likelihood-type objects such as square-root likelihood ratios and symmetric logarithmic derivatives. Sufficient real positive maps are shown to correspond to sufficient real *-subalgebras and real Jordan algebras, with the minimal sufficient real Jordan algebra generated by the likelihood-ratio set together with the projected reference state. The construction admits degenerate states, yields Koashi-Imoto-type decompositions in the real setting, and separates the likelihood-ratio part of sufficiency from its modular part.

Core claim

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. Minimal sufficient real *-subalgebras are characterized by the likelihood-ratio set together with rho-modular invariance, and the real Jordan algebra generated by the likelihood-ratio set and the projected reference state is the minimal sufficient real Jordan algebra. Koashi-Imoto type decompositions hold for both sufficient real *-subalgebras and sufficient real Jordan algebras. The formulation works with degenerate reference states and isolates the likelihood-ratio aspect of 0

What carries the argument

Real positive maps and the real Jordan algebra generated by the likelihood-ratio set plus the projected reference state, which together identify minimal sufficient structures for self-adjoint models.

If this is right

Square-root likelihood ratios and symmetric logarithmic derivatives serve as fundamental self-adjoint likelihood-type objects.
Ordinary quantum statistical models and local quantum statistical structures can be treated in a single unified setting.
Minimal sufficient real Jordan algebras are generated by the likelihood-ratio set and the projected reference state.
Koashi-Imoto type decompositions extend to sufficient real *-subalgebras and sufficient real Jordan algebras.
The framework works with degenerate reference states by separating the likelihood-ratio aspect from the modular aspect.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Real Jordan algebras may allow direct computation of sufficiency without first embedding into a complex algebra.
The separation of likelihood and modular aspects could clarify which parts of quantum statistics truly require non-commutative structure.
Classical statistical sufficiency might appear as the commutative special case of this real-algebraic construction.
The approach could be tested on finite-dimensional qubit models by checking whether real Jordan sufficiency matches known complex results for the same data.

Load-bearing premise

The introduced real positive maps and the real Jordan algebras they generate preserve all necessary statistical information for self-adjoint models without requiring additional complex structure or faithful states.

What would settle it

A concrete self-adjoint quantum statistical model in which the real Jordan algebra generated by the likelihood-ratio set fails to recover the same sufficiency relations that the corresponding complex *-subalgebra provides.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper builds a real Jordan algebra version of quantum sufficiency that works directly with general self-adjoint operators and degenerate states.

read the letter

The main takeaway is that Yamagata has constructed a framework for quantum sufficiency on real *-subalgebras and real Jordan algebras. It treats models made of general self-adjoint operators, including derivatives, and puts ordinary and local statistical structures in the same setting without forcing faithful reference states or complex CP maps. Square-root likelihood ratios and symmetric logarithmic derivatives appear as basic self-adjoint objects here. The paper introduces sufficient real positive maps and shows how they line up with the usual complex completely positive maps and real completely positive maps. It gives a characterization of minimal sufficient real Jordan algebras using the likelihood-ratio set together with rho-modular invariance, and it derives Koashi-Imoto type decompositions for the real structures. These steps separate the likelihood-ratio side of sufficiency from the modular side and allow degenerate states. That separation and the handling of non-faithful cases are the clearest practical gains over standard formulations. The correspondences between the different algebras and maps are stated cleanly, and the abstract indicates the derivations are carried through for self-adjoint models. The work looks internally consistent on the points described. A possible soft spot is whether the real positive maps really retain every piece of statistical information that the complex setting supplies, or whether some translation back to complex terms is still needed in practice. The stress-test note says the construction addresses this directly, but without the full proofs it is hard to judge how tight the preservation is in edge cases. Overall the claims are specific and the contrasts with prior literature are explicit. This paper is for people already working in quantum statistics who care about algebraic sufficiency and alternative operator structures. A reader who follows sufficiency results or Jordan-algebra approaches in quantum information would get direct value from the new characterizations and decompositions. It is coherent enough and addresses enough standing limitations in the usual setup to deserve referee time. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras for self-adjoint statistical models consisting of general self-adjoint operators (including derivatives of states). It introduces sufficient real positive maps, establishes correspondences between sufficient complex *-subalgebras, real *-subalgebras, and real Jordan algebras with complex CP, real CP, and real positive maps respectively, characterizes minimal sufficient real *-subalgebras via the likelihood-ratio set together with rho-modular invariance, proves that the real Jordan algebra generated by the likelihood-ratio set and the projected reference state is minimal sufficient, and derives Koashi-Imoto-type decompositions. The framework explicitly admits degenerate reference states and separates the likelihood-ratio aspect from the modular aspect.

Significance. If the derivations hold, the work is significant for extending quantum sufficiency beyond the standard complex *-algebraic setting with faithful states to a real Jordan-algebraic framework that unifies ordinary and local quantum statistical models. Strengths include the explicit treatment of general self-adjoint operators and degenerate states, the use of square-root likelihood ratios and symmetric logarithmic derivatives as fundamental self-adjoint objects, and the parameter-free characterizations that avoid fitted parameters or self-referential definitions. These features could broaden applicability in quantum statistics and information theory.

minor comments (3)

The abstract and introduction would benefit from a brief explicit example (e.g., a two-qubit model with a non-faithful reference state) illustrating how the real Jordan algebra generated by the likelihood-ratio set preserves statistical information without complex structure.
Notation for 'real positive maps' and 'rho-modular invariance' should be defined with a short comparison table or paragraph contrasting them directly to their complex CP counterparts to improve readability for readers familiar with standard quantum sufficiency.
In the section deriving the Koashi-Imoto-type decompositions, add a remark on how the separation of likelihood-ratio and modular aspects manifests in the decomposition when the reference state is degenerate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for the positive assessment of its significance. The recommendation for minor revision is noted. No specific major comments were raised in the report, so we will incorporate any minor editorial or clarification changes in the revised version while preserving the core results on real Jordan-algebraic sufficiency.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper develops an extension of quantum sufficiency to real *-subalgebras and real Jordan algebras by introducing real positive maps, likelihood-type operators, and explicit characterizations such as minimal sufficient objects via likelihood-ratio sets plus rho-modular invariance. These steps are presented as direct constructions contrasting with conventional complex CP maps and faithful-state Radon-Nikodym cocycles. No equations reduce results to fitted parameters, self-definitions, or load-bearing self-citations; the framework handles degenerate states and separates likelihood from modular aspects without importing uniqueness theorems or ansatzes from prior author work as unverified premises. The central claims rest on the introduced correspondences and decompositions, which are independent of the target statistical preservation result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper relies on standard properties of *-algebras and Jordan algebras from prior literature, introduces the new concept of sufficient real positive maps, and assumes that real structures can capture statistical sufficiency for self-adjoint operators.

axioms (2)

domain assumption Real *-subalgebras and real Jordan algebras admit completely positive and positive maps that correspond to sufficiency notions.
Invoked when establishing the correspondences between sufficient subalgebras and maps.
ad hoc to paper The likelihood-ratio set together with rho-modular invariance characterizes minimal sufficiency.
Central to the characterization of minimal sufficient real *-subalgebras.

invented entities (1)

sufficient real positive maps no independent evidence
purpose: To define sufficiency for real structures in a manner parallel to complex completely positive maps.
Introduced as the key new object linking real Jordan algebras to statistical sufficiency.

pith-pipeline@v0.9.0 · 5787 in / 1206 out tokens · 39685 ms · 2026-05-21T00:23:41.020249+00:00 · methodology

Review history (3 revisions) →

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 2 internal anchors

[1]

Sufficient subalgebras and the relative entropy of states of a von Neumann algebra,

D. Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra,” Communications in Mathematical Physics 105, 123–131 (1986)

work page 1986
[2]

Sufficiency in quantum statistical inference

Jenčová, A., and Petz, D., “Sufficiency in quantum statistical inference,” Communications in Mathematical Physics 263(1), 259–276 (2006). arXiv: math-ph/0412093

work page internal anchor Pith review Pith/arXiv arXiv 2006
[3]

Structure of sufficient quantum coarse-grainings,

M. Mosonyi and D. Petz, “Structure of sufficient quantum coarse-grainings,” Letters in Mathematical Physics 68, 19–30 (2004)

work page 2004
[4]

Quantum local asymptotic normality based on a new quantum likelihood ratio,

Yamagata, K., Fujiwara, A., and Gill, R. D. (2013). “Quantum local asymptotic normality based on a new quantum likelihood ratio,” Ann. Statist., 41, 2197–2217

work page 2013
[5]

Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics,

A. Fujiwara and K. Yamagata, “Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics,” Bernoulli, 26(3), 2105 (2020)

work page 2020
[6]

Efficiency of estimators for locally asymptotically normal quantum statistical models,

Akio Fujiwara and Koichi Yamagata, “Efficiency of estimators for locally asymptotically normal quantum statistical models,” The Annals of Statistics 51(3) 1159-1182 (2023)

work page 2023
[7]

Sufficient support size of measurements for quantum estimation

K. Yamagata, “Sufficient support size of measurements for quantum estimation,” arXiv:2604.21323 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[8]

Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)

work page 1976
[9]

Li, B. (2003). Real C*-Algebras. World Scientific Publishing, Singapore

work page 2003
[10]

Goodearl, K. R. (1982). Notes on Real and Complex C*-Algebras. Shiva Mathematics Series, 5. Shiva Publish- ing, Nantwich

work page 1982
[11]

Faraut and A

J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Euclidean Jordan algebra

work page 1994
[12]

McCrimmon, A Taste of Jordan Algebras, Springer, 2004/2006

K. McCrimmon, A Taste of Jordan Algebras, Springer, 2004/2006

work page 2004
[13]

and Størmer, E

Hanche-Olsen, H. and Størmer, E. (1984). Jordan Operator Algebras. Monographs and Studies in Mathematics,

work page 1984
[14]

E. G. Effros and E. Størmer, Positive projections and Jordan structure in operator algebras, Mathematica Scandinavica 45 (1979), 127–138. DOI: 10.7146/math.scand.a-11830

work page doi:10.7146/math.scand.a-11830 1979
[15]

Operations that do not disturb quantum states,

Koashi, M., and Imoto, N., “Operations that do not disturb quantum states,” Physical Review A 66(2), 022318 (2002). DOI: 10.1103/PhysRevA.66.022318

work page doi:10.1103/physreva.66.022318 2002
[16]

Probabilistic and Statistical Aspects of Quantum Theory,

A. S. Holevo, “Probabilistic and Statistical Aspects of Quantum Theory,” (2nd English edition) (Edizioni della Normale, Pisa., 2011). 21

work page 2011

[1] [1]

Sufficient subalgebras and the relative entropy of states of a von Neumann algebra,

D. Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra,” Communications in Mathematical Physics 105, 123–131 (1986)

work page 1986

[2] [2]

Sufficiency in quantum statistical inference

Jenčová, A., and Petz, D., “Sufficiency in quantum statistical inference,” Communications in Mathematical Physics 263(1), 259–276 (2006). arXiv: math-ph/0412093

work page internal anchor Pith review Pith/arXiv arXiv 2006

[3] [3]

Structure of sufficient quantum coarse-grainings,

M. Mosonyi and D. Petz, “Structure of sufficient quantum coarse-grainings,” Letters in Mathematical Physics 68, 19–30 (2004)

work page 2004

[4] [4]

Quantum local asymptotic normality based on a new quantum likelihood ratio,

Yamagata, K., Fujiwara, A., and Gill, R. D. (2013). “Quantum local asymptotic normality based on a new quantum likelihood ratio,” Ann. Statist., 41, 2197–2217

work page 2013

[5] [5]

Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics,

A. Fujiwara and K. Yamagata, “Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics,” Bernoulli, 26(3), 2105 (2020)

work page 2020

[6] [6]

Efficiency of estimators for locally asymptotically normal quantum statistical models,

Akio Fujiwara and Koichi Yamagata, “Efficiency of estimators for locally asymptotically normal quantum statistical models,” The Annals of Statistics 51(3) 1159-1182 (2023)

work page 2023

[7] [7]

Sufficient support size of measurements for quantum estimation

K. Yamagata, “Sufficient support size of measurements for quantum estimation,” arXiv:2604.21323 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[8] [8]

Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)

work page 1976

[9] [9]

Li, B. (2003). Real C*-Algebras. World Scientific Publishing, Singapore

work page 2003

[10] [10]

Goodearl, K. R. (1982). Notes on Real and Complex C*-Algebras. Shiva Mathematics Series, 5. Shiva Publish- ing, Nantwich

work page 1982

[11] [11]

Faraut and A

J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Euclidean Jordan algebra

work page 1994

[12] [12]

McCrimmon, A Taste of Jordan Algebras, Springer, 2004/2006

K. McCrimmon, A Taste of Jordan Algebras, Springer, 2004/2006

work page 2004

[13] [13]

and Størmer, E

Hanche-Olsen, H. and Størmer, E. (1984). Jordan Operator Algebras. Monographs and Studies in Mathematics,

work page 1984

[14] [14]

E. G. Effros and E. Størmer, Positive projections and Jordan structure in operator algebras, Mathematica Scandinavica 45 (1979), 127–138. DOI: 10.7146/math.scand.a-11830

work page doi:10.7146/math.scand.a-11830 1979

[15] [15]

Operations that do not disturb quantum states,

Koashi, M., and Imoto, N., “Operations that do not disturb quantum states,” Physical Review A 66(2), 022318 (2002). DOI: 10.1103/PhysRevA.66.022318

work page doi:10.1103/physreva.66.022318 2002

[16] [16]

Probabilistic and Statistical Aspects of Quantum Theory,

A. S. Holevo, “Probabilistic and Statistical Aspects of Quantum Theory,” (2nd English edition) (Edizioni della Normale, Pisa., 2011). 21

work page 2011