arxiv: 2604.08380 · v2 · submitted 2026-04-09 · 🪐 quant-ph · math-ph· math.MP· math.OA

Recognition: 2 theorem links

· Lean Theorem

Sufficiency and Petz recovery for positive maps

Lauritz van Luijk , Henrik Wilming

Authors on Pith no claims yet

Pith reviewed 2026-05-15 06:42 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPmath.OA

keywords sufficiencypositive trace-preserving mapsJordan algebrasdata processing inequalityrecovery mapsNeyman-Pearson testsquantum hypothesis testing

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The pith

Neyman-Pearson tests generate the minimal sufficient Jordan algebra for families of quantum states under positive maps.

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

The paper shows that for any family of quantum states, the Neyman-Pearson tests for distinguishing them generate a minimal sufficient Jordan algebra that is preserved under positive trace-preserving maps. This algebra generalizes the Koashi-Imoto decomposition, which was known only for completely positive maps, to the larger class of positive maps. A sympathetic reader would care because this gives a concrete algebraic way to determine when one statistical experiment can be converted into another using these maps, which are important in quantum information theory for modeling certain physical processes. It also proves that equality in the data-processing inequality for relative entropy or alpha-z Renyi divergence implies a recovery map exists even for positive maps. Finally, it establishes that for pairs of states, interconversion by positive maps is equivalent to interconversion by decomposable trace-preserving maps.

Core claim

The central discovery is that Neyman-Pearson tests generate the minimal sufficient Jordan algebra for families of states, and this structure determines interconvertibility under positive trace-preserving maps. Equality in data-processing inequalities for the relative entropy or the alpha-z quantum Renyi divergence implies the existence of a recovery map in the PTP case. Two dichotomies can be interconverted by PTP maps if and only if they can be interconverted by decomposable, trace-preserving maps. The result extends to approximately finite-dimensional von Neumann algebras via Frenkel's formula.

What carries the argument

The minimal sufficient Jordan algebra generated by Neyman-Pearson tests, which captures the information preserved by positive trace-preserving maps and generalizes the *-algebra in the Koashi-Imoto decomposition.

If this is right

Equality in the data-processing inequality for relative entropy or alpha-z Renyi divergence implies the existence of a recovery map for positive trace-preserving maps.
Two families of states (dichotomies) are interconvertible by positive trace-preserving maps exactly when they are interconvertible by decomposable trace-preserving maps.
The Koashi-Imoto decomposition for completely positive maps extends to positive maps via the Jordan algebra structure.
Frenkel's formula for the minimal sufficient algebra holds in the setting of approximately finite-dimensional von Neumann algebras.

Where Pith is reading between the lines

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

This algebraic characterization could simplify numerical searches for optimal positive maps in quantum state interconversion tasks.
In infinite-dimensional settings, the result suggests that sufficiency can be checked using algebraic conditions rather than direct map constructions.
The connection to Jordan algebras might link quantum sufficiency to classical statistical decision theory where similar structures appear in commutative cases.
One could test the result by constructing explicit small-dimensional examples of state families and verifying the generated algebra matches the preserved information.

Load-bearing premise

The minimal sufficient Jordan algebra fully captures the interconversion structure for arbitrary families of states in finite dimensions.

What would settle it

A pair of state families that share the same minimal sufficient Jordan algebra but cannot be interconverted by any positive trace-preserving map would disprove the main claim.

read the original abstract

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 $\alpha$-$z$ quantum R\'enyi 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.

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 gives a clean extension of sufficiency and recovery to positive trace-preserving maps via minimal sufficient Jordan algebras generated by Neyman-Pearson tests.

read the letter

The main contribution is showing that Neyman-Pearson tests generate the minimal sufficient Jordan algebra for families of states under PTP maps. This generalizes the Koashi-Imoto decomposition beyond CPTP maps and directly yields recovery maps from equality in the data-processing inequality for relative entropy or alpha-z Renyi divergences. They also prove that two dichotomies are interconvertible by PTP maps if and only if they are interconvertible by decomposable trace-preserving maps. The finite-dimensional claims rest on explicit constructions rather than assumptions, and the background review of Jordan algebras is thorough and helpful. The step to approximately finite-dimensional von Neumann algebras via a direct proof of Frenkel's formula is a useful addition beyond the main setting. The soft spots are minor and confined to the infinite-dimensional extension, where the paper does not claim full generality; the central finite-dimensional results appear tight with no visible circularity or hidden use of complete positivity. The work is for quantum information researchers focused on statistical experiments, sufficiency structures, or recovery maps. It shows clear, grounded thinking and deserves a serious referee because the new characterizations are explicit and the proofs are stated to be self-contained in finite dimensions.

Referee Report

1 major / 2 minor

Summary. The paper studies interconversion of families of quantum states via positive trace-preserving (PTP) maps, showing that this structure is captured by minimal sufficient Jordan algebras that generalize the Koashi-Imoto decomposition. It proves that Neyman-Pearson tests generate the minimal sufficient Jordan algebra (and hence the corresponding *-algebra), establishes that equality in the data-processing inequality for relative entropy or α-z Rényi divergence implies existence of a recovery map in the PTP case, and shows equivalence of interconvertibility for dichotomies under PTP maps versus decomposable trace-preserving maps. The work includes a review of Jordan algebra background and proves Frenkel's formula for approximately finite-dimensional von Neumann algebras.

Significance. If the central claims hold, the results supply a clean algebraic characterization of sufficiency for positive maps, with explicit constructions that connect directly to classical statistical tests. This strengthens the foundations for data-processing inequalities and recovery maps beyond the completely positive setting, and the finite-dimensional equivalences plus the AF von Neumann extension provide concrete tools for quantum statistics. The parameter-free nature of the derivations and the explicit generation of the algebra by Neyman-Pearson tests are notable strengths.

major comments (1)

[§4, Theorem 4.3] §4, Theorem 4.3: the equivalence between PTP interconvertibility and decomposable TP interconvertibility for dichotomies is stated to follow from the Jordan-algebra characterization, but the proof sketch does not explicitly verify that the generated algebra is invariant under the relevant maps when the states are not faithful; a short additional paragraph confirming this invariance would strengthen the claim.

minor comments (2)

[§2.2] §2.2: the definition of the minimal sufficient Jordan algebra could be accompanied by a one-line comparison to the classical sufficient statistic to aid readers outside operator algebras.
[Figure 1] Figure 1: the diagram illustrating the Koashi-Imoto versus PTP decomposition would benefit from labeling the arrows with the precise map classes (PTP vs. CPTP).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their careful reading and positive evaluation of our manuscript. We address the single major comment below.

read point-by-point responses

Referee: [§4, Theorem 4.3] §4, Theorem 4.3: the equivalence between PTP interconvertibility and decomposable TP interconvertibility for dichotomies is stated to follow from the Jordan-algebra characterization, but the proof sketch does not explicitly verify that the generated algebra is invariant under the relevant maps when the states are not faithful; a short additional paragraph confirming this invariance would strengthen the claim.

Authors: We thank the referee for pointing this out. Upon re-examination, we agree that explicitly confirming the invariance of the generated algebra under the relevant maps in the non-faithful case would clarify the argument. We will add a short paragraph in the proof of Theorem 4.3 to verify this invariance, thereby strengthening the claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit constructions

full rationale

The paper establishes its claims through explicit constructions (Neyman-Pearson tests generating the minimal sufficient Jordan algebra) and direct equivalences (DPI equality implying PTP recovery; PTP vs. decomposable interconvertibility) proven in finite dimensions, plus a self-contained proof of Frenkel's formula for the AF von Neumann extension. The reviewed mathematical background on Jordan algebras and Koashi-Imoto generalizations provides independent support without reducing to fitted inputs, self-definitions, or load-bearing self-citations. Self-references to prior Jordan algebra results are supplementary and not required for the new sufficiency characterizations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical background for Jordan algebras and von Neumann algebras without introducing new free parameters or invented entities. The central claims build on existing operator-algebraic structures rather than postulating new ones.

axioms (2)

standard math Standard properties of Jordan algebras and their relation to associative algebras in the context of quantum observables.
Invoked throughout the review of mathematical background and the generalization of Koashi-Imoto decomposition.
domain assumption Existence and properties of minimal sufficient *-algebras for statistical experiments in the finite-dimensional case.
Used as the foundation for extending to the PTP setting via Jordan algebras.

pith-pipeline@v0.9.0 · 5474 in / 1467 out tokens · 41867 ms · 2026-05-15T06:42:19.658269+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

minimal sufficient J*-algebra J(ρθ) … generated by the Neyman-Pearson tests … J(ρθ)=J*-alg(K(ρθ)) (Theorem C, §8)
Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

equality D(T*ρ∥T*σ)=D(ρ∥σ) iff PTP recovery exists (Theorem E, §9)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

90 extracted references · 90 canonical work pages · 3 internal anchors

[1]

Detection Theory and Quantum Mechanics

C. W. Helstrom. “Detection Theory and Quantum Mechanics”. In:Information and Control10.3 (1967), pp. 254–291.doi: 10.1016/S0019-9958(67)90302-6

work page doi:10.1016/s0019-9958(67)90302-6 1967
[2]

Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing

T. Ogawa and H. Nagaoka. “Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing”. In:IEEE Transactions on Information Theory46.7 (2000), pp. 2428–2433.doi: 10.1109/18.887855

work page doi:10.1109/18.887855 2000
[3]

Tomamichel

M. Tomamichel. Quantum Information Processing with Finite Resources. Vol. 5. SpringerBriefs in Mathematical Physics. Cham: Springer International Publishing,

work page
[4]

doi: 10.1007/978-3-319-21891-5

work page doi:10.1007/978-3-319-21891-5
[5]

F. Hiai. Quantum F-Divergences in von Neumann Algebras: Reversibility of Quan- tum Operations. Mathematical Physics Studies. Singapore: Springer, 2021.doi: 10.1007/978-981-33-4199-9

work page doi:10.1007/978-981-33-4199-9 2021
[6]

A Problem Relating to Positive Linear Maps on Matrix Algebras

P. M. Alberti and A. Uhlmann. “A Problem Relating to Positive Linear Maps on Matrix Algebras”. In:Reports on Mathematical Physics18.2 (1980), pp. 163–176. doi: 10.1016/0034-4877(80)90083-X

work page doi:10.1016/0034-4877(80)90083-x 1980
[7]

Matsumoto

K. Matsumoto. Reverse Test and Characterization of Quantum Relative Entropy

work page
[8]

Extending quantum operations

T. Heinosaari, M. A. Jivulescu, D. Reeb, and M. M. Wolf. “Extending Quantum Operations”. In:Journal of Mathematical Physics 53.10 (2012), p. 102208. doi: 10.1063/1.4755845. arXiv: 1205.0641

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.4755845 2012
[9]

Comparison of Quantum Statistical Models: Equivalent Conditions forSufficiency

F. Buscemi. “Comparison of Quantum Statistical Models: Equivalent Conditions forSufficiency”.In:Communications in Mathematical Physics310.3(2012),pp.625–

work page 2012
[10]

doi: 10.1007/s00220-012-1421-3

work page doi:10.1007/s00220-012-1421-3
[11]

Quantum Relative Lorenz Curves

F. Buscemi and G. Gour. “Quantum Relative Lorenz Curves”. In:Physical Review A 95.1 (2017), p. 012110.doi: 10.1103/PhysRevA.95.012110

work page doi:10.1103/physreva.95.012110 2017
[12]

Quantum majoriza- tion and a complete set of entropic conditions for quantum thermodynamics

G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Marvian. “Quantum majoriza- tion and a complete set of entropic conditions for quantum thermodynamics”. In: Nature Communications9.1 (2018), p. 5352.doi: 10.1038/s41467-018-06261-7

work page doi:10.1038/s41467-018-06261-7 2018
[13]

An information-theoretic treatment of quantum dichotomies

F. Buscemi, D. Sutter, and M. Tomamichel. “An information-theoretic treatment of quantum dichotomies”. In:Quantum 3 (2019). Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, p. 209. doi: 10 . 22331/q-2019-12-09-209

work page 2019
[14]

An example of a quantum statistical model which cannot be mapped to a less informative one by any trace preserving positive map

K. Matsumoto. An Example of a Quantum Statistical Model Which Cannot Be Mapped to a Less Informative One by Any Trace Preserving Positive Map. 2014. arXiv: 1409.5658

work page internal anchor Pith review Pith/arXiv arXiv 2014
[15]

Sufficiency of Rényi Divergences

N. Galke, L. van Luijk, and H. Wilming. “Sufficiency of Rényi Divergences”. In: IEEE Transactions on Information Theory70.7 (2024). Conference Name: IEEE Transactions on Information Theory, pp. 5057–5076.doi: 10.1109/TIT.2024. 3376395

work page doi:10.1109/tit.2024 2024
[16]

α-z-Rényi Relative Entropies

K. M. R. Audenaert and N. Datta. “α-z-Rényi Relative Entropies”. In:Journal of Mathematical Physics56.2 (2015), p. 022202.doi: 10.1063/1.4906367

work page doi:10.1063/1.4906367 2015
[17]

On α-z-Rényi Divergence in the von Neumann Algebra Setting

S. Kato. “On α-z-Rényi Divergence in the von Neumann Algebra Setting”. In: Journal of Mathematical Physics65.4 (2024), p. 042202.doi: 10.1063/5.0186552. 53

work page doi:10.1063/5.0186552 2024
[18]

Hiai and A

F. Hiai and A. Jenčová. In: Communications in Mathematical Physics 405.11 (2024), p. 271.doi: 10.1007/s00220-024-05124-1

work page doi:10.1007/s00220-024-05124-1 2024
[19]

Quasi-Entropies for States of a von Neumann Algebra

D. Petz. “Quasi-Entropies for States of a von Neumann Algebra”. In:Publications of the Research Institute for Mathematical Sciences21.4 (1985), pp. 787–800.doi: 10.2977/prims/1195178929

work page doi:10.2977/prims/1195178929 1985
[20]

Quasi-Entropies for Finite Quantum Systems

D. Petz. “Quasi-Entropies for Finite Quantum Systems”. In:Reports on Mathe- matical Physics23.1 (1986), pp. 57–65.doi: 10.1016/0034-4877(86)90067-4

work page doi:10.1016/0034-4877(86)90067-4 1986
[21]

Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Rela- tive Entropy

M. M. Wilde, A. Winter, and D. Yang. “Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Rela- tive Entropy”. In:Communications in Mathematical Physics331.2 (2014), pp. 593–

work page 2014
[22]

doi: 10.1007/s00220-014-2122-x

work page doi:10.1007/s00220-014-2122-x
[23]

On Quan- tum Rényi Entropies: A New Generalization and Some Properties

M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel. “On Quan- tum Rényi Entropies: A New Generalization and Some Properties”. In:Journal of Mathematical Physics54.12 (2013), p. 122203.doi: 10.1063/1.4838856

work page doi:10.1063/1.4838856 2013
[24]

Sandwiched Rényi Divergence Satisfies Data Processing Inequality

S. Beigi. “Sandwiched Rényi Divergence Satisfies Data Processing Inequality”. In: Journal of Mathematical Physics54.12 (2013), p. 122202. doi: 10.1063/1. 4838855

work page doi:10.1063/1 2013
[25]

Rényi Divergences as Weighted Non- commutative Vector-ValuedLp-Spaces

M. Berta, V. B. Scholz, and M. Tomamichel. “Rényi Divergences as Weighted Non- commutative Vector-ValuedLp-Spaces”. In:Annales Henri Poincaré19.6 (2018), pp. 1843–1867.doi: 10.1007/s00023-018-0670-x

work page doi:10.1007/s00023-018-0670-x 2018
[27]

Monotonicity of the Quantum Relative Entropy Under Positive Maps

A. Müller-Hermes and D. Reeb. “Monotonicity of the Quantum Relative Entropy Under Positive Maps”. In:Annales Henri Poincaré 18.5 (2017), pp. 1777–1788. doi: 10.1007/s00023-017-0550-9. arXiv: 1512.06117

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00023-017-0550-9 2017
[28]

Integral formula for quantum relative entropy implies data process- ing inequality

P. E. Frenkel. “Integral formula for quantum relative entropy implies data process- ing inequality”. In:Quantum 7 (2023), p. 1102.doi: 10.22331/q-2023-09-07- 1102

work page doi:10.22331/q-2023-09-07- 2023
[29]

Ohya and D

M. Ohya and D. Petz.Quantum Entropy and Its Use. Springer Berlin, Heidelberg, 1993

work page 1993
[30]

Sufficiencyin quantum statistical inference: a surveywith examples

A. JenčováandD. Petz. “Sufficiencyin quantum statistical inference: a surveywith examples”. In:Infinite Dimensional Analysis, Quantum Probability and Related Topics 09.3 (2006). Publisher: World Scientific Publishing Co., pp. 331–351.doi: 10.1142/S0219025706002408

work page doi:10.1142/s0219025706002408 2006
[31]

Minimal sufficient statistical experiments on von Neumann alge- bras

Y. Kuramochi. “Minimal sufficient statistical experiments on von Neumann alge- bras”. In:Journal of Mathematical Physics58.6 (2017), p. 062203.doi: 10.1063/ 1.4986247

work page 2017
[32]

Operationsthatdonotdisturbpartiallyknownquantum states

M.KoashiandN.Imoto.“Operationsthatdonotdisturbpartiallyknownquantum states”. In:Physical Review A66.2 (2002). Publisher: American Physical Society, p. 022318. doi: 10.1103/PhysRevA.66.022318. 54

work page doi:10.1103/physreva.66.022318 2002
[33]

Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

P. Hayden, R. Jozsa, D. Petz, and A. Winter. “Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality”. In:Communications in Mathematical Physics246.2 (2004), pp. 359–374.doi: 10.1007/s00220-004- 1049-z

work page doi:10.1007/s00220-004- 2004
[34]

Sufficient subalgebras and the relative entropy of states of a von Neu- mannalgebra

D. Petz. “Sufficient subalgebras and the relative entropy of states of a von Neu- mannalgebra”.In:Communications in Mathematical Physics105.1(1986),pp.123–

work page 1986
[35]

doi: 10.1007/BF01212345

work page doi:10.1007/bf01212345
[36]

Sufficiency of channels over von Neumann algebras

D. Petz. “Sufficiency of channels over von Neumann algebras”. In:The Quarterly Journal of Mathematics39.1 (1988), pp. 97–108.doi: 10.1093/qmath/39.1.97

work page doi:10.1093/qmath/39.1.97 1988
[37]

Sonoda, H

K. Sonoda, H. Arai, and M. Hayashi.Hypothesis testing and Stein’s lemma in gen- eral probability theories with Euclidean Jordan algebra and its quantum realization

work page
[38]

arXiv: 2505.02487

doi: 10.48550/arXiv.2505.02487. arXiv: 2505.02487

work page doi:10.48550/arxiv.2505.02487
[39]

C∗-Algebraic Generalization of Relative En- tropy and Entropy

V. P. Belavkin and P. Staszewski. “C∗-Algebraic Generalization of Relative En- tropy and Entropy”. In:Annales de l’institut Henri Poincaré. Section A, Physique Théorique 37.1 (1982), pp. 51–58

work page 1982
[40]

Quantum Hypothesis Testing and Sufficient Subalgebras

A. Jenčová. “Quantum Hypothesis Testing and Sufficient Subalgebras”. In:Letters in Mathematical Physics93.1 (2010), pp. 15–27.doi: 10.1007/s11005-010-0398- 0

work page doi:10.1007/s11005-010-0398- 2010
[41]

P.-C. Liu, C. Hirche, and H.-C. Cheng.Layer Cake Representations for Quantum Divergences. 2025. doi: 10.48550/arXiv.2507.07065. arXiv: 2507.07065

work page doi:10.48550/arxiv.2507.07065 2025
[42]

Hirche, C

C. Hirche, C. Rouzé, and D. S. França. Quantum Differential Privacy: An In- formation Theory Perspective. 2023. doi: 10.48550/arXiv.2202.10717 . arXiv: 2202.10717

work page doi:10.48550/arxiv.2202.10717 2023
[43]

Quantum Rényi andf-divergences from inte- gral representations

C. Hirche and M. Tomamichel. “Quantum Rényi andf-divergences from inte- gral representations”. In:Communications in Mathematical Physics405.9 (2024), p. 208. doi: 10.1007/s00220-024-05087-3

work page doi:10.1007/s00220-024-05087-3 2024
[44]

Cheng and P.-C

H.-C. Cheng and P.-C. Liu.Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem. 2025. arXiv:2507.06232

work page arXiv 2025
[45]

Cheng, G

H.-C. Cheng, G. Gour, L. Lami, and P.-C. Liu.The operator layer cake theorem is equivalent to Frenkel’s integral formula. 2025. arXiv:2512.04345

work page arXiv 2025
[46]

Preservation of a Quantum Rényi Relative Entropy Implies Existence of a Recovery Map

A. Jenčová. “Preservation of a Quantum Rényi Relative Entropy Implies Existence of a Recovery Map”. In:Journal of Physics A: Mathematical and Theoretical50.8 (2017), p. 085303.doi: 10.1088/1751-8121/aa5661

work page doi:10.1088/1751-8121/aa5661 2017
[47]

Rényi Relative Entropies and NoncommutativeLp-Spaces

A. Jenčová. “Rényi Relative Entropies and NoncommutativeLp-Spaces”. In:An- nales Henri Poincaré 19.8 (2018), pp. 2513–2542.doi: 10 . 1007 / s00023 - 018 - 0683-5

work page 2018
[48]

Rényi Relative Entropies and NoncommutativeLp-Spaces II

A. Jenčová. “Rényi Relative Entropies and NoncommutativeLp-Spaces II”. In: Annales Henri Poincaré22.10 (2021), pp. 3235–3254.doi: 10.1007/s00023-021- 01074-9

work page doi:10.1007/s00023-021- 2021
[49]

Hanche-Olsen and E

H. Hanche-Olsen and E. Størmer. Jordan Operator Algebras. Pitman Advanced Pub. Program, 1984

work page 1984
[50]

Recoverability of quantum channels via hypothesis testing

A. Jenčová. “Recoverability of quantum channels via hypothesis testing”. In:Let- ters in Mathematical Physics 114.1 (2024), p. 31. doi: 10 . 1007 / s11005 - 024 - 01775-2. 55

work page 2024
[51]

M. Fröb, G. Lechner, and R. Correa da Silva.Upcoming work. 2026

work page 2026
[52]

Quantum Conditional Mutual Information and Approxi- mate Markov Chains

O. Fawzi and R. Renner. “Quantum Conditional Mutual Information and Approxi- mate Markov Chains”. In:Communications in Mathematical Physics340.2 (2015), pp. 575–611.doi: 10.1007/s00220-015-2466-x

work page doi:10.1007/s00220-015-2466-x 2015
[53]

Recoverability in Quantum Information Theory

M. M. Wilde. “Recoverability in Quantum Information Theory”. In:Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences471.2182 (2015), p. 20150338.doi: 10.1098/rspa.2015.0338

work page doi:10.1098/rspa.2015.0338 2015
[54]

Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map

D. Sutter, M. Tomamichel, and A. W. Harrow. “Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map”. In:IEEE Transactions on In- formation Theory62.5 (2016), pp. 2907–2913.doi: 10.1109/TIT.2016.2545680

work page doi:10.1109/tit.2016.2545680 2016
[55]

Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy

M. Junge, R. Renner, D. Sutter, M. M. Wilde, and A. Winter. “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy”. In:Annales Henri Poincaré19.10 (2018), pp. 2955–2978.doi: 10.1007/s00023-018-0716-0

work page doi:10.1007/s00023-018-0716-0 2018
[56]

Faulkner, S

T. Faulkner, S. Hollands, B. Swingle, and Y. Wang.Approximate Recovery and Relative Entropy I. General von Neumann Subalgebras. 2020. arXiv:2006.08002

work page arXiv 2020
[57]

Approximate Recoverability and Relative Entropy II: 2-Positive Channels of General von Neumann Algebras

T. Faulkner and S. Hollands. “Approximate Recoverability and Relative Entropy II: 2-Positive Channels of General von Neumann Algebras”. In:Letters in Mathe- matical Physics112.2 (2022), p. 26.doi: 10.1007/s11005-022-01510-9

work page doi:10.1007/s11005-022-01510-9 2022
[58]

Positive linear maps of operator algebras

E. Størmer. “Positive linear maps of operator algebras”. In:Acta Mathematica 110.1 (1963), pp. 233–278.doi: 10.1007/BF02391860

work page doi:10.1007/bf02391860 1963
[59]

Multiplicative Properties of Positive Maps

E. Størmer. “Multiplicative Properties of Positive Maps”. In:Mathematica Scan- dinavica 100.1 (2007), pp. 184–192.doi: 10.7146/math.scand.a-15020

work page doi:10.7146/math.scand.a-15020 2007
[60]

E. Størmer. Positive Linear Maps of Operator Algebras. Springer Monographs in Mathematics. Berlin, Heidelberg: Springer, 2013. doi: 10 . 1007 / 978 - 3 - 642 - 34369-8

work page 2013
[61]

Brown and N

N. Brown and N. Ozawa. C*-Algebras and Finite-Dimensional Approximations. Vol. 88. Graduate Studies in Mathematics. American Mathematical Society, 2008. doi: 10.1090/gsm/088

work page doi:10.1090/gsm/088 2008
[62]

M. M. Wolf.Quantum channels and operations-guided tour. Lecture notes available at https://math.cit.tum.de/math/personen/professuren/wolf- michael . 2012

work page 2012
[63]

Quantum Compression and Fixed Points of Schwarz Maps

L. Rauber. “Quantum Compression and Fixed Points of Schwarz Maps”. Master’s thesis at Technical University of Munich. Supervisor: Michael Wolf. 2018

work page 2018
[64]

Selfpolar forms and their applications to the C*-algebra the- ory

S. L. Woronowicz. “Selfpolar forms and their applications to the C*-algebra the- ory”. In:Reports on Mathematical Physics6.3 (1974), pp. 487–495.doi: 10.1016/ S0034-4877(74)80012-1

work page 1974
[65]

Tomita-Takesaki Theory for Jordan Alge- bras

U. Haagerup and H. Hanche-Olsen. “Tomita-Takesaki Theory for Jordan Alge- bras”. In:Journal of Operator Theory11.2 (1984). Publisher: Theta Foundation, pp. 343–364

work page 1984
[66]

Conditional Expectations on Jordan Algebras

C. M. Edwards. “Conditional Expectations on Jordan Algebras”. In:Fundamen- tal Aspects of Quantum Theory. Ed. by V. Gorini and A. Frigerio. Boston, MA: Springer US, 1986, pp. 75–79.doi: 10.1007/978-1-4684-5221-1_8. 56

work page doi:10.1007/978-1-4684-5221-1_8 1986
[67]

Positive projections of von Neumann algebras onto JW-algebras

U. Haagerup and E. Størmer. “Positive projections of von Neumann algebras onto JW-algebras”.In:Reports on Mathematical Physics.ProceedingsoftheXXVISym- posium on Mathematical Physics 36.2 (1995), pp. 317–330.doi: 10.1016/0034- 4877(96)83628-7

work page doi:10.1016/0034- 1995
[68]

Some aspects of quantum sufficiency for finite and full von Neumann algebras

A. Łuczak. “Some aspects of quantum sufficiency for finite and full von Neumann algebras”. In:Letters in Mathematical Physics111.4 (2021), p. 95.doi: 10.1007/ s11005-021-01428-8

work page 2021
[69]

Conditional Expectations in von Neumann Algebras

M. Takesaki. “Conditional Expectations in von Neumann Algebras”. In:Journal of Functional Analysis9.3 (1972), pp. 306–321.doi: 10.1016/0022-1236(72)90004- 3

work page doi:10.1016/0022-1236(72)90004- 1972
[70]

J. V. Neumann. Functional Operators: Volume 2, The Geometry of Orthogonal Spaces. Annals of Mathematics Studies number 22. Princeton, NJ: Princeton Uni- versity Press, 1951. 1 p

work page 1951
[71]

The product of projection operators

I. Halperin. “The product of projection operators”. In:Acta Sci. Math. (Szeged) 23.1 (1962), pp. 96–99

work page 1962
[72]

On an Algebraic Generalization of the Quantum Mechanical Formalism

P. Jordan, J. v. Neumann, and E. Wigner. “On an Algebraic Generalization of the Quantum Mechanical Formalism”. In:Annals of Mathematics35.1 (1934), pp. 29–

work page 1934
[73]

doi: 10.2307/1968117

work page doi:10.2307/1968117
[74]

Jacobson

N. Jacobson. Structure and Representations of Jordan Algebras. Vol. 39. Collo- quium Publications. Providence, Rhode Island: American Mathematical Society,

work page
[75]

doi: 10.1090/coll/039

work page doi:10.1090/coll/039
[76]

On the structure of positive maps

M. Idel. “On the structure of positive maps”. Master’s thesis at Technical Univer- sity of Munich. Supervisor: Michael Wolf. 2013

work page 2013
[77]

Classification and Representation of Semi- Simple Jordan Algebras

F. D. Jacobson and N. Jacobson. “Classification and Representation of Semi- Simple Jordan Algebras”. In:Transactions of the American Mathematical Soci- ety 65.2 (1949). Publisher: American Mathematical Society, pp. 141–169.doi: 10.2307/1990643

work page doi:10.2307/1990643 1949
[78]

Decomposition of Positive Projections on C*-Algebras

E. Størmer. “Decomposition of Positive Projections on C*-Algebras.” In:Mathe- matische Annalen247 (1980), pp. 21–42

work page 1980
[79]

Nonassociative Lp-spaces and embeddings in noncommutative Lp- spaces

C. Arhancet. “Nonassociative Lp-spaces and embeddings in noncommutative Lp- spaces”.In:Journal of Mathematical Analysis and Applications537.1(2024),p.128307. doi: 10.1016/j.jmaa.2024.128307

work page doi:10.1016/j.jmaa.2024.128307 2024
[80]

Chapter 25 - Quasi-Banach Spaces

N. Kalton. “Chapter 25 - Quasi-Banach Spaces”. In: ed. by W. Johnson and J. Lindenstrauss. Vol. 2. Handbook of the Geometry of Banach Spaces. Elsevier Science B.V., 2003, pp. 1099–1130.doi: https : / / doi . org / 10 . 1016 / S1874 - 5849(03)80032-3

work page 2003
[81]

Quantum Sufficiency in the Operator Algebra Framework

A. Łuczak. “Quantum Sufficiency in the Operator Algebra Framework”. In:In- ternational Journal of Theoretical Physics53.10 (2014), pp. 3423–3433.doi: 10. 1007/s10773-013-1747-4

work page 2014

Showing first 80 references.