Relative Kubo-Ando Means of Completely Positive Maps

Mohsen Kian

arxiv: 2605.11701 · v2 · pith:OFUEZF7Jnew · submitted 2026-05-12 · 🧮 math.OA · math.FA

Relative Kubo-Ando Means of Completely Positive Maps

Mohsen Kian This is my paper

Pith reviewed 2026-05-20 22:09 UTC · model grok-4.3

classification 🧮 math.OA math.FA

keywords Kubo-Ando meanscompletely positive mapsArveson Radon-Nikodym theoremorder intervalsgeometric meanStinespring representationsmonotonicitydata processing inequality

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The pith

Relative Kubo-Ando means for pairs of completely positive maps dominated by an ambient map are independent of Stinespring representations.

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

The paper defines relative Kubo-Ando means for completely positive maps that are both dominated by a larger ambient map. It uses Arveson's Radon-Nikodym theorem to construct these means on order intervals and proves they remain unchanged under different choices of Stinespring dilation. The means satisfy the standard Kubo-Ando properties including monotonicity, transformer inequalities, Jensen-type inequalities, and data processing. A special intrinsic geometric mean is shown to be zero precisely when the two maps share no nonzero common completely positive submap. The construction recovers known means in finite dimensions and certain form-theoretic cases.

Core claim

For completely positive maps Φ and Ψ dominated by an ambient completely positive map Ω, the relative Kubo-Ando mean Φ σ_Ω Ψ is defined using the Radon-Nikodym derivative guaranteed by Arveson's theorem. This mean is independent of the chosen Stinespring representation. It satisfies monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and monotonicity with respect to the ambient map. When Ω equals Φ plus Ψ, the resulting intrinsic geometric mean vanishes if and only if Φ and Ψ have no nonzero common completely positive submap.

What carries the argument

The relative Kubo-Ando mean Φ σ_Ω Ψ constructed on the order interval of completely positive maps via Arveson's Radon-Nikodym theorem.

If this is right

The means increase when either argument map increases in the operator order.
They obey transformer inequalities under composition with auxiliary maps.
Jensen-type inequalities hold when convex functions are applied to the means.
Data processing inequalities are preserved under completely positive maps.
The intrinsic geometric mean is exactly zero in the absence of a nonzero common submap.

Where Pith is reading between the lines

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

The order-interval approach may allow similar means to be defined for other cones of positive maps that admit a Radon-Nikodym theorem.
The block-positivity characterization of the geometric mean could be used to test common submaps algorithmically in finite dimensions.
Agreement with the Choi-matrix mean suggests the construction is a natural infinite-dimensional extension of existing matrix means.

Load-bearing premise

The two completely positive maps must be dominated by a single common ambient completely positive map.

What would settle it

Two different Stinespring representations of the same pair of maps that produce unequal values for the relative mean would show the claimed independence fails.

read the original abstract

We develop a Kubo--Ando theory on order intervals of completely positive maps. Using Arveson's Radon--Nikodym theorem as a structural tool, we define relative Kubo--Ando means \(\Phi\sigma_\Omega\Psi\) for completely positive maps dominated by a common ambient map \(\Omega\). The special choice \(\Omega=\Phi+\Psi\) yields an intrinsic mean of two completely positive maps. We prove that these means are independent of the chosen Stinespring representation and satisfy the expected order-theoretic properties, including monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and monotonicity with respect to the ambient map. For the geometric mean, we obtain a block-positivity characterization and show that the intrinsic geometric mean vanishes exactly when the two maps have no nonzero common completely positive submap. Finally, we compare the construction with existing finite-dimensional and form-theoretic approaches: for maps between matrix algebras it agrees with the Choi-matrix mean, and in the geometric case it agrees with Okayasu's Pusz--Woronowicz mean on their common domain.

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

Kian defines relative Kubo-Ando means for CP maps below a common ambient map, proves independence from the Stinespring dilation, and gives a clean vanishing condition for the geometric case.

read the letter

The paper's main contribution is a definition of relative Kubo-Ando means for pairs of completely positive maps that are both below some dominating map Ω. By applying Arveson's Radon-Nikodym theorem in a Stinespring dilation of Ω, the author constructs these means and then proves they are independent of the particular dilation chosen. This independence is not automatic, but the work shows it holds, which makes the means well-defined on the original algebra. The special case where Ω is the sum of the two maps gives an intrinsic mean without an external reference. For the geometric mean, there is a block-positivity description and the result that it is zero exactly when the maps share no nonzero common completely positive submap. These are clean statements. The paper also verifies the standard properties one wants: monotonicity in each argument, transformer inequalities, Jensen-type inequalities, and data processing. In the finite-dimensional setting it recovers the mean coming from the Choi matrix, and in the geometric case it agrees with Okayasu's version of the Pusz-Woronowicz mean. Those consistency checks are valuable. The softer aspect is that most of the development stays at the level of abstract order properties. While the proofs are claimed to go through, a reader might want to see how the arguments handle the commutant or the specific form of the Radon-Nikodym derivatives in more detail. The lack of many worked examples in infinite dimensions means the practical difference from existing means is not yet fully visible, though that is common in this area. This paper is for people working on means of operators and maps in operator algebras. Someone interested in extending classical inequalities to the setting of CP maps will find the framework useful. It is not aimed at a broad audience outside the subfield. I think the work is solid enough on its own terms to merit a serious referee. The construction is new, the independence is addressed, and the comparisons to known cases are in place.

Referee Report

1 major / 2 minor

Summary. The manuscript develops relative Kubo-Ando means Φσ_ΩΨ for completely positive maps Φ and Ψ dominated by a common ambient map Ω, constructed via Arveson's Radon-Nikodym theorem applied to a Stinespring dilation of Ω. It asserts that these means are independent of the Stinespring representation, satisfy monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and monotonicity in Ω, and that the intrinsic geometric mean (Ω = Φ + Ψ) vanishes precisely when Φ and Ψ share no nonzero common completely positive submap. The construction is shown to agree with the Choi-matrix mean for maps on matrix algebras and with Okayasu's Pusz-Woronowicz mean in the geometric case, together with a block-positivity characterization for the geometric mean.

Significance. If the independence and order-theoretic properties hold, the work supplies a coherent extension of classical Kubo-Ando theory to order intervals of completely positive maps, furnishing intrinsic means and a vanishing criterion that links the geometric mean to the absence of common submaps. The explicit comparison with the Choi-matrix construction and with Okayasu's form-theoretic mean supplies useful consistency checks across finite- and infinite-dimensional settings. These results could serve as tools for studying structural properties of CP maps in operator algebras and quantum information.

major comments (1)

[§3 (Independence of Stinespring representation)] §3 (Independence of Stinespring representation): The proof that Φσ_ΩΨ is independent of the choice of Stinespring dilation (π, V) of Ω must explicitly address non-minimal dilations. When the dilation Hilbert space contains orthogonal summands outside the range of V, the Kubo-Ando mean of the lifted operators A_Φ and A_Ψ must be shown to compress to the same map after application of V^*; the current argument leaves open whether extra orthogonal components can alter the result, which is load-bearing for the well-definedness of the relative means and all subsequent monotonicity and data-processing claims.

minor comments (2)

[Abstract and §2] Notation for the mean is introduced as Φσ_ΩΨ in the abstract and should be used uniformly in all displayed equations and statements throughout the text to avoid subscript ambiguity.
[§4] The block-positivity characterization of the geometric mean would benefit from a short parenthetical reminder of the precise definition of block positivity employed, for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the independence proof. We address the point below.

read point-by-point responses

Referee: §3 (Independence of Stinespring representation): The proof that Φσ_ΩΨ is independent of the choice of Stinespring dilation (π, V) of Ω must explicitly address non-minimal dilations. When the dilation Hilbert space contains orthogonal summands outside the range of V, the Kubo-Ando mean of the lifted operators A_Φ and A_Ψ must be shown to compress to the same map after application of V^*; the current argument leaves open whether extra orthogonal components can alter the result, which is load-bearing for the well-definedness of the relative means and all subsequent monotonicity and data-processing claims.

Authors: We agree that the independence argument in §3 would benefit from an explicit treatment of non-minimal dilations to close the potential gap. In the revised version we will add a short paragraph (or lemma) showing that if the dilation space decomposes as H_Ω = ran(V) ⊕ K^⊥, then both A_Φ and A_Ψ vanish on K^⊥ because Φ, Ψ ≼ Ω; consequently any operator-monotone function of A_Φ and A_Ψ (in particular the Kubo-Ando mean) is likewise supported on ran(V). Compression by V^* therefore yields the same completely positive map irrespective of the orthogonal summand. This extension relies only on the support properties already implicit in Arveson’s Radon–Nikodym theorem and does not alter any other claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external theorem and explicit invariance proof keep derivation self-contained

full rationale

The paper applies Arveson's Radon-Nikodym theorem (an external structural result) to define the relative means on the order interval induced by a dominating ambient map Ω. It then proves independence from the choice of Stinespring representation by direct argument rather than by construction or self-reference. Monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and the block-positivity characterization for the geometric mean are all derived from the order-theoretic properties of the means and the ambient domination assumption. Special cases are shown to recover the Choi-matrix mean and Okayasu's Pusz-Woronowicz mean as consistency checks, not as definitional inputs. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same author. The construction is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction depends on Arveson's Radon-Nikodym theorem as the key external tool; no free parameters or new invented entities are indicated in the abstract.

axioms (1)

standard math Arveson's Radon-Nikodym theorem for completely positive maps
Invoked as the structural tool to define the relative means on order intervals.

pith-pipeline@v0.9.0 · 5708 in / 1227 out tokens · 66213 ms · 2026-05-20T22:09:48.659076+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.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We define the Ω-relative Kubo-Ando mean ... (Φσ_ΩΨ)(X) = V* π(X) (D^Ω_Φ σ D^Ω_Ψ) V ... independent of the chosen minimal Stinespring representation
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

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

For the geometric mean ... block-positivity characterization ... Φ#Ψ = 0 iff [0,Φ] ∩ [0,Ψ] = {0}

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

11 extracted references · 11 canonical work pages · 1 internal anchor

[1]

W. B. Arveson,Subalgebras ofC ∗-algebras, Acta Math.123(1969), 141–224

work page 1969
[2]

Bhatia,Positive Definite Matrices, Princeton University Press, Princeton, 2007

R. Bhatia,Positive Definite Matrices, Princeton University Press, Princeton, 2007

work page 2007
[3]

Jamio lkowski,Linear transformations which preserve trace and positive semidefiniteness of opera- tors, Rep

A. Jamio lkowski,Linear transformations which preserve trace and positive semidefiniteness of opera- tors, Rep. Math. Phys.3(1972), 275–278

work page 1972
[4]

Choi,Completely positive linear maps on complex matrices, Linear Algebra Appl.10(1975), 285–290

M.-D. Choi,Completely positive linear maps on complex matrices, Linear Algebra Appl.10(1975), 285–290

work page 1975
[5]

Gheondea and A

A. Gheondea and A. S. Kavruk,Absolute continuity for operator valued completely positive maps on C ∗-algebras, J. Math. Phys.50(2009), 022102. 28 M. KIAN

work page 2009
[6]

Hollands and A

S. Hollands and A. Ranallo,Channel divergences and complexity in algebraic QFT, Commun. Math. Phys.404(2023), 927–962

work page 2023
[7]

Kubo and T

F. Kubo and T. Ando,Means of positive linear operators, Math. Ann.246(1980), 205–224

work page 1980
[8]

P. E. Frenkel, M. Mosonyi, P. Vrana, and M. Weiner,Error bounds for composite quantum hypothesis testing and a new characterization of the weighted Kubo–Ando geometric means, arXiv:2503.13379, 2025

work page arXiv 2025
[9]

Geometric Means and Lebesgue-type Decomposition of Completely Positive Maps

R. Okayasu,Geometric Means and Lebesgue-type Decomposition of Completely Positive Maps, arXiv:2605.06019v1, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[10]

V. I. Paulsen,Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, Vol.78, Cambridge University Press, Cambridge, 2002

work page 2002
[11]

Pusz and S

W. Pusz and S. L. Woronowicz,Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys.8(1975), 159–170. Mohsen Kian: Department of Mathematics, University of Bojnord, Bojnord 94531, Iran Email address:kian@ub.ac.ir

work page 1975

[1] [1]

W. B. Arveson,Subalgebras ofC ∗-algebras, Acta Math.123(1969), 141–224

work page 1969

[2] [2]

Bhatia,Positive Definite Matrices, Princeton University Press, Princeton, 2007

R. Bhatia,Positive Definite Matrices, Princeton University Press, Princeton, 2007

work page 2007

[3] [3]

Jamio lkowski,Linear transformations which preserve trace and positive semidefiniteness of opera- tors, Rep

A. Jamio lkowski,Linear transformations which preserve trace and positive semidefiniteness of opera- tors, Rep. Math. Phys.3(1972), 275–278

work page 1972

[4] [4]

Choi,Completely positive linear maps on complex matrices, Linear Algebra Appl.10(1975), 285–290

M.-D. Choi,Completely positive linear maps on complex matrices, Linear Algebra Appl.10(1975), 285–290

work page 1975

[5] [5]

Gheondea and A

A. Gheondea and A. S. Kavruk,Absolute continuity for operator valued completely positive maps on C ∗-algebras, J. Math. Phys.50(2009), 022102. 28 M. KIAN

work page 2009

[6] [6]

Hollands and A

S. Hollands and A. Ranallo,Channel divergences and complexity in algebraic QFT, Commun. Math. Phys.404(2023), 927–962

work page 2023

[7] [7]

Kubo and T

F. Kubo and T. Ando,Means of positive linear operators, Math. Ann.246(1980), 205–224

work page 1980

[8] [8]

P. E. Frenkel, M. Mosonyi, P. Vrana, and M. Weiner,Error bounds for composite quantum hypothesis testing and a new characterization of the weighted Kubo–Ando geometric means, arXiv:2503.13379, 2025

work page arXiv 2025

[9] [9]

Geometric Means and Lebesgue-type Decomposition of Completely Positive Maps

R. Okayasu,Geometric Means and Lebesgue-type Decomposition of Completely Positive Maps, arXiv:2605.06019v1, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[10] [10]

V. I. Paulsen,Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, Vol.78, Cambridge University Press, Cambridge, 2002

work page 2002

[11] [11]

Pusz and S

W. Pusz and S. L. Woronowicz,Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys.8(1975), 159–170. Mohsen Kian: Department of Mathematics, University of Bojnord, Bojnord 94531, Iran Email address:kian@ub.ac.ir

work page 1975