Relative Kubo-Ando Means of Completely Positive Maps

Abstract: We introduce relative and intrinsic Kubo--Ando means for completely positive maps on (C^*)-algebras. These means extend the usual Kubo--Ando means of positive operators and are defined using Arveson's Radon--Nikodym theorem for completely positive maps. We prove their basic order-theoretic properties, including monotonicity, transformer and Jensen inequalities, data processing, and monotonicity with respect to the ambient map. In the geometric case, 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. We further prove agreement with the Choi-matrix mean for maps between matrix algebras and with Okayasu's Pusz--Woronowicz geometric mean on their common domain.

• The paper constructs a canonical, representation-independent Kubo–Ando calculus for completely positive maps by applying operator means to Arveson Radon–Nikodym derivatives in a commutant.
• It proves transformer, Jensen, ambient-monotonicity, and finite-Kraus data-processing inequalities, establishing order-theoretic tools relevant to operator algebras and quantum channel theory.
• It characterizes the geometric mean through completely positive block maps, shows that its vanishing detects disjointness in the CP order, and recovers Choi-matrix and Okayasu constructions in their respective settings.

Overview

This paper, by Mohsen Kian, constructs a Kubo–Ando calculus for completely positive (CP) maps on $C^*$-algebras. The central observation is that Arveson's Radon–Nikodym theorem identifies the order interval $[0,\Omega]$ of CP maps dominated by a fixed ambient map $\Omega:\mathscr A\to B(\mathcal H)$ with the operator interval of positive contractions in the commutant $\pi(\mathscr A)'$ of a minimal Stinespring representation. Applying any Kubo–Ando operator mean $\sigma$ to the Radon–Nikodym derivatives and transporting back yields a well-defined binary operation $\Phi\sigma_\Omega\Psi$ on $[0,\Omega]$, called the relative mean; taking $\Omega=\Phi+\Psi$ gives the intrinsic mean $\Phi\sigma\Psi$.

The construction is canonical: it is independent of the choice of minimal Stinespring representation (by unitary equivalence of such representations and congruence invariance of $\sigma$), and it recovers the classical Kubo–Ando mean when $[0,\Omega]$0. The paper establishes the expected order-theoretic axioms, proves a block-positivity characterization of the geometric mean, shows that the intrinsic geometric mean detects disjointness in the CP order, and verifies agreement with two existing constructions: the Choi-matrix mean in finite dimensions and Okayasu's Pusz–Woronowicz geometric mean for normal CP maps on von Neumann algebras.

The relative mean and its basic properties

Given $[0,\Omega]$1 and a minimal Stinespring representation $[0,\Omega]$2, Arveson's theorem supplies unique positive contractions $[0,\Omega]$3 with $[0,\Omega]$4 and similarly for $[0,\Omega]$5. The relative mean is defined by

$[0,\Omega]$6

Since $[0,\Omega]$7 lie in the commutant, so does $[0,\Omega]$8, hence the result is again CP; monotonicity of $[0,\Omega]$9 gives $\Omega:\mathscr A\to B(\mathcal H)$0. A key structural fact is that the Radon–Nikodym correspondence is an order isomorphism under which the relative mean is exactly the pullback of the operator mean:

$\Omega:\mathscr A\to B(\mathcal H)$1

Consequently all fixed-ambient Kubo–Ando properties transfer verbatim: closure, joint monotonicity, positive homogeneity $\Omega:\mathscr A\to B(\mathcal H)$2, idempotence, symmetry for symmetric means, and the sharp bounds

$\Omega:\mathscr A\to B(\mathcal H)$3

where $\Omega:\mathscr A\to B(\mathcal H)$4 for the representing function $\Omega:\mathscr A\to B(\mathcal H)$5 of $\Omega:\mathscr A\to B(\mathcal H)$6. Two illustrative examples are worked out: tensor-amplification maps $\Omega:\mathscr A\to B(\mathcal H)$7 satisfy $\Omega:\mathscr A\to B(\mathcal H)$8 for every Kubo–Ando mean, and for positive functionals on $\Omega:\mathscr A\to B(\mathcal H)$9 the intrinsic geometric mean is the functional whose density is the pointwise scalar geometric mean $\pi(\mathscr A)'$0.

Transformer inequality, Jensen inequality, and data processing

The paper then develops results comparing means across different ambient maps. The transformer inequality states that for every bounded operator $\pi(\mathscr A)'$1,

$\pi(\mathscr A)'$2

The proof compresses the minimal Stinespring space to the reducing subspace $\pi(\mathscr A)'$3, observes that the derivatives of $\pi(\mathscr A)'$4 and $\pi(\mathscr A)'$5 with respect to $\pi(\mathscr A)'$6 are the compressed operators $\pi(\mathscr A)'$7 and $\pi(\mathscr A)'$8, and applies the classical transformer inequality to the inclusion map. Equality holds whenever $\pi(\mathscr A)'$9 reduces both $\sigma$0 and $\sigma$1 — a clean equality criterion stated as a separate proposition.

Two consequences follow. First, a $\sigma$2-Jensen inequality: for finite families $\sigma$3 and bounded operators $\sigma$4,

$\sigma$5

proved via direct sums plus the transformer inequality; this relies on the fact that the relative mean respects finite orthogonal direct sums. Taking each ambient map to be the sum of its arguments yields intrinsic $\sigma$6-concavity. Second, a data-processing inequality: for any CP post-processing $\sigma$7 with finite Kraus form,

$\sigma$8

which is the natural contractivity statement relevant to quantum channel divergences. Note that the data-processing result is restricted to finite Kraus representations; an extension to general normal CP maps is not addressed.

Finally, the mean is monotone in the ambient map: if $\sigma$9 and $\Phi\sigma_\Omega\Psi$0, then $\Phi\sigma_\Omega\Psi$1. The proof builds a minimal Stinespring representation of $\Phi\sigma_\Omega\Psi$2 from that of $\Phi\sigma_\Omega\Psi$3 using the support projection of the derivative $\Phi\sigma_\Omega\Psi$4, and applies the transformer inequality to $\Phi\sigma_\Omega\Psi$5.

Finite-dimensional formula

For $\Phi\sigma_\Omega\Psi$6 with minimal Kraus representation $\Phi\sigma_\Omega\Psi$7, every dominated CP map has a unique matrix representative: $\Phi\sigma_\Omega\Psi$8 with $\Phi\sigma_\Omega\Psi$9. The relative mean is then computed coefficientwise,

$[0,\Omega]$0

which follows from the commutant identification $[0,\Omega]$1 and the tensor identity $[0,\Omega]$2.

The relative geometric mean

Specializing to $[0,\Omega]$3, the paper proves that $[0,\Omega]$4 is the largest CP map $[0,\Omega]$5 such that the block map

$[0,\Omega]$6

is completely positive. The forward direction uses positivity of the commutant element $[0,\Omega]$7; the converse shows that any admissible $[0,\Omega]$8 satisfies $[0,\Omega]$9 (via compression by $\Omega=\Phi+\Psi$0), so its derivative $\Omega=\Phi+\Psi$1 exists, and minimality of the doubled Stinespring representation forces the block matrix $\Omega=\Phi+\Psi$2 to be positive, whence $\Omega=\Phi+\Psi$3.

An important corollary, stated by the author as a remark: since the admissible class of $\Omega=\Phi+\Psi$4 depends only on $\Omega=\Phi+\Psi$5 and $\Omega=\Phi+\Psi$6, the relative geometric mean is independent of the dominating map, i.e., $\Omega=\Phi+\Psi$7. This independence fails for general Kubo–Ando means, where the ambient-map monotonicity theorem shows genuine dependence.

The second main geometric result is a disjointness criterion: $\Omega=\Phi+\Psi$8 if and only if $\Omega=\Phi+\Psi$9. When $\Phi\sigma\Psi$0, the derivatives commute and sum to $\Phi\sigma\Psi$1, so $\Phi\sigma\Psi$2, which vanishes exactly when $\Phi\sigma\Psi$3 is a projection; and $\Phi\sigma\Psi$4 is a projection exactly when no nonzero CP map is dominated by both. An explicit example makes this concrete: on $\Phi\sigma\Psi$5, the identity map and conjugation by $\Phi\sigma\Psi$6 are both faithful unital CP maps, yet their intrinsic geometric mean vanishes. Thus zero geometric mean reflects disjointness in the CP order rather than orthogonality of the values $\Phi\sigma\Psi$7 and $\Phi\sigma\Psi$8 — a point worth emphasizing, since it distinguishes this notion sharply from the operator-level intuition.

Comparison with existing constructions

Choi-matrix mean. For CP maps $\Phi\sigma\Psi$9, define $\sigma$0 via the Choi–Jamiołkowski correspondence. The paper proves the exact identity

$\sigma$1

using the Kraus formula and a congruence argument with the injective operator $\sigma$2. Consequently, in the full matrix-algebra case the relative mean is independent of $\sigma$3 for every Kubo–Ando mean, not just the geometric one, and coincides with the Choi-matrix construction — recovering in particular the weighted geometric means used by Frenkel–Mosonyi–Vrana–Weiner in composite quantum channel discrimination. The advantage of the present formulation is that it remains meaningful for arbitrary $\sigma$4-algebras, where no Choi matrix exists.

Okayasu's Pusz–Woronowicz mean. For normal CP maps $\sigma$5 from a von Neumann algebra into $\sigma$6, the intrinsic geometric mean agrees with Okayasu's construction based on the Pusz–Woronowicz functional calculus. The proof is short and rests on the fact that both means admit the same maximality characterization as the largest off-diagonal corner of a CP $\sigma$7 block map. The author notes that the frameworks are complementary: Okayasu's approach develops parallel sums and Lebesgue-type decompositions, while the Radon–Nikodym framework provides relative means for arbitrary Kubo–Ando means over arbitrary order intervals.

Limitations and open questions

Several restrictions should be noted. The transformer and Jensen inequalities require unitality of $\sigma$8 in some statements, and the data-processing inequality is proved only for CP maps with finite Kraus representations, leaving the general case open. The agreement with Okayasu's mean is established only for normal CP maps on von Neumann algebras, and the comparison there covers only the geometric mean, not the full Kubo–Ando family. Independence of the dominating map holds for the geometric mean (and, in matrix algebras, for all means), but for general Kubo–Ando means on general $\sigma$9-algebras the relative mean genuinely depends on $[0,\Omega]$00, and the paper does not characterize when this dependence disappears. Whether the intrinsic means satisfy additional structural properties known at the operator level — such as Ando–Hiai type log-majorization inequalities or connections to Riemannian geometry on the CP cone — is not addressed.

Conclusion

The paper supplies a canonical, representation-independent extension of the Kubo–Ando axiomatic theory from positive operators to completely positive maps, built on Arveson's Radon–Nikodym theorem. It verifies the standard order-theoretic properties (monotonicity, transformer and Jensen inequalities, data processing, ambient-map monotonicity), characterizes the geometric mean by block positivity, links vanishing of the intrinsic geometric mean to disjointness in the CP order, and proves exact agreement with both the Choi-matrix mean in finite dimensions and Okayasu's Pusz–Woronowicz geometric mean where the latter is defined. The result positions the Radon–Nikodym transport principle as a unifying device for mean-theoretic constructions on cones of CP maps, with potential relevance to channel divergence theory in quantum information.