Relative Kubo-Ando Means of Completely Positive Maps

TL;DR

This paper introduces a new theory of relative Kubo--Ando means for completely positive maps, extending existing concepts with order-theoretic properties and geometric interpretations.

Contribution

It develops a novel framework for relative Kubo--Ando means of completely positive maps, including intrinsic and geometric means, with proven properties and comparisons to existing methods.

Findings

Means are independent of Stinespring representation.

Intrinsic geometric mean vanishes when maps lack common submaps.

Construction aligns with known matrix algebra means in special cases.

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.