Tomiyama-type maps with a diagonal perturbation

TL;DR

This paper characterizes the positivity conditions of a two-parameter family of linear maps on matrix algebras, using the Choi matrix method to provide explicit criteria and geometric insights.

Contribution

It introduces a new class of Tomiyama-type maps with diagonal perturbations and derives explicit positivity conditions across all dimensions.

Findings

Explicit necessary and sufficient conditions for positivity, complete positivity, and k-positivity.

Geometric characterization of positivity regions in parameter space.

Application of Choi matrix and block-positivity techniques.

Abstract

We investigate a two-parameter family of linear maps on matrix algebras, constructed as diagonal perturbations of classical Tomiyama maps. Employing the Choi matrix method alongside block-positivity techniques, we derive explicit necessary and sufficient conditions for positivity, complete positivity, and k-positivity across arbitrary dimensions. These conditions provide a transparent geometric characterization of the positivity regions within the parameter space.