Sufficient conditions for the Kadison--Schwarz property of unital positive maps on $M_3$

TL;DR

This paper derives explicit analytic sufficient conditions for the Kadison--Schwarz property of unital positive maps on 3x3 matrices, using algebraic and structural analysis without numerical methods.

Contribution

It provides the first explicit analytic criteria for the KS property on M_3, expanding understanding beyond low-dimensional or symmetric cases.

Findings

Conditions depend on Bloch parameters and Lie algebra structure

Reduces problem to estimates involving symmetric tensor d_{ijk}

Clarifies weaker conditions than complete positivity for KS property

Abstract

Kadison--Schwarz (KS) maps form a natural class of positive linear maps lying strictly between positivity and complete positivity. Despite their relevance in operator algebras and quantum dynamics, explicit analytic sufficient conditions for the KS property remain scarce beyond low-dimensional or highly symmetric settings. In this work we analyze unital positive linear maps on $M_3$ within the Bloch--Gell--Mann representation and derive explicit analytic sufficient conditions ensuring the Kadison--Schwarz property. The approach exploits unitary equivalence together with structural properties of the Lie algebra $\mathfrak{su}(3)$ and does not rely on numerical optimization or semidefinite-programming methods. A key mechanism is the cancellation of contributions associated with antisymmetric structure constants, which reduces the problem to estimates governed solely by the symmetric tensor $d_{ijk}$. The results clarify how the Kadison--Schwarz property can hold under assumptions substantially weaker than complete positivity and yield a structural criterion for the KS property on $M_3$ in terms of Bloch parameters.