Optimizing positive maps in the matrix algebra $M_n$

TL;DR

This paper develops an optimization method for positive maps in matrix algebras, extending known results to cases where the greatest common divisor of parameters is 2 or 3, combining analytical and numerical techniques.

Contribution

It introduces a new optimization approach for positive maps in matrix algebras and addresses cases with GCD(n,k)=2 or 3, expanding previous results.

Findings

Optimization procedure for GCD(n,k)=2 derived analytically.

Numerical analysis for GCD(n,k)=3 provided.

Conjecture on optimality for GCD(n,k)=2 and 3 supported.

Abstract

We present an optimization procedure for a seminal class of positive maps $\tau_{n,k}$ in the algebra of $n \times n$ complex matrices introduced and studied by Tanahasi and Tomiyama, Ando, Nakamura and Osaka. Recently, these maps were proved to be optimal whenever the greatest common divisor $GCD(n,k)=1$. We attain a general conjecture how to optimize a map $\tau_{n,k}$ when $GCD(n,k)=2$ or 3. For $GCD(n,k)=2$, a series of analytical results are derived and for $GCD(n,k)=3$, we provide a suitable numerical analysis.