On a family of a linear maps from $M_{n}(\mathbb{C})$ to $M_{n^{2}}(\mathbb{C})$

TL;DR

This paper introduces a parametric family of linear maps from $M_{n}(C)$ to $M_{n^{2}}(C)$, analyzing their positivity properties and identifying parameter ranges for various positivity levels, especially for the case $n=3$.

Contribution

It generalizes previous work on equivariant maps, providing a detailed study of positivity, complete positivity, and decomposability for a new family of maps between matrix algebras.

Findings

Identifies parameter ranges where maps are positive for all $n \\geq 3$.

Determines conditions for 2-positivity and non-complete positivity at $n=3$.

Provides explicit examples of maps with specific positivity properties.

Abstract

Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $\Phi_{\alpha, \beta,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$ and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,n}$ is positive for any natural number $n \geq 3$. We focus on the case of $n=3,$ that is, $\Phi_{\alpha, \beta,3}$ and study the properties of $2$-positivity, completely positivity and decomposability. In particular, we give values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,3}$ is $2$-positive and not completely positive.