Choi matrices revisited. III

TL;DR

This paper explores the structure of linear isomorphisms and bilinear pairings between mapping spaces and tensor products of matrices, focusing on their preservation of $k$-positivity, superpositivity, and Schmidt number dualities.

Contribution

It characterizes all linear isomorphisms and bilinear pairings that preserve key positivity and duality properties in quantum information theory.

Findings

Identifies all linear isomorphisms mapping $k$-superpositive maps to states with Schmidt number $ extless= k$.

Describes all bilinear pairings maintaining duality between $k$-positivity and Schmidt numbers.

Provides a comprehensive framework for understanding positivity-preserving transformations in quantum maps.

Abstract

We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send $k$-superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to $k$. They also send $k$-positive maps onto $k$-block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between $k$-positivity and Schmidt numbers $\le k$. They also retain the duality between $k$-superpositivity and $k$-block-positivity.