Equality condition for a matrix inequality by partial transpose

TL;DR

This paper investigates the equality conditions for a matrix inequality involving partial transpose, providing explicit constructions and extensions for specific matrix cases relevant to quantum information theory.

Contribution

It explicitly characterizes when the inequality becomes equality, extending results to various matrix sizes and forms, including rectangular matrices.

Findings

Explicit equality conditions for vectors and 2x2 matrices.

Extension of results to square and rectangular matrices.

Characterization of matrix structures for the case K=2.

Abstract

The partial transpose map is a linear map widely used quantum information theory. We study the equality condition for a matrix inequality generated by partial transpose, namely $\rank(\sum^K_{j=1} A_j^T \otimes B_j)\le K \cdot \rank(\sum^K_{j=1} A_j \otimes B_j)$, where $A_j$'s and $B_j$'s are respectively the matrices of the same size, and $K$ is the Schmidt rank. We explicitly construct the condition when $A_i$'s are column or row vectors, or $2\times 2$ matrices. For the case where the Schmidt rank equals the dimension of $A_j$, we extend the results from $2\times 2$ matrices to square matrices, and further to rectangular matrices. In detail, we show that $\sum^K_{j=1} A_j \otimes B_j$ is locally equivalent to an elegant block-diagonal form consisting solely of identity and zero matrices. We also study the general case for $K=2$, and it turns out that the key is to characterize the expression of matrices $A_j$'s and $B_j$'s.