Rank in Matrix Analysis: On the Preservers of Maximally Entangled States and Fractional Minimal Rank

Abstract

For Hilbert spaces \s X, \s Y, the set of maximally entangled states, \MES_{\s X, \s Y}, is a set of rank-1 positive semidefinite operators over the space \s X \otimes \s Y. In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces \s X, \s Y. After classifying these linear maps in the case where \dim \s X divides \dim \s Y, we consider possible avenues of extending these results and consider the set \WMES_{\s X, \s Y}, which is a set of low-rank positive semidefinite operators over \s X \otimes \s Y. We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices. We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the ''minimal rank'' and ''triangular minimal ranks'' differ in general.Ph.D., Mathematics -- Drexel University, 201