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arXiv:1909.05086

On the Preservers of Maximally Entangled States

Benjamin Großmann, Hugo J. Woerdeman

Published Sep 11, 2019• 0 citations
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Abstract

We characterize the linear maps that preserve maximally entangled states in $L(\mathcal X \otimes \mathcal Y)$ in the case where $\dim(\mathcal X)$ divides $\dim(\mathcal Y)$.

DOI
10.48550/arxiv.1909.05086

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We characterize the linear maps that preserve maximally entangled states in L(X⊗Y)L(\mathcal X \otimes \mathcal Y)L(X⊗Y) in the case where dim⁡(X)\dim(\mathcal X)dim(X) divides dim⁡(Y)\dim(\mathcal Y)dim(Y).
For Hilbert spaces X , Y, the set of maximally entangled states, MES X ,Y , is a set of rank-1 positive semidefinite operators over the space X ⊗ 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 X , Y.After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES X ,Y , which is a set of low-rank positive semidefinite operators over X ⊗ 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.
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Unextendible maximally entangled bases in Cpd⊗Cqd{\mathbb {C}}^{pd}\otimes {\mathbb {C}}^{qd}Cpd⊗Cqd C pd ⊗ C qd