Dissertation
Rank in matrix analysis: on the preservers of maximally entangled states and fractional minimal rank
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2019
DOI:
https://doi.org/10.17918/etd-9532
Abstract
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 [o times] 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 [o times] 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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Details
- Title
- Rank in matrix analysis
- Creators
- Benjamin Wilhelm Grossmann - DU
- Contributors
- Hugo J. Woerdeman (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- vi, 73 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 9532; 991014632195904721