Citation
Daftuar, Sumit Kumar (2004) Eigenvalue inequalities in quantum information processing. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd03312004100014
Abstract
This thesis develops restrictions governing how a quantum system, jointly held by two parties, can be altered by the local actions of those parties, under assumptions about how they may communicate. These restrictions are expressed as constraints involving the eigenvalues of the density matrix of one of the parties. The thesis is divided into two parts.
Part I (Chapters 14) explores what is possible if the two parties may use only classical communication. A wellknown result by M. Nielsen says that this is intimately connected to the mathematical notion of majorization. If entanglement catalysis is permitted, then the relevant notion is an extension of majorization known as the trumping relation. In Part I, we study the structure of the trumping relation.
Part II (Chapters 59) considers the question of how a state can change as a result of quantum communication between the parties; i.e., one party sends the other a portion of the jointly held quantum system. Given the spectrum of the initial state, it turns out that the possible spectra of the final state are given by the solutions to linear inequalities. We develop a method for deriving these inequalities, using a variational principle. In order to apply this principle, we need to know when certain subvarieties of a Grassmannian variety intersect, which can be regarded as a problem in Grassmannian cohomology. We discuss this cohomology and derive the conditions for nontrivial intersections. Finally, we illustrate how these intersections give rise to the desired inequalities.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  ELOCC; entanglement catalysis; Grassmannian; LOCC; majorization; partial trace; quantum communication; quantum information; Schubert calculus; trumping; variational principle 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  25 September 2003 
NonCaltech Author Email:  daftuar (AT) post.harvard.edu 
Record Number:  CaltechETD:etd03312004100014 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd03312004100014 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  1224 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  02 Apr 2004 
Last Modified:  26 Dec 2012 02:36 
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