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Eigenvalue inequalities in quantum information processing

Citation

Daftuar, Sumit Kumar (2004) Eigenvalue inequalities in quantum information processing. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-03312004-100014

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 1-4) explores what is possible if the two parties may use only classical communication. A well-known 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 5-9) 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):
  • Preskill, John P.
Thesis Committee:
  • Preskill, John P. (chair)
  • Damanik, David
  • Lorden, Gary A.
  • Wilson, Richard M.
Defense Date:25 September 2003
Author Email:daftuar (AT) post.harvard.edu
Record Number:CaltechETD:etd-03312004-100014
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-03312004-100014
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 ETD-db
Deposited On:02 Apr 2004
Last Modified:26 Dec 2012 02:36

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