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Type of Document	Dissertation
Author	Daftuar, Sumit Kumar
Author's Email Address	daftuar AT post.harvard.edu
URN	etd-03312004-100014
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-03312004-100014
Title	Eigenvalue inequalities in quantum information processing
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title John Preskill Committee Chair David Damanik Committee Member Gary Lorden Committee Member Richard Wilson Committee Member
Keywords	variational principle Schubert calculus partial trace quantum communication quantum information Grassmannian majorization entanglement catalysis trumping ELOCC LOCC
Date of Defense	2003-09-25
Availability	unrestricted
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.
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