| 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 Michael Nielsen says that this is intimately connected to the majorization relation: if x is the vector of eigenvalues of the initial state, then y can be the vector of eigenvalues of the final state if and only if x is majorized by y. It was recently observed that it is possible for x ⊗ z to be majorized by y ⊗ z, even if x is not majorized by y; physically, this means that the presence of a state with eigenvalues z is a catalyst that allows a certain transformation to occur. If such a z exists, then x is said to be trumped by y. Part I is mainly a study of the structure of this trumping relation, an extension of the majorization relation. Notably, we show that for almost all probability vectors y ∈ Rd where d ≥ 4, there is no finite dimension n such that the set of vectors trumped by y can be determined by restricting attention to catalysts of dimension n. We also study some concrete examples to illustrate various aspects 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 a regarded as a problem in Grassmannian cohomology. We discuss this cohomology and derive the conditions for nontrivial intersection. Finally, we illustrate how these intersections give rise to the desired inequalities. |
