Computing finite-dimensional bipartite quantum separability

Ever since entanglement was identified as a computational and cryptographic resource, effort has been made to find an efficient way to tell whether a given density matrix represents an unentangled, or separable, state. Essentially, this is the quantum separability problem. In Chapter 1, I begin with a brief introduction to quantum states, entanglement, and a basic formal definition of the quantum separability problem. I conclude the first chapter with a summary of one-sided tests for separability, including those involving semidefinite programming. In Chapter 2, I apply polyhedral theory to prove easily that the set of separable states is not a polytope; for the sake of completeness, I then review the role of polytopes in nonlocality. Next, I give a novel treatment of entanglement witnesses and define a new class of entanglement witnesses, which may prove to be useful beyond the examples given. In the last section, I briefly review the five basic convex body problems given in [1], and their application to the quantum separability problem. In Chapter 3, I treat the separability problem as a computational decision problem and motivate its approximate formulations. After a review of basic complexity-theoretic notions, I discuss the computational complexity of the separability problem: I discuss the issue of NP-completeness, giving an alternative definition of the separability problem as an NP-hard problem in NP. I finish the chapter with a comprehensive survey of deterministic algorithmic solutions to the separability problem, including one that follows from a second NP formulation. Chapters 1 to 3 motivate a new interior-point algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witness in the span of the subset of observables. When all the expected values are known, the algorithm solves the separability problem. In Chapter 4, I give the motivation for the algorithm and show how it can be used in a particular physical scenario to detect entanglement (or decide separability) of an unknown quantum state using as few quantum resources as possible. I then explain the intuitive idea behind the algorithm and relate it to the standard algorithms of its kind. I end the chapter with a comparison of the complexities of the algorithms surveyed in Chapter 3. Finally, in Chapter 5, I present the details of the algorithm and discuss its performance relative to standard methods.