Berlin 2005 – scientific programme
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Q: Quantenoptik und Photonik
Q 52: Quanteninformation III
Q 52.1: Talk
Tuesday, March 8, 2005, 17:00–17:15, HU Audimax
Complete hierarchies of efficient approximations to problems in entanglement theory — •Otfried Gühne1, Jens Eisert2, Philipp Hyllus3, and Marcos Curty4 — 1Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, A-6020 Innsbruck — 2Institut für Physik, Universität Postdam, Am Neuen Palais 10, D-14469 Potsdam — 3Institut für Theoretische Physik, Universität Hannover, Appelstraße 2, D-30167 Hannover — 4Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, Staudtstraße 7/B2, D-91058 Erlangen
In entanglement theory, many problems can be reduced to optimization problems. Examples of these problems are the decision whether a state is entangled or not and the minimization of expectation values of witnesses with respect to product states. In this contribution, we investigate this type of problems from the perspective of convex optimization. We show that these problems can be formulated as certain optimization problems: as optimization problems of a linear function with polynomial constraints on the variables, employing polynomials of degree three or less. We then apply known methods from the theory of semi-definite relaxations to these problems, notably a method due to Lasserre. By this we arrive at a hierarchy of efficiently solvable approximations to the solution, approximating the exact solution as closely as desired, in a way that is asymptotically complete. For example, this results in a hierarchy of sufficient criteria for entanglement, such that every entangled state will be detected in some step of the hierarchy. Finally, we present numerical examples to demonstrate the practical accessibility of this approach.