Complete hierarchies of efficient approximations to problems in entanglement theory

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Abstract
We investigate several problems in entanglement theory from the perspective of convex optimization. This list of problems comprises (A) the decision whether a state is multiparty entangled, (B) the minimization of expectation values of entanglement witnesses with respect to pure product states, (C) the closely related evaluation of the geometric measure of entanglement to quantify pure multiparty entanglement, (D) the test whether states are multiparty entangled on the basis of witnesses based on second moments and on the basis of linear entropic criteria, and (E) the evaluation of instances of maximal output purities of quantum channels. We show that these problems can be formulated as certain optimization problems: as polynomially constrained problems employing polynomials of degree 3 or less. We then apply very recently established known methods from the theory of semidefinite relaxations to the formulated optimization problems. By this construction 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 efficiently decidable sufficient criteria for multiparticle entanglement, such that every entangled state will necessarily be detected in some step of the hierarchy. Finally, we present numerical examples to demonstrate the practical accessibility of this approach.

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