Navascués, M.Feix, A.Araújo, M.Vértesi, T.2016-02-082016-02-08201510502947http://hdl.handle.net/11693/26696We study and extend the semidefinite programming (SDP) hierarchies introduced in Navascués and Vértesi [Phys. Rev. Lett. 115, 020501 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.020501] for the characterization of the statistical correlations arising from finite-dimensional quantum systems. First, we introduce the dimension-constrained noncommutative polynomial optimization (NPO) paradigm, where a number of polynomial inequalities are defined and optimization is conducted over all feasible operator representations of bounded dimensionality. Important problems in device-independent and semi-device-independent quantum information science can be formulated (or almost formulated) in this framework. We present effective SDP hierarchies to attack the general dimension-constrained NPO problem (and related ones) and prove their asymptotic convergence. To illustrate the power of these relaxations, we use them to derive a number of dimension witnesses for temporal and Bell-type correlation scenarios, and also to bound the probability of success of quantum random access codes. © 2015 American Physical Society.EnglishQuantum opticsAsymptotic convergenceFinite-dimensional quantum systemsNoncommutative polynomialsPolynomial inequalitiesProbability of successQuantum information scienceSemi-definite programmingStatistical correlationConstrained optimizationArticle10.1103/PhysRevA.92.042117