Characterizing finite-dimensional quantum behavior

Navascués, M. and Feix, A. and Araújo, M. and Vértesi, Tamás (2015) Characterizing finite-dimensional quantum behavior. PHYSICAL REVIEW A, 92 (4). ISSN 2469-9926

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Abstract

We study and extend the semidefinite programming (SDP) hierarchies introduced in [Phys. Rev. Lett. 115, 020501] for the characterization of the statistical correlations arising from finite dimen- sional quantum systems. First, we introduce the dimension-constrained noncommutative polynomial optimization (NPO) paradigm, where a number of polynomial inequalities are defined and optimiza- tion 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 at- tack 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 new dimension wit- nesses for temporal and Bell-type correlation scenarios, and also to bound the probability of success of quantum random access codes.

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