Characterizing finite-dimensional quantum behavior

TL;DR

This paper develops semidefinite programming hierarchies to characterize finite-dimensional quantum correlations, introducing a new optimization framework and deriving novel dimension witnesses for quantum information scenarios.

Contribution

It introduces the dimension-constrained noncommutative polynomial optimization paradigm and effective SDP hierarchies with proven convergence for quantum correlation characterization.

Findings

Derived new dimension witnesses for temporal and Bell correlations.

Bound the success probability of quantum random access codes.

Extended SDP hierarchies to finite-dimensional quantum systems.

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 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 new dimension witnesses for temporal and Bell-type correlation scenarios, and also to bound the probability of success of quantum random access codes.