2017/08/31 by Sander Gribling, Gribling, Sander, David de Laat +3 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Optimization and Control (math.OC) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #math.OC #quant-ph
paper · pdf · doi:10.48550/arxiv.1708.09696
26 pages
openalex publication_date 2017/08/31 · arxiv created 2018/01/09 · arxiv updated 2018/01/10 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
In this paper we study bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. We construct a hierarchy of semidefinite programming lower bounds on the minimal entanglement dimension of a bipartite correlation. This hierarchy converges to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. For synchronous correlations, we show a correspondence between the minimal entanglement dimension and the completely positive semidefinite rank of an associated matrix. We then study optimization over the set of synchronous correlations by investigating quantum graph parameters. We unify existing bounds on the quantum chromatic number and the quantum stability number by placing them in the framework of tracial optimization. In particular, we show that the projective packing number, the projective rank, and the tracial rank arise naturally when considering tracial analogues of the Lasserre hierarchy for the stability and chromatic number of a graph. We also introduce semidefinite programming hierarchies converging to the commuting quantum chromatic number and commuting quantum stability number.