Bounds on Entanglement Dimensions and Quantum Graph Parameters via Noncommutative Polynomial Optimization

Sander Gribling, David de Laat, Monique Laurent

Research output: Working paperOther research output

Abstract

In this paper we study bipartite quantum correlations using techniques from tracial 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.
Original languageEnglish
Place of PublicationIthaca
PublisherCornell University Library
Number of pages29
Publication statusPublished - 31 Aug 2017

Publication series

NamearXiv
Volume1708.09696

Fingerprint

Quantum Graphs
Entanglement
Stability number
Polynomial
Optimization
Chromatic number
Semidefinite Programming
Positive semidefinite
Randomness
Packing
Correspondence
Lower bound
Analogue
Converge
Hierarchy
Graph in graph theory

Keywords

  • Optimization and Control
  • Quantum Physics

Cite this

Gribling, S., de Laat, D., & Laurent, M. (2017). Bounds on Entanglement Dimensions and Quantum Graph Parameters via Noncommutative Polynomial Optimization. (arXiv; Vol. 1708.09696). Ithaca: Cornell University Library.
Gribling, Sander ; de Laat, David ; Laurent, Monique. / Bounds on Entanglement Dimensions and Quantum Graph Parameters via Noncommutative Polynomial Optimization. Ithaca : Cornell University Library, 2017. (arXiv).
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Gribling, S, de Laat, D & Laurent, M 2017 'Bounds on Entanglement Dimensions and Quantum Graph Parameters via Noncommutative Polynomial Optimization' arXiv, vol. 1708.09696, Cornell University Library, Ithaca.

Bounds on Entanglement Dimensions and Quantum Graph Parameters via Noncommutative Polynomial Optimization. / Gribling, Sander; de Laat, David; Laurent, Monique.

Ithaca : Cornell University Library, 2017. (arXiv; Vol. 1708.09696).

Research output: Working paperOther research output

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N2 - In this paper we study bipartite quantum correlations using techniques from tracial 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.

AB - In this paper we study bipartite quantum correlations using techniques from tracial 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.

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Gribling S, de Laat D, Laurent M. Bounds on Entanglement Dimensions and Quantum Graph Parameters via Noncommutative Polynomial Optimization. Ithaca: Cornell University Library. 2017 Aug 31. (arXiv).