Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

Sander Gribling, David de Laat, Monique Laurent

Research output: Contribution to journalArticleScientificpeer-review

Abstract

In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence 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. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.
Original languageEnglish
Pages (from-to)5-42
JournalMathematical Programming
Volume170
Issue number1
DOIs
Publication statusPublished - Jul 2018

Fingerprint

Quantum Graphs
Entanglement
Polynomials
Polynomial
Optimization
Semidefinite Programming
Stability number
Optimization Problem
Randomness
Lower bound

Keywords

  • entanglement dimension
  • polynomial optimization
  • quantum graph parameters
  • quantum correlations

Cite this

@article{6b1a64833f754661b808f30735b964ab,
title = "Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization",
abstract = "In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence 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. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.",
keywords = "entanglement dimension, polynomial optimization, quantum graph parameters, quantum correlations",
author = "Sander Gribling and {de Laat}, David and Monique Laurent",
year = "2018",
month = "7",
doi = "10.1007/s10107-018-1287-z",
language = "English",
volume = "170",
pages = "5--42",
journal = "Mathematical Programming",
issn = "0025-5610",
publisher = "Springer",
number = "1",

}

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization. / Gribling, Sander; de Laat, David; Laurent, Monique.

In: Mathematical Programming , Vol. 170, No. 1, 07.2018, p. 5-42.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

AU - Gribling, Sander

AU - de Laat, David

AU - Laurent, Monique

PY - 2018/7

Y1 - 2018/7

N2 - In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence 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. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.

AB - In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence 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. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.

KW - entanglement dimension

KW - polynomial optimization

KW - quantum graph parameters

KW - quantum correlations

U2 - 10.1007/s10107-018-1287-z

DO - 10.1007/s10107-018-1287-z

M3 - Article

VL - 170

SP - 5

EP - 42

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1

ER -