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

Sander Gribling1, David de Laat1, Monique Laurent1,2

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This study introduces new methods for optimizing quantum correlations, defining the minimal average entanglement dimension and unifying bounds for quantum graph parameters using tracial polynomial optimization.

Keywords:
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Area of Science:

  • Quantum Information Theory
  • Noncommutative Polynomial Optimization
  • Quantum Computing

Background:

  • Bipartite quantum correlations are fundamental to quantum information processing.
  • Understanding entanglement and quantum correlations is crucial for developing quantum technologies.
  • Existing methods for analyzing quantum correlations have limitations in certain optimization problems.

Purpose of the Study:

  • To investigate optimization problems related to bipartite quantum correlations.
  • To introduce and analyze the minimal average entanglement dimension.
  • To unify and extend bounds for quantum graph parameters within a novel optimization framework.

Main Methods:

  • Application of tracial noncommutative polynomial optimization techniques.
  • Construction of a hierarchy of semidefinite programming lower bounds.
  • Development of new semidefinite programming hierarchies for quantum graph parameters.

Main Results:

  • Convergence to the minimal average entanglement dimension, quantifying entanglement needed for quantum correlations with free shared randomness.
  • Unification of existing bounds on quantum chromatic and quantum stability numbers.
  • Establishment of a novel framework for studying optimization problems over synchronous quantum correlations.

Conclusions:

  • The developed methods provide powerful tools for analyzing quantum correlations and entanglement.
  • The minimal average entanglement dimension offers a new perspective on quantifying entanglement resources.
  • The tracial polynomial optimization framework successfully unifies and extends key concepts in quantum graph theory.