Bounding the Set of Finite Dimensional Quantum Correlations

Area of Science:

• Quantum Information Theory
• Mathematical Physics
• Computational Complexity

Background:

• Analyzing finite-dimensional quantum systems requires advanced computational tools.
• Semidefinite programming (SDP) offers powerful relaxations for optimization problems.
• Understanding quantum nonlocality and system dimensionality is crucial for quantum technologies.

Purpose of the Study:

• To present a simple, flexible, and programmable method for deriving high-performance SDP relaxations.
• To apply this method to bound quantum nonlocality in finite-dimensional Bell scenarios.
• To introduce a new dimension witness capable of distinguishing classical, real, and complex systems.

Main Methods:

• Derivation of semidefinite programming relaxations for operator algebras.
• Application of SDP to bound quantum nonlocality in Bell scenarios with bounded dimensions.
• Development and analysis of prepare-and-measure dimension witnesses.

Main Results:

• A flexible and efficient method for analyzing finite-dimensional quantum systems.
• New bounds on the strength of quantum nonlocality in specific Bell scenarios.
• Proof of soundness for existing prepare-and-measure dimension witnesses.
• Proposal of a novel dimension witness distinguishing classical, real, and complex systems.

Conclusions:

• The developed SDP method provides a versatile tool for studying quantum systems.
• The findings advance our understanding of quantum nonlocality and communication complexity.
• The new dimension witness offers a practical way to characterize quantum system properties.