Bounding the set of finite dimensional quantum correlations

TL;DR

This paper introduces a flexible semidefinite programming approach to analyze finite dimensional quantum correlations, enabling bounds on quantum nonlocality, communication complexity, and system dimension distinctions.

Contribution

The authors develop a simple, adaptable SDP relaxation method for finite dimensional quantum systems, providing new bounds and dimension witnesses for quantum correlations.

Findings

Bounded quantum nonlocality in bipartite and tripartite scenarios.

Validated the effectiveness of dimension witnesses in quantum systems.

Proposed a new dimension witness distinguishing classical, real, and complex qubits.

Abstract

We describe a simple method to derive high performance semidefinite programming relaxations for optimizations over complex and real operator algebras in finite dimensional Hilbert spaces. The method is very flexible, easy to program and allows the user to assess the behavior of finite dimensional quantum systems in a number of interesting setups. We use this method to bound the strength of quantum nonlocality in bipartite and tripartite Bell scenarios where the dimension of a subset of the parties is bounded from above. We derive new results in quantum communication complexity and prove the soundness of the prepare-and-measure dimension witnesses introduced in [Phys. Rev. Lett. 105, 230501 (2010)]. Finally, we propose a new dimension witness that can distinguish between classical, real and complex two-level systems.