Bounding the set of finite dimensional quantum correlations

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.

References in corpus (9)

• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Bounding the set of quantum correlations
• Testing the Hilbert space dimension
• Device-independent tests of classical and quantum dimensions
• Semi-device-independent security of one-way quantum key distribution
• A lower bound on the dimension of a quantum system given measured data
• Dimension witnesses and quantum state discrimination
• Efficiency of higher dimensional Hilbert spaces for the violation of Bell inequalities
• Properties of dimension witnesses and their semi-definite programming relaxations

Cited by in corpus (5)

• Qutrit witness from the Grothendieck constant of order four
• Shared randomness and device-independent dimension witnessing
• Towards minimal self-testing of qubit states and measurements in prepare-and-measure scenarios
• Quantum dimension test using the uncertainty principle
• NPA Hierarchy and Extremal Criterion in the Simplest Bell Scenario