Semidefinite programming relaxations for quantum correlations

Abstract

Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science. Many otherwise intractable fundamental and applied problems can be successfully approached by means of relaxation to a semidefinite program. Here, we review such methodology in the context of quantum correlations. We discuss how the core idea of semidefinite relaxations can be adapted for a variety of research topics in quantum correlations, including nonlocality, quantum communication, quantum networks, entanglement, and quantum cryptography.

Published in Reviews of Modern Physics. v4: minor changes

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• A note on the inner products of pure states and their antidistinguishability
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• What is the minimum CHSH score certifying that a state resembles the singlet?
• Quantum LOSR networks cannot generate graph states with high fidelity
• Post-quantum nonlocality in the minimal triangle scenario
• A hierarchy of efficient bounds on quantum capacities exploiting symmetry
• Certifying Quantum Separability with Adaptive Polytopes
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• Generalizing optimal Bell inequalities
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• Quantum Preparation Games
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• Quantum Random Access Code in Noisy Channels
• A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces
• Interplays between classical and quantum entanglement-assisted communication scenarios
• The SWAP Imposter: Bidirectional Quantum Teleportation and its Performance
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• Truncated moment sequences and a solution to the channel separability problem
• Separation of quantum, spatial quantum, and approximate quantum correlations
• Quasi-polynomial time algorithms for free quantum games in bounded dimension
• More randomness from noisy sources
• The 7 faces of quantum NP
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• Optimal single-shot discrimination of optical modes
• Comment on "Geometry of the quantum set on no-signaling faces"
• Optimisation of time-ordered processes in the finite and asymptotic regime
• The Binary-Outcome Detection Loophole
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