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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• Multiple-Access Channel Coding with Non-Signaling Correlations
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• Entanglement transitivity problems
• A note on the inner products of pure states and their antidistinguishability
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• Optimal universal quantum circuits for unitary complex conjugation
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• Quantum LOSR networks cannot generate graph states with high fidelity
• What is the minimum CHSH score certifying that a state resembles the singlet?
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• A hierarchy of efficient bounds on quantum capacities exploiting symmetry
• Certifying Quantum Separability with Adaptive Polytopes
• Generalizing optimal Bell inequalities
• Quantum Preparation Games
• Geometrical self-testing of partially entangled two-qubit states
• Connector tensor networks: a renormalization-type approach to quantum certification
• Quantum Random Access Code in Noisy Channels
• The SWAP Imposter: Bidirectional Quantum Teleportation and its Performance
• A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces
• Interplays between classical and quantum entanglement-assisted communication scenarios
• Incompatibility of local measurements provide advantage in local quantum state discrimination
• Truncated moment sequences and a solution to the channel separability problem
• Device-independent and semi-device-independent entanglement certification in broadcast Bell scenarios
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• Optimisation of time-ordered processes in the finite and asymptotic regime
• Comment on "Geometry of the quantum set on no-signaling faces"
• Bell inequalities with overlapping measurements
• The Binary-Outcome Detection Loophole

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