[Paper Review] Semidefinite programming relaxations for quantum correlations

Armin Tavakoli, Alejandro Pozas-Kerstjens+2 more|arXiv (Cornell University)|Jul 5, 2023

Quantum Information and Cryptography8 citations

TL;DR

A comprehensive review of how semidefinite programming (SDP) relaxations and hierarchies are used to characterize quantum correlations across entanglement, nonlocality, quantum communication, networks, and cryptography.

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.

Motivation & Objective

• Introduce semidefinite programming basics and their role in quantum information.
• Present a general framework for SDP relaxation hierarchies applicable to quantum correlations.
• Discuss SDP methods in entanglement theory, nonlocality, and device-independent scenarios.
• Explain SDP applications in quantum communication, randomness, and quantum cryptography.
• Provide practical guidance and software resources for implementing SDP approaches.

Proposed method

• Describe SDP primal and dual formulations and conditions for strong duality.
• Outline polynomial and noncommutative polynomial optimization techniques (moment and sum-of-squares approaches).
• Introduce entanglement and nonlocality hierarchies (Doherty–Parrilo–Spedalieri; Navascués–Pironio–Acín).
• Explain device-independent and semi-device-independent certifications via SDP relaxations.
• Apply SDP frameworks to networks via inflation techniques and related relaxations.
• Offer implementation guidance and links to public SDP solvers and software.

Experimental results

Research questions

• RQ1 How can SDP relaxations approximate the set of quantum correlations in various scenarios?
• RQ2 What hierarchies (e.g., NPA, DPS, inflation) provide convergent sequences to the quantum set in entanglement, nonlocality, and networks?
• RQ3 How can SDP tools be used to certify entanglement, dimension, and nonlocality in device-dependent and device-independent settings?
• RQ4 What are the practical computational strategies and limitations for applying SDP relaxations to quantum communication and cryptography tasks?
• RQ5 How do dual formulations yield witnesses and bounds for quantum properties such as entanglement witnesses and Bell inequalities?

Key findings

• SDP relaxations provide tractable outer approximations to hard quantum-correlation problems and can converge to exact solutions in the infinite limit.
• Entanglement, steering, and nonlocality can be studied via SDP hierarchies such as DPS and NPA, with duals yielding witnesses and inequalities.
• SDP methods enable device-independent certification, entanglement dimension tests, and quantum-information tasks in communication and cryptography.
• The framework extends to networks through inflation techniques and related SDP relaxations, allowing analysis of complex causal structures.
• Practical guidance, including solver recommendations and software packages, is provided to implement these SDP approaches.