Semidefinite block-matrix relaxations for computing quantum correlations

TL;DR

This paper introduces a versatile semidefinite relaxation method that incorporates various constraints to better compute quantum correlations, demonstrated through five diverse quantum information applications.

Contribution

It generalizes the Navascués-Pironio-Acín hierarchy by allowing a broader set of constraints in semidefinite relaxations for quantum correlation problems.

Findings

Effective in addressing five quantum information problems

Balances computational cost with relaxation accuracy

Demonstrates versatility across different quantum tasks

Abstract

Bounding the correlations predicted by quantum theory is an important challenge in quantum information science. Today's leading approach is semidefinite programming relaxations, but existing methods still cannot account for many relevant types of constraints. Here, we propose a semidefinite relaxation methodology that can incorporate a breadth of constraints needed in various quantum correlation problems, thereby generalising the seminal Navascu\'es-Pironio-Ac\'in hierarchy. It yields useful results at reasonable computational cost. We showcase the methodology and its features by using it to address five different quantum information problems. These are (i) entanglement witnessing from imperfect measurement devices, (ii) certifying measurements from fidelity-constrained sources, (iii) computing dimensionality in genuine multi-particle entangled states, (iv) benchmarking dimensionality for state preparation devices, and (v) finding uncertainty relations for nearly anti-commuting observables. These applications reflect both the usefulness and versatility of the methodology, as well as its potential for broader relevance in the field.