Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semidefinite block-matrix relaxations for computing quantum correlations

Nicola D'Alessandro, Carles Roch i Carceller, Armin Tavakoli

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 Navascués-Pironio-Acín 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.

Semidefinite block-matrix relaxations for computing quantum correlations

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 Navascués-Pironio-Acín 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.
Paper Structure (33 sections, 71 equations, 2 figures, 5 tables)

This paper contains 33 sections, 71 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Adding an objective function
  4. An illustrative example
  5. Entanglement detection with imperfect measurements
  6. Description of Problem 1
  7. BMM method for solving Problem 1
  8. Results for Problem 1
  9. Quantum certification in the prepare-and-measure scenario
  10. Description of Problem 2
  11. BMM method for solving Problem 2
  12. Results for Problem 2
  13. Dimensionality of multipartite entanglement
  14. Description of Problem 3
  15. BMM method for solving Problem 3
  16. ...and 18 more sections

Figures (2)

  • Figure 1: Upper bound on the entanglement witness in Eq \ref{['eq:Pauli_wit2']} for three-dimensional systems and imperfection parameter $\varepsilon$. The dashed line represent the maximal quantum value $W=2$. At $\varepsilon=0$ the witness bound is $W=1+1/3$.
  • Figure 2: Upper bounds on the classical value of the witness $W_{d}$ for dimensions $d=2,\dots, 5$ (left axis) and success probability for discriminating the Hesse SIC $\textbf{E}_{\text{Hesse}}$ (right axis) as function of the distrust. No assumption is made on the communication channel's dimension. The dashed gray line marks the witness' quantum bound.