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Characterizing and Harnessing Correlations Featuring Independent Qubit Devices

Liang-Liang Sun, Xiang Zhou, Chengjie Zhang, Zizhu Wang, Yong-Shun Song, Sixia Yu

TL;DR

This work tackles the problem of characterizing and exploiting correlations generated by independent qubit devices in both PM and Bell scenarios. It introduces a gap-crossing, uncertainty-relation–based framework that yields explicit non-convex correlation boundaries and can even infer measurement parameters and states from observed data without a priori device information. By embedding these insights into the NPA hierarchy, the authors optimize entanglement detection and show that certain local correlations can certify entanglement when analyzed with inferred measurement structure. The approach also provides practical device-referring protocols and tight bounds for Bell and PM correlations, with potential broad impact on device characterization and quantum information tasks where non-extreme correlations are relevant.

Abstract

We propose a framework to characterize the correlations in qubit systems for Bell and prepare-and-measure scenarios with independent devices -- a typically non-convex problem. Based on this result, we introduce protocols for referring devices and detecting entanglement with correlation that are not necessarily extreme or nonlocal, as required by common linear approach. } Specifically, our correlation criterion, derived from uncertainty relation specific to qubit systems, can capture the non-convex nature of the set of correlations arising from Bell and prepare-and-measure scenarios, as demonstrated through concrete examples. Conversely, when given an observed correlation, our framework can refer potential measurements and quantum states -- which are sometimes uniquely determined -- even with correlations that are not extreme. This extends common protocols that merely verify devices using extreme correlations. We then enhance entanglement detection for qubit system by incorporating the inferred information in Navascués-Pironio-Acín (NPA) hierarchy, showing that some local correlations can also verify entanglement. Since the scenarios considered here are standard platforms for most quantum information protocols and device inference is a central issue in quantum information science, our methodology, which is well-suited to these tasks, may provide a foundation for a broad range of applications.

Characterizing and Harnessing Correlations Featuring Independent Qubit Devices

TL;DR

This work tackles the problem of characterizing and exploiting correlations generated by independent qubit devices in both PM and Bell scenarios. It introduces a gap-crossing, uncertainty-relation–based framework that yields explicit non-convex correlation boundaries and can even infer measurement parameters and states from observed data without a priori device information. By embedding these insights into the NPA hierarchy, the authors optimize entanglement detection and show that certain local correlations can certify entanglement when analyzed with inferred measurement structure. The approach also provides practical device-referring protocols and tight bounds for Bell and PM correlations, with potential broad impact on device characterization and quantum information tasks where non-extreme correlations are relevant.

Abstract

We propose a framework to characterize the correlations in qubit systems for Bell and prepare-and-measure scenarios with independent devices -- a typically non-convex problem. Based on this result, we introduce protocols for referring devices and detecting entanglement with correlation that are not necessarily extreme or nonlocal, as required by common linear approach. } Specifically, our correlation criterion, derived from uncertainty relation specific to qubit systems, can capture the non-convex nature of the set of correlations arising from Bell and prepare-and-measure scenarios, as demonstrated through concrete examples. Conversely, when given an observed correlation, our framework can refer potential measurements and quantum states -- which are sometimes uniquely determined -- even with correlations that are not extreme. This extends common protocols that merely verify devices using extreme correlations. We then enhance entanglement detection for qubit system by incorporating the inferred information in Navascués-Pironio-Acín (NPA) hierarchy, showing that some local correlations can also verify entanglement. Since the scenarios considered here are standard platforms for most quantum information protocols and device inference is a central issue in quantum information science, our methodology, which is well-suited to these tasks, may provide a foundation for a broad range of applications.

Paper Structure

This paper contains 10 sections, 2 theorems, 30 equations, 3 figures.

Table of Contents

  1. Characterizing correlation featuring Qubit systems
  2. Characterizing Correlations in PM scenario
  3. Characterizing correlations in Bell Scenarios
  4. Inference of quantum measurements and states with correlations
  5. Optimization of entanglement detection
  6. Bounding correlation with uncertainty relation
  7. Correlation with and without public randomness
  8. Bounding correlation ${\rm Q}^{Bell}_{xy}$ in minimal Bell's scenario
  9. Bounding correlation in prepare-and-measure scenario $Q^{PM}_{xy}$.
  10. Referring devices with correlation $Q^{Bell}_{xy}$

Key Result

Theorem 1

If a correlation $\{p(a|A_{i,j}, \rho)\mid \rho\in \Omega\}_{a}$ arising from a PM scenario uses binary PVMs $A_{i}$ and $A_{j}$ on two-dimensional systems with independent devices, one has where $g_{\pm }( A_{i}, A_{j}, \rho)\equiv {\Pi}_{l=i, j}A_{l|\rho}\pm {\Pi}_{l=i, j}\sqrt{1- A_{i|\rho}^{2}}$. Otherwise, correlation cannot be generated by independent qubit systems.

Figures (3)

  • Figure 1: Characterizing correlations $Q^{Bell}_{xy}$ featuring independent qubit systems using criteria: Sikora-Varvitsiotis-Wei (SVW) witness, Navascués-Pironio-Acín (NPA) criterion, and uncertainty relation considering PVMs (UR-PVM) and POVMs (UR-POVM), and correlations that are local, where the regions above these curves are forbidden by these criteria.
  • Figure 2: Characterizing correlations $Q^{PM}_{xy}$ featuring independent qubit systems with the criteria: the Bowles-Quintino-Brunner (BQB) witness and uncertainty relation considering POVMs, where the regions above these curves are forbidden by these criteria.
  • Figure 3: Figures of $r_{1}$ and $c_{01}$. The measurements of $A_{0}$ and $A_{1}$ are uniquely determined up to unitary operation as $A_{0}=\sigma_{z}$, $A_{1}=r_{1}c_{01}\sigma_{z}+r_{1}{\sqrt{1-c^{2}_{01}}}\sigma_{y}+(1-r_{1})\openone$.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2