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Superactivation of Bell nonlocality in pure anyonic states

Cheng-Qian Xu, Wenhao Ye, Li You

TL;DR

This work addresses how Bell nonlocality relates to entanglement in anyonic systems lacking a tensor product structure. By analyzing Fibonacci anyons and defining multi-copy states through tangled braiding, it shows that pure states with nonzero $E_{ m AEE}$ can be local in single-copy tests but nonlocal under collective measurements (superactivation). The authors decompose anyonic entanglement into ACE and CE components, prove asymptotic results (Theorems on AEE, ACE, CE), and establish that $E_{ m CE}=0$ iff a state is local, while $E_{ m ACE}>0$ enables superactivation. The findings illuminate how entanglement and nonlocality interplay in non-tensor-product spaces and advance understanding of anyonic quantum information processing. Overall, the work reveals a nuanced landscape where non-tensorial entanglement can be converted into tensor-product entanglement to reveal hidden nonlocal correlations.

Abstract

Standard quantum information theory is founded on the assumption that multi-party state space possesses a tensor product structure. Anyons, as quasiparticles in two-dimensional systems, exhibit unique entanglement properties that differ from the conventional quantum systems, resulting from the absence of a tensor product structure in their state spaces. This motivates us to investigate the relationship between Bell nonlocality and entanglement in anyonic states. Specifically, we find that certain pure anyonic states with non-zero anyonic entanglement entropy (AEE) are local, yet exhibit nonlocality when subjected to collective measurements on multiple copies-a phenomenon known as superactivation of nonlocality, which is typically observed in conventional mixed states. To analyze this, we decompose the total entanglement of anyonic states into two components: one from the tensor product structure and the other representing residual contributions. By studying their asymptotic behavior, we find that the former gradually increases and approaches the AEE while the latter diminishes with the number of copies. Crucially, the entanglement component associated with the tensor product structure demonstrates a significant correlation with nonlocality, which explains the observed superactivation of nonlocality. Our findings provide new insights into the connection between entanglement and nonlocality in anyonic systems.

Superactivation of Bell nonlocality in pure anyonic states

TL;DR

This work addresses how Bell nonlocality relates to entanglement in anyonic systems lacking a tensor product structure. By analyzing Fibonacci anyons and defining multi-copy states through tangled braiding, it shows that pure states with nonzero can be local in single-copy tests but nonlocal under collective measurements (superactivation). The authors decompose anyonic entanglement into ACE and CE components, prove asymptotic results (Theorems on AEE, ACE, CE), and establish that iff a state is local, while enables superactivation. The findings illuminate how entanglement and nonlocality interplay in non-tensor-product spaces and advance understanding of anyonic quantum information processing. Overall, the work reveals a nuanced landscape where non-tensorial entanglement can be converted into tensor-product entanglement to reveal hidden nonlocal correlations.

Abstract

Standard quantum information theory is founded on the assumption that multi-party state space possesses a tensor product structure. Anyons, as quasiparticles in two-dimensional systems, exhibit unique entanglement properties that differ from the conventional quantum systems, resulting from the absence of a tensor product structure in their state spaces. This motivates us to investigate the relationship between Bell nonlocality and entanglement in anyonic states. Specifically, we find that certain pure anyonic states with non-zero anyonic entanglement entropy (AEE) are local, yet exhibit nonlocality when subjected to collective measurements on multiple copies-a phenomenon known as superactivation of nonlocality, which is typically observed in conventional mixed states. To analyze this, we decompose the total entanglement of anyonic states into two components: one from the tensor product structure and the other representing residual contributions. By studying their asymptotic behavior, we find that the former gradually increases and approaches the AEE while the latter diminishes with the number of copies. Crucially, the entanglement component associated with the tensor product structure demonstrates a significant correlation with nonlocality, which explains the observed superactivation of nonlocality. Our findings provide new insights into the connection between entanglement and nonlocality in anyonic systems.

Paper Structure

This paper contains 14 sections, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fibonacci Anyon Model and Pure Anyonic States
  3. Fibonacci Anyon Model
  4. Pure Anyonic States
  5. Bell Inequality Violation with Pure Anyonic States
  6. Multiple Copies of an Anyonic State
  7. Superactivation of Nonlocality
  8. The Asymptotic Behavior of Anyonic Entanglement
  9. Proof to show that $E_{\rm CE}$ characterizes Bell nonlocality
  10. Summary
  11. The Entanglement Symmetry of Pure Anyonic States
  12. The Tangled Braiding and Multiple Copies of an Anyoinc State
  13. Anyonic Pythagorean Theorem for Anyonic Relative Entropy
  14. Proof of Theorem 2: Asymptotic Behavior of Anyonic Entanglement

Figures (1)

  • Figure 1: The scatter plots of various average entanglement measures (\ref{['eq:AEM']}) for finite $n$-copy of anyonic state $\ket{\tau, \tau; 1}$. Black triangles, red dots, and blue dots denote the average $E_{\rm AREE}$, $E_{\rm ACE}$, and $E_{\rm CE}$ respectively. Dashed lines indicate the vertical coordinate $y=\log_2 \text{d}_\tau \approx 0.69$, which is the asymptote for the average $E_{\rm AREE}$ and $E_{\rm CE}$.