Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Network nonlocality breaking channels

Kaushiki Mukherjee, Nirman Ganguly

TL;DR

This work introduces network nonlocality breaking channels, extending the concept of locality-breaking noise to quantum networks with linear and star topologies. It provides practical, inequality-based criteria to determine when single-qubit channels destroy or preserve non-$n$-locality under $k$ uses, covering unital and non-unital channels and applying them to depolarizing and dephasing models. The study yields explicit bounds and regions in channel-parameter space that guarantee breaking of non-$n$-locality and, in some cases, preservation of network nonlocality, including full-network nonlocality in non-standard setups. These results offer guidance for designing robust quantum networks and understanding how noise affects complex networked quantum correlations. The framework connects Bloch-geometry, CPTP maps, and network nonlocality inequalities to quantify resource loss or conservation under channel-induced noise.

Abstract

Network nonlocality, a recently noted form of nonlocality has been shown to have distinctive features, marking a significant departure from the notion of standard Bell nonlocality in the context of quantum correlations. On a pragmatic front, it has gained significant importance as researchers worldwide actively engage in the study on quantum networks. However, as typical to any quantum resource, network nonlocality is also vulnerable to environmental noise, which sometimes prove to be detrimental. Environmental interactions are modeled in terms of quantum channels. In the present study, we introduce and characterize network nonlocality breaking channels. Network nonlocality breaking channels model environmental influences which results in the loss of resource, i.e., the system loses its nonlocal resource due to such interactions. The study is done in the ambit of some suitably chosen inequalities in (i) linear networks and (ii) star-shaped networks. Further, the loss in full network nonlocality is also studied. Furthermore, we also characterize quantum channels according to their ability in preserving quantum resources, i.e., they do not break network nonlocality, which enables one to identify useful quantum channels in networks. The study is vindicated by illustrations from various noise models like depolarizing and dephasing channels.

Network nonlocality breaking channels

TL;DR

This work introduces network nonlocality breaking channels, extending the concept of locality-breaking noise to quantum networks with linear and star topologies. It provides practical, inequality-based criteria to determine when single-qubit channels destroy or preserve non--locality under uses, covering unital and non-unital channels and applying them to depolarizing and dephasing models. The study yields explicit bounds and regions in channel-parameter space that guarantee breaking of non--locality and, in some cases, preservation of network nonlocality, including full-network nonlocality in non-standard setups. These results offer guidance for designing robust quantum networks and understanding how noise affects complex networked quantum correlations. The framework connects Bloch-geometry, CPTP maps, and network nonlocality inequalities to quantify resource loss or conservation under channel-induced noise.

Abstract

Network nonlocality, a recently noted form of nonlocality has been shown to have distinctive features, marking a significant departure from the notion of standard Bell nonlocality in the context of quantum correlations. On a pragmatic front, it has gained significant importance as researchers worldwide actively engage in the study on quantum networks. However, as typical to any quantum resource, network nonlocality is also vulnerable to environmental noise, which sometimes prove to be detrimental. Environmental interactions are modeled in terms of quantum channels. In the present study, we introduce and characterize network nonlocality breaking channels. Network nonlocality breaking channels model environmental influences which results in the loss of resource, i.e., the system loses its nonlocal resource due to such interactions. The study is done in the ambit of some suitably chosen inequalities in (i) linear networks and (ii) star-shaped networks. Further, the loss in full network nonlocality is also studied. Furthermore, we also characterize quantum channels according to their ability in preserving quantum resources, i.e., they do not break network nonlocality, which enables one to identify useful quantum channels in networks. The study is vindicated by illustrations from various noise models like depolarizing and dephasing channels.

Paper Structure

This paper contains 37 sections, 10 theorems, 87 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bloch Matrix Representation
  4. Quantum channels
  5. $n$-local Linear Network Kau
  6. $n$-local Star NetworkTava
  7. Full Network Nonlocality
  8. Detection of full network nonlocality for $n$$=$$3$
  9. Standard and Non Standard Networks
  10. Non $n$-locality Breaking Channels
  11. Non $n$-local Resource
  12. Non $n$-locality Breaking Channel
  13. Characterizing $k$-Use Non $n$-Locality Breaking Channel
  14. Characterization in Linear $n$-local Network($\textbf{N}_{lin}$)
  15. Unital Channel
  16. ...and 22 more sections

Key Result

Theorem 1

If any single-qubit unital channel $\mathcal{N}_{U}$ be such that both channel parameters $\alpha,\beta$ are proper complex numbers satisfying then $\mathcal{N}_{U}$ is $k$-use non $n$-locality breaking channel with respect to $n$-local inequality Eq.(ineqb) in any linear $n$-local network. In particular, if $\mathcal{N}_{U}$ be such that any three of $Re(\alpha),Im(\alpha),Re(\beta),Im(\beta)$ a

Figures (10)

  • Figure 1: Schematic representation of linear $n$-local network
  • Figure 2: Schematically representing $n$-local network in star topology
  • Figure 3: Single-qubit channel $\mathcal{C}$ acting on some of the qubits in a linear $n$-local network $\textbf{N}_{lin}.$ As shown here only one qubit from each of the sources $\mathbf{S}_1$ and $\mathbf{S}_n$ and both qubits from $\mathbf{S}_2$ are passing through $\mathcal{C}.$
  • Figure 4: Visualizing the depolarizing channel $\mathcal{N}_{dep}$ as $k$-use non $n$-locality breaking channel for different values of $k.$ Variation of the criterion for $\mathcal{N}_{dep},$ to be non $n$-locality breaking with respect to the parameter characterizing the channel, is plotted. Vertical axis gives the criterion(Eq.(\ref{['depo1']})) imposed over $q$ for $\mathcal{N}_{dep}$ to be $k$-use non $n$-locality breaking. For any value of $q$ for which the curve lies below the horizontal axis, corresponding channel acts as $k$-use non $n$-locality breaking with $k$ taking some fixed values. No definite conclusion can be given for any portion of the curve above the horizontal axis.
  • Figure 5: Shaded region gives subspace in the parameter space $(t,\lambda_1,\lambda_3)$ of the class of non unital channels $\mathcal{N}_{NU}$ specified by Eqs.(\ref{['nu1']},\ref{['nu2']}). For any tuple of parameter values lying in this region, corresponding non unital channel acts as $k$-use non $n$-locality breaking channel in $\mathbf{N}_{lin}.$
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Corollary 2.1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 3 more