Maximal qubit violations of $n$-locality in star and chain networks

Maximal qubit violations of $n$ -locality in star and chain networks

Brian Doolittle and Eric Chitambar

Phys. Rev. A 108, 042409 – Published 9 October 2023

Abstract

The nonlocal correlations of noisy quantum systems are important for both understanding nature and developing quantum technology. We consider the correlations of star and chain quantum networks in which noisy entanglement sources are measured by nonsignaling parties. We derive the necessary and sufficient conditions for when a broad family of star and chain $n$ -locality inequalities achieve their maximal quantum violation assuming that each party's dichotomic observables are separable across qudit systems. When pairs of local dichotomic observables are considered on qubit systems, we derive maximal $n$ -local violations that are larger and more robust to noise than the maximal $n$ -locality violations reported previously. To obtain these larger values, we consider observables that we have not found in previous studies. Thus, we gain insights into self-testing entanglement sources and measurements in star and chain networks.

Received 13 January 2023
Revised 4 May 2023
Accepted 20 September 2023

DOI:https://doi.org/10.1103/PhysRevA.108.042409

Physics Subject Headings (PhySH)

Entanglement detection Nonlocality Quantum communication, protocols & technology Quantum correlations in quantum information Quantum correlations, foundations & formalism Quantum networks

Quantum Information, Science & Technology

Authors & Affiliations

Brian Doolittle ^1,* and Eric Chitambar ²

¹Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
²Department of Electrical and Computer Engineering, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

^*Corresponding author: brian.d.doolittle@gmail.com

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Issue

Vol. 108, Iss. 4 — October 2023

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Images

Figure 1
Star and chain networks of $n$ sources (green ellipses) and $m = n + 1$ nonsignaling parties (rectangles). External parties are labeled as $A$ (red) or $B$ (blue). Central parties are labeled as $C$ (purple). The input to each party is specified in the subscript while the superscript indexes the linked sources. Parties $A, B$ , and $C$ have the respective classical inputs and alphabets $x \in X, y \in Y$ , and $z \in Z$ . All parties are assumed to output binary values.
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Figure 2
Quantum sources (ellipses) distribute entangled qudits to linked parties (rectangles). Qudit observables are labeled using $A$ (red) or $B$ (blue), while multiqudit observables are labeled using $C$ group together separable qudit observables (purple dashed). The $n$ -locality inequalities bounding the chain network assume that all multiqudit observables are conditioned on the same value, $z = z_{j} \in Z$ , which is passed to each local qudit observable contained by the multiqubit observable, e.g., $C_{z}^{1, 2} = B_{z}^{1} \otimes A_{z}^{2}$ .
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Figure 3
We compare $S_{n -CHSH}^{★} (ρ_{[n]})$ (solid) with ${\hat{S}}_{n -CHSH}^{★} (ρ_{[n]})$ (dashed). In both plots, the $n$ -local bound is shown by the blue dash-dotted line and the solid or dashed curves are ordered from top to bottom as shown by the legend. (left) We consider $k < n = 12$ noisy sources that have $τ_{i, 0} = 1$ and $τ_{i, 1} \in [0, 1]$ . The remaining $(n - k)$ sources have $τ_{i, 0} = τ_{i, 1} = 1$ . Note that, for each $k$ , each solid line has a corresponding dashed line that lies just below it. (right) For all $i \in [n]$ , we consider $\frac{1}{2} G_{CHSH}^{★} (ρ_{i}) = β^{★} \in [1.0, 1.1]$ while $τ_{i, 1}^{2} = {(β^{★})}^{2} - τ_{i, 0}^{2}$ where $τ_{i, 0}^{2}$ are evenly spaced in $[\frac{1}{2} (τ_{i, 0}^{2} + τ_{i, 1}^{2}), min {1, τ_{i, 0}^{2} + τ_{i, 1}^{2}}]$ . Note that, for all $n, S_{n -CHSH}^{★} (ρ_{[n]})$ achieves the solid orange line while the dashed lines plotting ${\hat{S}}_{n -CHSH}^{★} (ρ_{[n]})$ achieve smaller scores.
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Figure 4
For various $n, K_{n -CHSH}^{★} (ρ_{[n]})$ (solid) is compared with ${\hat{K}}_{n -CHSH}^{★} (ρ_{[n]})$ (dashed). In both plots, the $n$ -local bound is given by the dash-dotted blue line and, when $n = 2, K_{2 -CHSH}^{★} (ρ_{[n]}) = {\hat{K}}_{2 -CHSH}^{★} (ρ_{[n]})$ both follow the solid orange line plotting the largest $n$ -local violation. Furthermore, the curves are ordered from top to bottom as shown by the legend. (left) We set $τ_{i, 0}^{2} = 1$ and vary $τ_{i, 1}^{2} \in [0, 1]$ . Note that for all $n, K_{n -CHSH}^{★} (ρ_{[n]})$ achieves the solid orange line while ${\hat{K}}_{n -CHSH}^{★} (ρ_{[n]})$ is necessarily smaller for all $n > 2$ . (right) We set $τ_{i, 0}^{2} = \frac{3}{4} + \frac{1}{4} τ_{i, 1}^{2}$ and vary $τ_{i, 1}^{2} \in [0, 1]$ . Note that for each $n$ , the dashed line lies just below the solid line while having the same value at $τ_{i, 1}^{2} = {0, 1}$ .
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