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Linear Program Witness for Network Nonlocality in Arbitrary Networks

Salome Hayes-Shuptar, Daniel Bhatti, Ana Belen Sainz, David Elkouss

TL;DR

This work develops a linear programming witness for network nonlocality by formulating five linear constraint classes on an auxiliary distribution q(a,λ), balancing network-agnostic and network-specific structure. By enumerating deterministic LV strategies and isolating an amenable subset of outcomes, the LP tests network-local feasibility without requiring nonconvex optimization, providing a sufficient certificate of nonlocality. The framework is demonstrated on a family of ring networks, culminating in a detailed 6-party, 4-source case where infeasibility is observed for a range of beam-splitter transmissivities and validated across multiple solvers. The method offers a scalable alternative to inflation or SDP-based approaches, enabling efficient witnesses for diverse quantum network architectures and guiding experimental parameter choices for robust network-nonlocality certification.

Abstract

Network nonlocality extends Bell nonlocality to settings with multiple independent sources and parties. Certifying it in quantum information processing tasks requires suitable witnesses. However, in contrast to local correlations, the set of network-local correlations is non-convex. This non-convexity makes certifying network nonlocality a highly non-trivial task. Existing approaches involve leveraging network-specific properties, or inflation-based methods whose constraints grow combinatorially in the number of local variables. In this work, we introduce a linear programming witness for network nonlocality built from five classes of linear constraints. These classes are network-agnostic, although the explicit forms of the constraints must be tailored to a specific network's structure. We use the procedure to construct network nonlocality witnesses for a family of ring networks and certify network nonlocality for a concrete example, relying only on observed probabilities and a tunable experimental parameter. Our work advances the search for efficient witnesses to certify network nonlocality across diverse quantum network architectures.

Linear Program Witness for Network Nonlocality in Arbitrary Networks

TL;DR

This work develops a linear programming witness for network nonlocality by formulating five linear constraint classes on an auxiliary distribution q(a,λ), balancing network-agnostic and network-specific structure. By enumerating deterministic LV strategies and isolating an amenable subset of outcomes, the LP tests network-local feasibility without requiring nonconvex optimization, providing a sufficient certificate of nonlocality. The framework is demonstrated on a family of ring networks, culminating in a detailed 6-party, 4-source case where infeasibility is observed for a range of beam-splitter transmissivities and validated across multiple solvers. The method offers a scalable alternative to inflation or SDP-based approaches, enabling efficient witnesses for diverse quantum network architectures and guiding experimental parameter choices for robust network-nonlocality certification.

Abstract

Network nonlocality extends Bell nonlocality to settings with multiple independent sources and parties. Certifying it in quantum information processing tasks requires suitable witnesses. However, in contrast to local correlations, the set of network-local correlations is non-convex. This non-convexity makes certifying network nonlocality a highly non-trivial task. Existing approaches involve leveraging network-specific properties, or inflation-based methods whose constraints grow combinatorially in the number of local variables. In this work, we introduce a linear programming witness for network nonlocality built from five classes of linear constraints. These classes are network-agnostic, although the explicit forms of the constraints must be tailored to a specific network's structure. We use the procedure to construct network nonlocality witnesses for a family of ring networks and certify network nonlocality for a concrete example, relying only on observed probabilities and a tunable experimental parameter. Our work advances the search for efficient witnesses to certify network nonlocality across diverse quantum network architectures.
Paper Structure (11 sections, 2 theorems, 82 equations, 5 figures, 1 table)

This paper contains 11 sections, 2 theorems, 82 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Standard Bell Nonlocality
  4. Network Nonlocality
  5. General Linear Program Procedure
  6. Ring Network Construction
  7. LP Procedure for the Ring Network
  8. Special Case: 6 Party 4 Source Network
  9. Conclusion
  10. Proof of Theorem \ref{['thm:lv_value']}
  11. Proof of Theorem \ref{['thm:prob_lv_subset']}

Key Result

Theorem 1

Consider outcomes $\mathbf{a}\in\mathcal{O}_S$ with $a_n=\texttt{X}$. Suppose the two LVs $A_n$ receives have target lists Both lists contain $A_n$, and for sources distributing to $P$ parties, their maximum overlap can extend to $P-1$ parties. Now consider the joint target list minus $A_n$, $\{A_u,A_v,A_{u'},A_{v'}\}$. If we observe the outcomes in this joint target list are compatible with exac

Figures (5)

  • Figure 1: (a) Bipartite Scenario with one source $S_1$ characterized by one LV $\lambda_1$, distributing states to two parties, $A$ and $B$. (b) Bilocal scenario with two sources $S_1$ and $S_2$, characterized by two LVs $\lambda_1$ and $\lambda_2$, distributing states to three parties $A$, $B$, and $C$. The parties receive inputs $(x,y,z)$, determining their measurement settings and return measurement outcomes $(a,b,c)$.
  • Figure 2: $\mathcal{D}$ denotes the LV domain and $\mathcal{O}$ denotes the output space, which are related via the map $F(\mathcal{D})\mapsto \mathcal{O}$. $\mathcal{S}_p^{(1)}$ and $\mathcal{S}_p^{(2)}$ denote subregions in the LV space, $\mathcal{O}_p=F(\mathcal{S}_p^{(1)}\sqcup \mathcal{S}_p^{(2)})$ denotes a subregion in output space, and $\boldsymbol{\lambda}_j$ and $\boldsymbol{\lambda}_j'$ denote specific strategies in $\mathcal{S}_p^{(1)}$ and $\mathcal{S}_p^{(2)}$ respectively.
  • Figure 3: Ring network with $M$ sources, $S_1,\dots,S_M$, distributing tripartite single-photon W states to $N$ parties, $A_1,\dots,A_N$. Each party is equipped with a tunable beamsplitter and a non-photon-number-resolving detector in each optical mode. Each source $S_m$ has an associated LV $\lambda_m$. For this specific source-to-party configuration, odd-numbered sources signal to the next-nearest-neighbor to the left of their middle party and the nearest-neighbor to the right of their middle party, and vice versa for even-numbered sources.
  • Figure 4: 6-party ring network with 4 sources, characterized by their LVs, $\lambda_1$, $\lambda_2$, $\lambda_3$, and $\lambda_4$, distributing 3-party W states. Each party is equipped with a tunable beamsplitter and two detectors.
  • Figure 5: Numerical witness of network nonlocality. $T> 0$ corresponds to violation of network-locality conditions. (a) Minimized sum of tolerances against transmissivity for step size $\delta t=0.001$, for the 6-party, 4-source ring network in Fig. \ref{['fig:6p4s_ring_network']}. (b) Difference in sum of tolerances between the ECOS solver, and SCS and GLPK solvers.

Theorems & Definitions (7)

  • Definition 1: Strategy
  • Definition 2: Outcome pattern
  • Definition 3: Outcome Subset
  • Definition 4: Outcome Substring
  • Definition 5: Remaining LVs
  • Theorem 1: LV Value
  • Theorem 2: Probability of an LV Subset