arxiv: 2605.00981 · v1 · submitted 2026-05-01 · 🪐 quant-ph

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The minimal example of quantum network Bell nonlocality

Erwan Don , Jessica Bavaresco , Patryk Lipka-Bartosik , Nicolas Gisin , Nicolas Brunner , Alejandro Pozas-Kerstjens

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Pith reviewed 2026-05-09 19:06 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum nonlocalityBell nonlocalityquantum networkstriangle networkhigher-order quantum operationsminimal scenariobinary outcomes

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The pith

Quantum nonlocality is possible in the triangle network with no inputs and binary outcomes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that in the triangle network—three parties linked by independent sources, each performing one binary measurement without any input choice—quantum resources can generate correlations that no local model can explain. A family of target distributions is proven to be nonlocal, and an explicit quantum model is constructed using a parameterization method inspired by higher-order quantum operations that reproduces the distributions to machine precision. This setup is the smallest possible in terms of the number of observed variables and the size of their outcome sets that allows quantum network nonlocality. Readers should care as it identifies the minimal configuration for observing this phenomenon and offers insights into the underlying mechanism.

Core claim

Quantum nonlocality is possible in the triangle network where the parties have no input choices and produce only binary-valued outcomes. We start by identifying a family of target distributions and proving their nonlocality. Next, we construct an explicit quantum model that reproduces the target distributions to machine precision using an efficient method for parameterizing quantum distributions in networks inspired by higher-order quantum operations. This constitutes the smallest scenario possible that supports quantum nonlocality in networks.

What carries the argument

An explicit quantum model for distributions in the triangle network, constructed via a parameterization inspired by higher-order quantum operations that matches target nonlocal distributions to machine precision.

Load-bearing premise

The parameterized quantum model corresponds to a valid physical realization with independent sources and local measurements.

What would settle it

Showing that no valid quantum states and measurements in the triangle network can reproduce the target distributions or that the model violates source independence.

Figures

Figures reproduced from arXiv: 2605.00981 by Alejandro Pozas-Kerstjens, Erwan Don, Jessica Bavaresco, Nicolas Brunner, Nicolas Gisin, Patryk Lipka-Bartosik.

**Figure 1.** Figure 1: FIG. 1. (a) The binary-outcome triangle network. Parties view at source ↗

**Figure 2.** Figure 2: FIG. 2. The quantum model that reproduces the target dis view at source ↗

**Figure 4.** Figure 4: FIG. 4. Euclidean distance between the distributions given by view at source ↗

**Figure 3.** Figure 3: FIG. 3. Values of the parameters of the quantum model view at source ↗

read the original abstract

In recent years, the study of Bell nonlocality has been generalized to quantum networks, where multiple independent sources distribute physical systems to distant parties who perform local measurements. In this context, a central open question is to identify the minimal network configuration in which quantum resources produce Bell nonlocal correlations. Here we address this question and show that quantum nonlocality is possible in the triangle network where the parties have no input choices and produce only binary-valued outcomes. To do so, we start by identifying a family of target distributions and proving their nonlocality. Next, we construct an explicit quantum model that reproduces the target distributions to machine precision. For this, we develop an efficient method for parameterizing quantum distributions in networks, inspired by the formalism of higher-order quantum operations. When considering the number of observed variables and their cardinality, this constitutes the smallest scenario possible that supports quantum nonlocality in networks. Moreover, by analyzing the explicit quantum model, we obtain new insights into how nonlocal distributions can be generated in quantum networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper closes the minimal open case for quantum network nonlocality by proving nonlocality in the no-input binary triangle and supplying a numerical quantum realization via a new parameterization.

read the letter

The key takeaway is that the triangle network with no inputs and binary outputs now has an explicit quantum model that produces nonlocal correlations, and the authors prove those correlations lie outside the local set. This settles the smallest scenario by number of parties, inputs, and output cardinalities that was still open. They first pick target distributions, show they cannot be reproduced by any local model in the network, then optimize a parameterization inspired by higher-order operations to match the targets to machine precision. The parameterization itself is a practical addition that should help others explore network correlations more efficiently. The nonlocality proof stands on its own and does not depend on the quantum construction. The numerical match is tight enough to be convincing for existence, and the authors extract some intuition about how the nonlocality arises from the independent sources. The main limitation is that the quantum model remains numerical rather than analytic. Without a closed-form map from the optimized parameters back to concrete states and measurements, a reviewer will want to confirm that the construction respects the independent-source constraint and does not introduce effective correlations through the parameterization. That check is doable but not automatic from the abstract alone. The paper is aimed at people working on quantum networks and Bell nonlocality. Anyone tracking minimal examples or looking for concrete constructions will get value from it. It is worth sending to peer review because it answers a well-defined open question with both a proof and a working model, even if the model is numerical.

Referee Report

2 major / 2 minor

Summary. The paper claims to identify the minimal scenario for quantum network Bell nonlocality: the triangle network with no inputs and binary outcomes. It selects a family of target distributions, proves they lie outside the local set, and constructs an explicit quantum model via a parameterization inspired by higher-order quantum operations that reproduces the targets to machine precision. Analysis of this model is said to yield insights into nonlocal correlation generation in networks.

Significance. If the quantum model is a valid realization with independent sources, the result would establish the smallest known scenario (by number of observed variables and outcome cardinality) exhibiting quantum nonlocality in networks, which is a significant advance. The parameterization technique could be reusable for other network scenarios, and the explicit model provides concrete insights beyond abstract existence proofs.

major comments (2)

[Quantum model construction] In the construction of the explicit quantum model (the section following the nonlocality proof), the higher-order parameterization is optimized to match the target distributions numerically to machine precision. However, the manuscript does not provide an explicit mapping from the optimized parameters back to concrete three independent bipartite source states and the local measurements each party performs on its two incoming systems. Without this mapping and a verification that the resulting objects satisfy the network constraints (independent sources, no communication between parties), it is not guaranteed that the numerical match corresponds to a physically valid quantum network realization rather than an effective model that encodes disallowed correlations.
[Nonlocality proof] The nonlocality proof for the target distributions is load-bearing for the central claim. The manuscript should specify the exact method (e.g., the linear program or inequality used) and confirm that it correctly incorporates the triangle network structure with three independent sources; any relaxation or approximation in the proof would undermine the separation from the local set.

minor comments (2)

[Abstract] The abstract states that the model matches 'to machine precision' but does not report the numerical tolerance, the optimization algorithm, or the number of parameters optimized; adding these details would improve reproducibility.
[Notation and definitions] Notation for the observed distributions and the network elements (sources, parties) should be introduced once and used consistently; minor inconsistencies in variable names appear in the early sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of our results. We address each major comment below and will revise the manuscript accordingly to strengthen the exposition and explicitness of the constructions.

read point-by-point responses

Referee: [Quantum model construction] In the construction of the explicit quantum model (the section following the nonlocality proof), the higher-order parameterization is optimized to match the target distributions numerically to machine precision. However, the manuscript does not provide an explicit mapping from the optimized parameters back to concrete three independent bipartite source states and the local measurements each party performs on its two incoming systems. Without this mapping and a verification that the resulting objects satisfy the network constraints (independent sources, no communication between parties), it is not guaranteed that the numerical match corresponds to a physically valid quantum network realization rather than an effective model that encodes disallowed correlations.

Authors: We appreciate the referee highlighting this important point regarding the explicitness of our quantum model. The parameterization we employ is specifically designed within the framework of higher-order quantum operations to ensure that it corresponds to a valid quantum network with three independent sources and local measurements. Each optimized parameter set directly defines the source states and the measurement operators in a way that respects the network topology and independence constraints. Nevertheless, to address the concern and make the construction fully transparent, we will include in the revised manuscript the explicit extraction of the three bipartite source states and the local measurements from the optimized parameters, along with a direct verification that they satisfy the required conditions (no communication between parties and independent sources). This will confirm that the numerical reproduction corresponds to a physically valid realization. revision: yes
Referee: [Nonlocality proof] The nonlocality proof for the target distributions is load-bearing for the central claim. The manuscript should specify the exact method (e.g., the linear program or inequality used) and confirm that it correctly incorporates the triangle network structure with three independent sources; any relaxation or approximation in the proof would undermine the separation from the local set.

Authors: We agree that providing the precise details of the nonlocality proof is essential for the rigor of our central claim. The proof proceeds by formulating and solving a linear program that determines whether the target distribution lies outside the set of local correlations in the triangle network. The LP is constructed using the standard convex characterization of network-local distributions, with variables representing the joint distributions from each independent source and constraints enforcing the independence of the three sources as well as the locality of the measurements. We will add a dedicated subsection or appendix in the revised version that explicitly states the LP formulation, including all variables, objective function, and constraints, and confirm that it fully incorporates the triangle structure without any relaxations or approximations. The separation from the local set is established exactly (up to numerical precision of the solver). revision: yes

Circularity Check

0 steps flagged

No circularity: nonlocality proved independently before numerical quantum model construction

full rationale

The paper identifies a family of target distributions, proves they lie outside the local set for the triangle network, and separately constructs a quantum realization via a new parameterization method that matches the targets to machine precision. Neither step reduces to the other by definition or construction; the nonlocality proof does not rely on the quantum parameters, and the parameterization is presented as an existence witness rather than a fitted prediction or self-referential ansatz. No self-citation is load-bearing for the central claim, and the derivation remains self-contained against the stated network constraints.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the chosen target distributions being nonlocal (proved) and on the higher-order quantum operation parameterization being able to represent all relevant quantum strategies in the network. No free parameters are explicitly named in the abstract, but the numerical match to machine precision implies at least some optimization parameters whose independence from the target is not stated.

axioms (1)

domain assumption The triangle network with independent sources and no inputs admits a quantum description via higher-order operations that can be parameterized efficiently.
Invoked to justify the construction of the explicit model.

pith-pipeline@v0.9.0 · 5492 in / 1254 out tokens · 23460 ms · 2026-05-09T19:06:18.595714+00:00 · methodology

discussion (0)

Reference graph

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