Towards a minimal example of quantum nonlocality without inputs

Sadra Boreiri, Antoine Girardin, Bora Ulu, Patryk Lipka-Bartosik, Nicolas Brunner, and Pavel Sekatski

Phys. Rev. A 107, 062413 – Published 13 June 2023

Abstract

The network scenario offers interesting new perspectives on the phenomenon of quantum nonlocality. Notably, when considering networks with independent sources, it is possible to demonstrate quantum nonlocality without the need for measurement inputs, i.e., with all parties performing a fixed quantum measurement. Here we aim to find minimal examples of this effect. Focusing on the minimal case of the triangle network, we present examples involving output cardinalities of $3 − 3 − 3$ and $3 − 3 − 2$ . A key element is a rigidity result for the parity token counting distribution, which represents a minimal example of rigidity for a classical distribution. Finally, we discuss the prospects of finding an example of quantum nonlocality in the triangle network with binary outputs and point out a connection to the Lovasz local lemma.

Received 14 March 2023
Accepted 17 May 2023

DOI:https://doi.org/10.1103/PhysRevA.107.062413

Physics Subject Headings (PhySH)

Nonlocality Quantum correlations in quantum information Quantum correlations, foundations & formalism Quantum information theory Quantum networks

Quantum Information, Science & TechnologyNetworks

Authors & Affiliations

Sadra Boreiri^*, Antoine Girardin ^*, Bora Ulu, Patryk Lipka-Bartosik, Nicolas Brunner, and Pavel Sekatski

Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland

^*These authors contributed equally to this work.

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Vol. 107, Iss. 6 — June 2023

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Images

Figure 1
The triangle network features three distant parties. Each pair of parties is connected via a bipartite source. Importantly here, all parties perform a fixed measurement, i.e., they receive no input. In this work, we look for a minimal example (in terms of output cardinality) of quantum nonlocality.
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Figure 2
Noise robustness of the $3 − 3 − 3$ quantum distribution. The plot shows the minimal distance $d (P_{target}, P_{NN})$ found by the neural network (LHV-Net) between the target distribution and the closest local distribution, for different levels of noise, as quantified by the visibility parameter $V$ . Here, the measurements are given by setting $u^{2} = 0.84$ , and we consider two possible coarse grainings, the blue curve is for the measurements given in Eq. (5), the orange curve coarse grain as $0 : | 00 〉 〈 00 | + | {\bar{1}}_{0} 〉 〈 {\bar{1}}_{0} |, 1 : | 11 〉 〈 11 |$ , and $2 : | {\bar{1}}_{1} 〉 〈 {\bar{1}}_{1} |$ . The vertical line represents the critical visibilities for the original four-output RGB4 distribution as found in [22]. Note that the coarse-grained $3 − 3 − 3$ distributions appear to be much less robust to noise. The inset shows the distance for different values of the measurement parameter $u^{2}$ . The local distributions found by LHV-Net are very close to the three-output RGB4 distributions except around $u^{2} = 0.84$ .
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Figure 3
Noise robustness for the coarse-grained elegant distribution. The plot shows the distance $d (P_{target}, P_{NN})$ between the target distribution and the closest local distribution found by LHV-Net, for different values of the visibility $V$ . The vertical line represents the critical visibility for the original four-output distributions as found in [22]. Again, it seems that coarse graining reduces robustness to noise.
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Figure 4
Noise robustness of the coarse-grained Fritz distributions. The plot shows the distance $d (P_{target}, P_{NN})$ between the target distribution and the closest local distribution found by LHV-Net, for different values of the visibility $V$ . Both $3 − 3 − 2$ and $3 − 3 − 3$ cases are shown. The vertical line represents the critical visibility ( $V^{*} = 1 / \sqrt{2}$ ) for the original four-output distributions (see [22]). The colored regions indicate the visibilities detected via the web inflation (see Appendix pp5 for details).
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Figure 5
In uniform token counting strategy over the triangle, when all parties obtain $\bar{1}$ , the token paths are undetermined between $t = ⥀$ and $⥁$ . In a quantum scenario the system could be in a superposition state of $t = ⥀$ and $⥁$ . However, in a classical scenario the underlying trajectory of tokens is either $t = ⥀$ or $t = ⥁$ .
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Figure 6
The inflated triangle scenario where $α, β$ , and $γ$ are the sources and the rest are observable variables. The inflation order is set to be 2. The copies are created such that $A_{α = x, β = y, γ = z} = A_{x y z}$ where $x, y, z \in {1, 2}$ and the symbol $•$ means that the variable in question does not have a dependency on that particular source.
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