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arXiv:2209.09921 · Physical Review Letters

Partial Self-Testing and Randomness Certification in the Triangle Network

Pavel Sekatski, Sadra Boreiri, Nicolas Brunner

Published Sep 5, 2023• 22 citations
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Abstract

Quantum nonlocality can be demonstrated without inputs (i.e., each party using a fixed measurement setting) in a network with independent sources. Here we consider this effect on ring networks, and show that the underlying quantum strategy can be partially characterized, or self-tested, from observed correlations. Applying these results to the triangle network allows us to show that the nonlocal distribution of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140401] requires that (i) all sources produce a minimal amount of entanglement, (ii) all local measurements are entangled, and (iii) each local outcome features a minimal entropy. Hence we show that the triangle network allows for genuine network quantum nonlocality and certifiable randomness.

DOI
10.1103/physrevlett.131.100201

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Quantum nonlocality can be demonstrated without inputs (i.e., each party using a fixed measurement setting) in a network with independent sources. Here we consider this effect on ring networks, and show that the underlying quantum strategy can be partially characterized, or self-tested, from observed correlations. Applying these results to the triangle network allows us to show that the nonlocal distribution of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140401] requires that (i) all sources produce a minimal amount of entanglement, (ii) all local measurements are entangled, and (iii) each local outcome features a minimal entropy. Hence we show that the triangle network allows for genuine network quantum nonlocality and certifiable randomness.
Correlations in quantum networks with independent sources exhibit a completely novel form of nonclassicality in the sense that the nonlocality of such correlations can be demonstrated in fixed local input scenarios. Before the pioneering work of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)], the nonlocal feature of such network correlations was directly attributable to standard Bell nonlocality. In the above-mentioned reference, the authors provided some of the first examples of triangle network correlations, whose nonlocality cannot be deduced from Bell-CHSH nonlocality. To date, a complete characterization of such scenarios has yet to be provided. The present work characterizes correlations arising due to fixed local measurements in a triangle network under a source independence assumption. Precisely speaking, a set of criteria is framed in the form of Bell-type inequalities, each of which is necessarily satisfied by trilocal correlations. Possible quantum violation of at least one criterion from the set is analyzed, which in turn points out the utility of the set of criteria to detect nonlocality (nontrilocality) in quantum triangle networks. Interestingly, measurement on a local product state basis turns out to be sufficient to generate nontrilocal correlations in some quantum networks. Noise tolerance of the detection criteria is discussed followed by a generalization of the framework for demonstrating correlations in any nnn-sided polygon where nnn is finite.
In the last decade, it was understood that quantum networks involving several independent sources of entanglement which are distributed and measured by several parties allowed for completely novel forms of nonclassical quantum correlations, when entangled measurements are performed. Here, we experimentally obtain quantum correlations in a triangle network structure and provide solid evidence of its nonlocality. Specifically, we first obtain the Elegant distribution proposed in [Gisin et al., Entanglement 25 years after quantum teleportation: Testing joint measurements in quantum networks, Entropy 21, 325 (2019)ENTRFG1099-430010.3390/e21030325] by performing a six-photon experiment. Then, we justify its nonlocality based on machine learning tools to estimate the distance of the experimentally obtained correlation to the local set.
Quantum correlations in networks with independent sources have revealed novel forms of nonclassical behavior. While entanglement in the sources is a necessary ingredient, the role played by entanglement in the measurements remains largely unexplored. In particular, all existing demonstrations of full network nonlocality, certifying the nonclassicality of every source in the network, have relied on entangled measurements performed at a central node with no inputs. In this work, we construct an explicit strategy that does not rely on entangled measurements, yet still achieves full network nonlocality. Our approach is based on separable measurements augmented with bidirectional classical feedforward. We further show that this same class of measurements can give rise to another recently proposed form of network nonlocality, the minimal network nonclassicality, which ensures that the observed correlations cannot be attributed to any fixed subset of nonclassical sources within the network. Finally, building on a recently developed certification framework, we quantify the amount of device-independent randomness that can be extracted from full network nonlocal correlations under different measurement strategies. Beyond their foundational significance, our results also offer a practically attractive route toward experimental implementations of network nonlocality, as they remove the need for entangled measurements.
Quantum networks with multiple sources allow the observation of quantum nonlocality without inputs. Consequently, the incompatibility of measurements is not a necessity for observing quantum nonlocality when one has access to multiple quantum sources. Here we investigate the minimal scenario without inputs where one can observe any form of quantum nonlocality. We show that even two parties with two sources that might be classically correlated can witness a form of quantum nonlocality, in particular quantum steering, in networks without inputs if one of the parties is trusted, that is, performs a fixed known measurement. We term this effect as swap-steering. The scenario presented in this work is minimal to observe such an effect. Consequently, a scenario exists where one can observe quantum steering but not Bell non-locality. We further construct a linear witness to observe swap-steering. Interestingly, this witness enables self-testing of the quantum states generated by the sources and the local measurement of the untrusted party. This in turn allows certifying two bits of randomness that can be obtained from the measurement outcomes of the untrusted device without the requirement of initially feeding the device with randomness.
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.
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 $3\text{\ensuremath{-}}3\text{\ensuremath{-}}3$ and $3\text{\ensuremath{-}}3\text{\ensuremath{-}}2$. 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.