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Laser Physics Letters

Probability distribution of three-parameter symmetric joint measurement on a triangular network

Ding Dong, Yu-Yan Zhang, Zan-Jia Li, Li-Ya Hong, Hongwei Zhu, Ying-Qiu He

Published Feb 1, 2026• 0 citations
Publisher

Abstract

Abstract Quantum networks relying on entanglement swapping offer practical resources for quantum communication, generally assisted by Bell state measurements (BSM). In contrast to conventional BSM, the novel symmetric joint measurements not only subsume standard BSM but also feature parameterization advantages. We investigate the probability distribution of three-parameter symmetric joint measurements on a triangular network. A primary result is that the output probabilities generally depend on the quantum state distributed by the sources to the three parties. We consider four Bell states serving as initial states, exploring the permutation symmetry of the measurement probabilities while applying them to nonlocality tests in a triangular network. The results demonstrate that the three-parameter symmetric joint measurement is highly effective for network nonlocality testing. Specifically, when using three singlet states as the initial states, the network nonlocality is robustly exhibited regardless of the selection of measurement parameters z , ϕ , and θ .

DOI
10.1088/1612-202x/ae3e58

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Abstract Quantum networks relying on entanglement swapping offer practical resources for quantum communication, generally assisted by Bell state measurements (BSM). In contrast to conventional BSM, the novel symmetric joint measurements not only subsume standard BSM but also feature parameterization advantages. We investigate the probability distribution of three-parameter symmetric joint measurements on a triangular network. A primary result is that the output probabilities generally depend on the quantum state distributed by the sources to the three parties. We consider four Bell states serving as initial states, exploring the permutation symmetry of the measurement probabilities while applying them to nonlocality tests in a triangular network. The results demonstrate that the three-parameter symmetric joint measurement is highly effective for network nonlocality testing. Specifically, when using three singlet states as the initial states, the network nonlocality is robustly exhibited regardless of the selection of measurement parameters z , ϕ , and θ .
Traditional Bell state measurement (BSM) and product basis measurements (PBM) have been integral to nearly the entire development of quantum computing. Unlike the BSM and the PBM, a recently proposed two-qubit joint measurement called the elegant joint measurement (EJM) exhibits novel tetrahedral symmetry in its single-qubit reduced states. In [Phys.Rev.Lett.126:220401], a parameterized two-qubit iso-entangled basis was proposed, with concurrence between 1/2 and 1, perfectly spanning the original EJM and conventional BSM. We present a two-qubit symmetric joint measurement having concurrence from 0 to 1/2, which is complementary to [Phys.Rev.Lett.126:220401] and contains the PBM and the original EJM. We investigate the symmetry of the current structure and its application in triangular networks. The results indicate that the reduction vectors of the current basis states exhibit rotational symmetry, rather than the aforementioned mirror symmetry; moreover, the output probability distributions of three parties in the network explicitly demonstrate the expected permutation symmetry. Furthermore, we generalize the two-qubit symmetric joint measurement to the multiqubit systems with an even number of qubits.
We use hyperentanglement to experimentally realize deterministic entanglement swapping based on quantum elegant joint measurements. These are joint projections of two qubits onto highly symmetric, isoentangled bases. We report measurement fidelities no smaller than 97.4%. We showcase the applications of these measurements by using the entanglement swapping procedure to demonstrate quantum correlations in the form of proof-of-principle violations of both bilocal Bell inequalities and more stringent correlation criteria corresponding to full network nonlocality. Our results are a foray into entangled measurements and nonlocality beyond the paradigmatic Bell state measurement and they show the relevance of more general measurements in entanglement swapping scenarios.
When quantum systems are shared by multiple parties in a network, the measurement outcomes of the parties can exhibit non-classical correlations, i.e., correlations that cannot be obtained if the parties shared classical systems instead. This phenomenon is known as quantum nonlocality and is typically demonstrated in the Bell scenario. However, the Bell scenario is fundamentally simpler to investigate than general networks, since the latter come with non-convex optimization problems that are often intractable. The triangle network is one of the simplest networks exhibiting this non-convexity due to the presence of three independent sources. Although some special cases of quantum nonlocality are known in the triangle network, general methods to certify classical incompatibility are still lacking, which suggests that our understanding of networks is still rather limited. For instance, the Elegant Joint Measurement (EJM) distribution is a simple and highly-symmetric outcome distribution that can be obtained with quantum systems and measurements in the triangle network. This distribution was conjectured to be non-classical eight years ago. In this article, we provide the first proof of non-classicality of the EJM distribution. To do so, we show how to combine inflation, a causal inference technique, with powerful symmetry reductions and Frank-Wolfe algorithms for large-scale optimization. We then use these methods to obtain computer-assisted proofs of non-classicality in exact arithmetic.
Abstract Defining nonlocality in a no‐input closed quantum network scenario is a new area of interest nowadays. Gisin, in [Entropy 21, 325 (2019)], proposed a possible condition for non‐tri‐locality of the trivial no‐input closed network scenario, triangle network, by introducing a new kind of joint measurement bases and a probability bound. In [npj Quantum Information (2020) 6:70] they found a shred of numerical evidence in support of Gisin's probability bound. Now based on that probability bound, it finds the nature of the correlation in a triangle network scenario. This study observes how far the probability lies from that Gisin's bound with every possible combination of entangled and local pure states distributed from three independent quantum sources. Here, it uses the generalized Elegant Joint Measurements bases for each party and find that there is a dependency of non‐locality on the entanglement of these joint measurement bases. It also checks the probability bound for the polygon structure.
Defining nonlocality in a no-input closed quantum network scenario is a new area of interest nowadays. Gisin, in[Entropy 21, 325 (2019)], proposed a possible condition for non-tri-locality of the trivial no-input closed network scenario, triangle network, by introducing a new kind of joint measurement bases and a probability bound. In[npj Quantum Information (2020) 6:70] they found a shred of numerical evidence in support of Gisin's probability bound. Now based on that probability bound, we find the nature of the correlation in a triangle network scenario. We here observe how far the probability lies from that Gisin's bound with every possible combination of entangled and local pure states distributed from three independent quantum sources. Here we use the generalized Elegant Joint Measurements bases for each party and find that there is a dependency of non-locality on the entanglement of these joint measurement bases. We also check the probability bound for the polygon structure.
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.