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arXiv:2602.04173 · Physical Review A

Symmetric joint measurement as a complement to the elegant joint measurement

Ying-Qiu He, Yu-Yan Zhang, Dong Ding, Ting Gao, Feng-Li Yan

quant-phSubmitted Feb 4, 2026• Announced Feb 5, 2026• 0 citations
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Abstract

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.

Journal reference
Physical Review A 114, 032402 (2026)
DOI
10.1103/dmhd-b7cd
Comments
8 pages, 4 figures

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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.
A standard two-qubit joint measurement is the well-known Bell state measurement, in which each reduced state (traced out one qubit) is the completely mixed state. Recently, a novel quantum joint measurement known as the elegant joint measurement (EJM) has been proposed, where the reduced states of the EJM basis have tetrahedral symmetry. In this paper, we first propose a five-parameter entangled state and reveal its inherent symmetry. Based on this, we define a generalized EJM parametrized by zzz, ${\ensuremath{\varphi}}_{}$, and ${\ensuremath{\theta}}_{}$, and provide the quantum circuits for preparing and detecting these basis states. There are three main results: (i) The previous single-parameter EJM can be directly derived by specifying the parameters zzz and ${\ensuremath{\varphi}}_{}$; (ii) the initial unit vectors related to the four vertices of the regular tetrahedron are not limited to the original choice and not all the unit vectors in cylindrical coordinates are suitable for forming the EJM basis; and (iii) the reduced states of the present EJM basis can always form two mirror-image tetrahedra, robustly preserving its elegant properties. We aim to determine what kind of states the EJM basis belongs to and to provide a method for constructing the generalized three-parameter EJM, which may contribute to the multisetting measurement and the potential applications for quantum information processing.
The elegant joint measurement (EJM) is a highly symmetric, partially entangled two-qubit measurement whose local marginals form a regular tetrahedron on the Bloch sphere and which has a low entanglement cost for local implementation. It plays a central role in quantum networks exhibiting nonclassical correlations and serves as a paradigmatic example of an entangled measurement with local structure. Despite its significance, generalizing the EJM beyond two qubits has remained unresolved. Here, we extend the EJM to the multipartite setting by identifying all tetrahedrally symmetric, efficiently localizable multiqubit bases. For two qubits, these criteria uniquely select the EJM. For three or more, they yield a discrete set of equivalence classes, reflecting the richer structure of multiparticle entanglement.
We give a closed-form construction of the nnn-qubit Elegant Joint Measurement (EJM) proposed in [PRL \textbf{136}, 190201 (2026)] and show that it is part of a tunable family of measurements with tetrahedrally arranged Bloch vectors. The construction is based on the interference pattern implied by a single phase polynomial built from the elementary symmetric functions. It realises a regular tetrahedral measurement for every nnn, and the corresponding measurement unitary lies at level n+1n{+}1n+1 of the Clifford hierarchy. Starting from this measurement, we ask whether the size of the local tetrahedron -- and hence the entanglement of the basis -- can be varied while preserving its symmetry. For every even nnn the answer is yes, and remarkably the size follows the same one-parameter law that governs the known two-qubit family, interpolating down to a 111-uniform basis. For n=3n=3n=3 the EJM is locally isolated, while for odd n≥5n\ge5n≥5 we do not know an analogous closed-form family. We also give an analogous construction, valid for every n≥3n \geq3n≥3, with square local geometry.
We investigate the generalization of symmetric quantum joint measurements on multiple qubits. We first describe a method for constructing a symmetric joint measurement basis for three qubits by utilizing single-qubit states corresponding to the four vertices of an isohedral tetrahedron on the Bloch sphere. We then demonstrate the expected tetrahedral symmetry of the current measurement basis and discuss its application in a trilocal star-shaped network. This architecture enables us to generalize the two-qubit symmetric joint measurement to an nnn-qubit version, preserving the tetrahedral or hexahedral symmetry.
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 .
While entanglement between distant parties has been extensively studied, entangled measurements have received relatively little attention despite their significance in understanding nonlocality and their central role in quantum computation and networks. We present a systematic study of entangled measurements, providing a complete classification of all equivalence classes of iso-entangled bases for projective joint measurements on two qubits. The application of this classification to the triangular network reveals that the elegant joint measurement, along with white noise, is the only measurement resulting in output permutation invariant probability distributions when the nodes are connected by Werner states. The paper concludes with a discussion of partial results in higher dimensions. Published by the American Physical Society 2024