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Physical Review A

Sequential sharing of entanglement via probabilistic projective measurements

Yanling Wang, Shi-Min Ye, Wei Chen, Zhi‐Xiang Jin, Mao-Sheng Li

Published Nov 5, 2025• 3 citations
Publisher

Abstract

Recently, Sasmal et al. [Phys. Rev. Lett. 133, 170201 (2024)] introduced probabilistic projective measurements (PPMs) for sequential sharing of nonlocality. Here we develop a general framework for sequentially witnessing quantum correlations, including bipartite and genuine multipartite entanglement, via PPMs. We show that a class of positive-operator-valued measures admits a unified PPM realization and present an explicit formula linking the initial state, intermediate channels, and witness operators, providing both computational efficiency and structural insight. Using this framework, we demonstrate unbounded sequential entanglement sharing for pure and certain mixed states, as well as detection of genuine multipartite entanglement in Greenberger-Horne-Zeilinger-type states. Our results highlight the nontrivial role of PPMs in sequential entanglement witnessing and address generalized multipartite sharing, as discussed in a recent review article [Phys. Rep. 1098, 1 (2025)].

DOI
10.1103/pck8-pwgq

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Recently, Sasmal et al. [Phys. Rev. Lett. 133, 170201 (2024)] introduced probabilistic projective measurements (PPMs) for sequential sharing of nonlocality. Here we develop a general framework for sequentially witnessing quantum correlations, including bipartite and genuine multipartite entanglement, via PPMs. We show that a class of positive-operator-valued measures admits a unified PPM realization and present an explicit formula linking the initial state, intermediate channels, and witness operators, providing both computational efficiency and structural insight. Using this framework, we demonstrate unbounded sequential entanglement sharing for pure and certain mixed states, as well as detection of genuine multipartite entanglement in Greenberger-Horne-Zeilinger-type states. Our results highlight the nontrivial role of PPMs in sequential entanglement witnessing and address generalized multipartite sharing, as discussed in a recent review article [Phys. Rep. 1098, 1 (2025)].
Sequential sharing of quantum nonlocality enables multiparty collaborative information processing, which is typically realized through unsharp measurements in previous studies. By contrast, we experimentally demonstrate that the sequential sharing of quantum nonlocality can be observed using only projective measurements, with the introduction of classical randomness. We achieve sequential violations of the Clauser-Horne-Shimony-Holt inequality in a bipartite quantum system by sharing classical randomness or using a probabilistic projective measurement. Extending the first protocol to a tripartite quantum system, sequential tripartite nonlocality is observed with the Greenberger-Horne-Zeilinger state, where violations of the Mermin inequality are observed in sequential measurements. We experimentally demonstrate these phenomena through the preparation and measurement of a three-photon entangled state. Our work reveals the key role of projective measurements in the sequential sharing of quantum nonlocality, providing a new experimental complement for measuring quantum nonlocality and opening new technological pathways for distributed utilization nonlocality in quantum networks.
The prevailing consensus is that the sequential sharing of nonlocality in a Bell experiment requires generalized unsharp measurements, given that a sharp measurement inevitably destroys the entanglement of the shared state. In contrast, a recent work [A. Steffinlongo and A. Tavakoli, Projective measurements are sufficient for recycling nonlocality, Phys. Rev. Lett. 129, 230402 (2022)PRLTAO0031-900710.1103/PhysRevLett.129.230402] demonstrated the sharing of nonlocality up to two sequential observers by employing projective measurement aided with local randomness. Here, we introduce a form of local randomness-assisted qubit projective measurement protocol that enables the sharing of nonlocality by an arbitrary number of sequential observers (Bobs) with a single spatially separated observer (Alice). We achieve this by inspecting the diversity involved in implementing generalized measurements that harness nonlocality by preserving a sufficient amount of entanglement of the shared two-qubit entangled state. Furthermore, we highlight the crucial interplay between the degrees of measurement incompatibility of Alice and Bob in demonstrating the unbounded sharing of nonlocality.
Bell nonlocality is the key quantum resource in some device-independent quantum information processing. It is of great importance to study the efficient sharing of this resource. Unsharp measurements are widely used in sharing the nonlocality of an entangled state shared among several sequential observers. Recently, the authors in [Phys. Rev. Lett. 129, 230402 (2022)] showed that the Bell nonlocality of two-qubit pure states can be shared even when one only uses projective measurements and local randomness. We demonstrate that projective measurements are also sufficient for sharing the Bell nonlocality of arbitrary high-dimensional pure bipartite states. Our results promote further understanding of the nonlocality sharing of high-dimensional quantum states under projective measurements.
Bipartite entangled states whose violations of the Clauser-Horne-Shimony-Holt Bell inequality can be observed by a single Alice and arbitrarily many sequential Bobs exist [Brown and Colbeck, Phys. Rev. Lett. 125, 090401 (2020)]. Here we consider their analogs for tripartite systems: a tripartite entangled state is shared among Alice, Bob, and multiple Charlies. The first Charlie measures his qubit and then passes his qubit to the next Charlie, who measures again with other measurements, and so on. The goal is to maximize the number of Charlies that can observe some kind of nonlocality with the single Alice and Bob. It has been shown that at most two Charlies can share genuine nonlocality of the Greenberger-Horne-Zeilinger state via the violation of the Svetlichny inequality with Alice and Bob [S. Saha et al., Quantum Inf. Process. 18, 42 (2019); Zhang and Fei, Phys. Rev. A 103, 032216 (2021)]. In this work, we show that arbitrarily many Charlies can have standard nonlocality (via violations of the Mermin inequality) and some other kind of genuine nonlocality (which is known as genuinely nonsignal nonlocality) with a single Alice and single Bob.
Sequential sharing of quantum nonlocality (SSQN) is crucial for device-independent tasks in quantum information processing, wherein relaying the post-measurement qubit through a local quantum channel to a subsequent observer constitutes an essential operational step. Here we present a theoretical analysis of noise robustness of sequential sharing for bipartite Bell and tripartite Mermin nonlocality under the influence of local phase-flip, bit-flip, and depolarizing quantum channels. It is proved that arbitrarily many independent observers can sequentially share the quantum nonlocality of Bell, Greenberger-Horne-Zeilinger, and W states via respective noise-immune channels, whereas such unbound feature of SSQN is lost under other local noisy quantum channels. Furthermore, we demonstrate that the noise-immune channel enabling unbounded SSQN can be switched by employing our newly designed measurement strategies assisted by local unitary operations on the initial entangled states. Moreover, as illustrative examples of noise robustness, we propose two concrete schemes for sharing Bell and Mermin nonlocality with two sequential local observers on one side subject to local noisy channels. Our work establishes a practical framework for realizing the SSQN under noisy quantum channels, and reveals the connection between noise robustness and measurement strategies.
Abstract While previous studies on sequential quantum sharing rely on either sharp measurements for ‘non-local’ entangled states or weak measurements for ‘local’ entangled states, we bridge this gap by investigating ‘local’ entangled states under a robust projective measurement framework. We demonstrate that probabilistic projective measurements with fixed probability can enable entanglement sharing for certain ‘local’ entangled states. Furthermore, we show that by probabilistic combination of certain different projective measurement strategies, the entanglement sharing can be expanded even under strict probability constraints. Subsequently, we prove that under projective measurements with deterministic probabilities, the maximum number of observers capable of sharing entanglement for ‘local’ entangled states is explicitly derived in terms of the entanglement of the initial state. Finally, the relationship among the sharing capacity, the initial state’s entanglement and the measurement strength of projective measurements is analytically presented.