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.
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.
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.
Abstract In the standard Bell experiment, two parties perform local projective measurements on a shared pair of entangled qubits to generate nonlocal correlations. However, these measurements completely destroy the entanglement, rendering the post-measurement state unable for subsequent use. For a long time, it was believed that only unsharp measurements can be used to share quantum correlations. Remarkably, recent research has shown that classical randomness assisted projective measurements are sufficient for sharing nonlocality (Steffinlongo and Tavakoli 2022 Phys. Rev. Lett. 129 230402). Here, by stochastically combining no more than two different projective measurement strategies, we report an experimental observation of double Clauser–Horne–Shimony–Holt inequality violations with two measurements in a sequence made on each pair of maximally and partially entangled polarization photons. Our results reveal that the double violation achieved by partially entangled states can be 11 standard deviations larger than that achieved by maximally entangled ones. Our scheme eliminates the requirement for entanglement assistance in previous unsharp-measurement-based sharing schemes, making it experimentally easier. Our work provides possibilities for sharing other types of quantum correlations in various physical systems with projective measurements.
As a way of saving quantum resources, recycling of Bell nonlocality has been experimentally studied, but restricted to sequential unsharp measurements. However, it has been theoretically shown recently that projective measurements are sufficient for recycling nonlocality [Phys. Rev. Lett. \textbf{129}, 230402 (2022)]. Here, we go beyond unsharp measurement scenarios and experimentally demonstrate the recycling of nonlocal resources with projective measurements. By verifying the violation of Clauser-Horne-Shimony-Holt (CHSH) inequality, we find that three independent parties can recycle the Bell nonlocality of a two-qubit state, whether it is maximally or partially entangled. Furthermore, in the double violation region, the optimal trade-off for partially entangled states can be 11 standard deviations better than that for maximally entangled states. Our results experimentally eliminate the common misconception that projective measurements are incompatible with the recycling of quantum correlations. In addition, our nonlocality recycling setup does not require entanglement assistance, which is much more experimentally friendly, thus paving the way for the reuse of other kinds of quantum correlations.
Abstract Bell nonlocality is a valuable resource in quantum information processing tasks. Scientists are interested in whether a single entangled state can generate a long sequence of nonlocal correlations. Previous work has accomplished sequential tripartite nonlocality sharing through unsharp measurements. In this paper, we investigate the sharing of tripartite nonlocality using only projective measurements and sharing classical randomness. For the generalized GHZ state, we have demonstrated that using unbiased measurement choices, two Charlies can share the standard tripartite nonlocality with a single Alice and a single Bob, while at most one Charlie can share the genuine tripartite nonlocality with a single Alice and a single Bob. However, with biased measurement choices, the number of Charlies sharing the genuine tripartite nonlocality can be increased to two. Nonetheless, we find that using biased measurements does not increase the number of sequential observers sharing the standard tripartite nonlocality. Moreover, we provide the feasible range of double violation for the parameters of the measurement combination probability with respect to the state.
Recently the authors of Phys. Rev. Lett. 125, 090401 (2020) considered the following scenario: Alice and Bob each have half of a pair of entangled qubit states. Bob measures his half and then passes his part to a second Bob who measures again, and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality with the single Alice. By taking the maximally entangled pure two-qubit state $|\ensuremath{\phi}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|00\ensuremath{\rangle}+|11\ensuremath{\rangle})$ as an example, it has been constructively proved that arbitrarily many independent Bobs can share the nonlocality with the single Alice. Here we demonstrate that arbitrarily many independent observers can share the nonlocality of a single arbitrary dimensional bipartite entangled but not necessary two-qubit entangled state. Furthermore, taking the generalized Greenberger-Horne-Zeilinger (GHZ) states as an example, we show that at most two Charlies can share the genuine nonlocality of a single generalized GHZ state with an Alice and a Bob.
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.