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Incompatibility of quantum measurements and quantum steering in the entanglement-swapping network

Nanqun Cui, Kan He, Shuyuan Yang

Published Feb 7, 2026• 0 citations
Publisher
DOI
10.1007/s11128-026-05075-7

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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.
Einstein-Podolsky-Rosen (EPR) steering is a quantum nonlocal effect that characterizes the ability to remotely control quantum states of one system via local measurements on another distant system entangled with the controlled one. Entanglement swapping is an effective way to realize distant entanglement, which is useful for building quantum networks. Previous studies show that the presence of entanglement of subsystems in mixed states is merely necessary for achieving entanglement swapping. In this paper, we find that for Gaussian entanglement swapping with two pairs of entangled modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}\phantom{\rule{4pt}{0ex}}(j=1,2)$, the existence of directional EPR steering from the modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}$ under homodyne detection to the other modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ is a sufficient and necessary condition for achieving swapped entanglement between modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{2}$, and moreover this achievement is independent of the reverse steering from modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}$. We further reveal that this is because the steering in that direction enables the amplitude and phase squeezing of the two composite modes of modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{2}$ via homodyne detection, which is necessary and sufficient for swapped entanglement. As a concrete example, we next investigate light-mechanical steering in a dispersively or dissipatively coupled optomechanical system, and we consider the generation of entanglement between two distant optomechanical oscillators via entanglement swapping by utilizing steering. It is interesting to find that even without the constraint that the cavity linewidth should be smaller than the mechanical frequency, robust mechanical entanglement can be obtained in the steady-state regime.
Quantum nonlocality and its asymmetric counterpart, quantum steering, are among the most intriguing manifestations of quantum mechanics. From a theoretical perspective, they are not only of fundamental significance but also hold promise for a wide range of applications. However, the stringent technological requirements for their experimental observation in a loophole-free way have largely restricted their realization to foundational demonstrations. A major challenge in these setups is the limited detection efficiency of current detectors as to observe nonlocality or steering in the standard scenarios, one requires detectors above a certain critical efficiency which can only be achieved with superconducting detectors. Considering the simplest quantum network, we demonstrate here that quantum steering between two parties, can be demonstrated for any non-zero detection efficiency, if the sources in the network generate states above a critical visibility well-within the current practical limits. This form of quantum steering in networks, is termed swap-steering. Moreover, when the sources are perfect, swap-steering can be observed using any entangled measurement on the untrusted side, even with arbitrarily low detection efficiency. Consequently, two major loopholes, the detection-loophole as well as free-will loophole can be closed easily in quantum steering experiments, when implemented using networks.