The development of large-scale quantum networks promises to bring a multitude of technological applications as well as shed light on foundational topics, such as quantum nonlocality. It is particularly interesting to consider scenarios where sources within the network are statistically independent, which leads to so-called network nonlocality, even when parties perform fixed measurements. Here we promote certain parties to be trusted and introduce the notion of network steering and network local hidden state (NLHS) models within this paradigm of independent sources. In one direction, we show how the results from Bell nonlocality and quantum steering can be used to demonstrate network steering. We further show that it is a genuinely novel effect by exhibiting unsteerable states that nevertheless demonstrate network steering based upon entanglement swapping yielding a form of activation. On the other hand, we provide no-go results for network steering in a large class of scenarios by explicitly constructing NLHS models.
The development of large-scale quantum networks promises to bring a multitude of technological applications as well as shed light on foundational topics, such as quantum nonlocality. It is particularly interesting to consider scenarios where sources within the network are statistically independent, which leads to so-called network nonlocality, even when parties perform fixed measurements. Here we promote certain parties to be trusted and introduce the notion of network steering and network local hidden state (NLHS) models within this paradigm of independent sources. In one direction, we show how the results from Bell nonlocality and quantum steering can be used to demonstrate network steering. We further show that it is a genuinely novel effect by exhibiting unsteerable states that nevertheless demonstrate network steering based upon entanglement swapping yielding a form of activation. On the other hand, we provide no-go results for network steering in a large class of scenarios by explicitly constructing NLHS models.
Abstract Network quantum steering (NQS) arises from network models required with classification of trusted and untrusted parties. The network local hidden state (NLHS) models have first been proposed to define the NQS in a linear network with end points being trusted. In the paper, we devote to establishing the NLHS model to define the NQS in a kind of more complex and applied-extensively networks, namely, the two-forked tree-shaped network. Here we assume that the parties at the last layer are trusted while the remaining parties are untrusted in this network. According to the NLHS model, we observe that network nonlocality implies network steerability. Furthermore, we pay more attentions to discovering the relationship between the network quantum unsteerability and separability/unsteerability of bipartite sources in this two-forked tree-shaped network. Moreover, we generalize two kinds of bipartite steering inequality criteria as the NQS criteria. They are built based on statistical quantities, which can be directly evaluated in experiments.
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.
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.
In this work, we will consider the star network scenario where the central party is trusted while all the edge parties (with a number of n) are untrusted. Network steering is defined with an n local hidden state model which can be viewed as a special kind of n local hidden variable model. Two different types of sufficient criteria, nonlinear steering inequality and linear steering inequality will be constructed to verify the quantum steering in a star network. Based on the linear steering inequality, how to detect the network steering with a fixed measurement will be discussed.
Network quantum steering plays a pivotal role in quantum information science, enabling robust certification of quantum correlations in scenarios with asymmetric trust assumptions among network parties. The intricate nature of quantum networks, however, poses significant challenges for the detection and quantification of steering. In this work, we develop a neural network-based method for measuring network quantum steerability, which can be generalized to arbitrary quantum networks and naturally applied to standard steering scenarios. Our method provides an effective framework for steerability analysis, demonstrating remarkable accuracy and efficiency in standard bipartite and multipartite steering scenarios. Numerical simulations involving isotropic states and noisy Greenberger-Horne-Zeilinger states yield results that are consistent with established findings in these respective scenarios. Furthermore, we demonstrate its utility in the bilocal network steering scenario, where an untrusted central party shares two-qubit isotropic states of different visibilities, $\ensuremath{\nu}$ and $\ensuremath{\omega}$, with trusted end-point parties and performs a single Bell state measurement. Through explicit construction of a network local hidden state model derived from numerical results and incorporation of the entanglement properties of network assemblages, we analytically demonstrate that the network steering thresholds are determined by the curve $\ensuremath{\nu}\ensuremath{\omega}=1/3$ under the corresponding configuration.
Quantum steering is an asymmetric form of quantum nonlocality where one can detect whether a measurement on one system can steer or change another distant system. It is well-known that there are quantum states that are entangled but unsteerable in the standard quantum steering scenario. Consequently, a long-standing open problem in this regard is whether the steerability of every entangled state can be activated in some way. In this work, we consider quantum networks and focus on the swap-steering scenario without inputs and find linear witnesses of network steerability corresponding to any negative partial transpose (NPT) bipartite state and a large class of bipartite states that violate the computable cross-norm (CCN) criterion. Furthermore, by considering that the trusted party can perform tomography of the incoming subsystems, we construct linear inequalities to witness swap-steerability of every bipartite entangled state. Consequently, for every bipartite entangled state one can now observe a form of quantum steering.