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Physica Scripta

Quantum steering in two-forked tree-shaped networks

Shuyuan Yang, Kan He, Xiaofei Qi, Jinchuan Hou

Published Oct 18, 2023• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ad049f

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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.
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.
Quantum network correlations are crucial for long-distance quantum communication, quantum cryptography, and distributed quantum computing. Detecting network steering is particularly challenging in complex network structures. We have studied the steering inequality criteria for a 2-forked 3-layer tree-shaped network. Assuming the first and third layers are trusted and the second layer is untrusted, we derived a steering inequality criterion using the correlation matrix between trusted and untrusted observables. In particular, we apply the steering criterion to three classes of measurements which are of special significance: local orthogonal observables, mutually unbiased measurements, and general symmetric informationally complete measurements. We further illustrate the effectiveness of our method through an example.
Abstract 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.
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 nnn) are untrusted. Network steering is defined with an nnn local hidden state model which can be viewed as a special kind of nnn 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.
In the last decade, much attention has been focused on examining the nonlocality of various quantum networks, which are fundamental for long-distance quantum communications. In this paper, we consider the nonlocality of any forked tree-shaped network, where each node, respectively, shares arbitrary number of bipartite sources with other nodes in the next "layer". The Bell-type inequalities for such quantum networks are obtained, which are, respectively, satisfied by all (tn-1)-local correlations and all local correlations, where tn denotes the total number of nodes in the network. The maximal quantum violations of these inequalities and the robustness to noise in these networks are also discussed. Our network can be seen as a generalization of some known quantum networks.