Bipartite and multipartite entangled states are basic ingredients for constructing quantum networks and their accurate verification is crucial to the functioning of the networks, especially for untrusted networks. Here we propose a simple approach for verifying the Bell state in an untrusted network in which one party is not honest. Only local projective measurements are required for the honest party. It turns out each verification protocol is tied to a probability distribution on the Bloch sphere and its performance has an intuitive geometric meaning. This geometric picture enables us to construct the optimal and simplest verification protocols, which are also very useful to detecting entanglement in the untrusted network. Moreover, we show that our verification protocols can achieve almost the same sample efficiencies as protocols tailored to standard quantum state verification. Furthermore, we establish an intimate connection between the verification of Greenberger–Horne–Zeilinger states and the verification of the Bell state. By virtue of this connection we construct the optimal protocol for verifying Greenberger–Horne–Zeilinger states and for detecting genuine multipartite entanglement.

二分和多分纠缠态是构建量子网络的基本要素，它们的准确验证对于网络的功能至关重要，尤其是对于不受信任的网络。在这里，我们提出了一种在一方不诚实的不可信网络中验证贝尔状态的简单方法。诚实方只需要局部投影测量。事实证明，每个验证协议都与 Bloch 球体上的概率分布相关，其性能具有直观的几何意义。这个几何图使我们能够构建最优和最简单的验证协议，这对于检测不可信网络中的纠缠也非常有用。而且，我们表明，我们的验证协议可以实现与为标准量子状态​​验证量身定制的协议几乎相同的样本效率。此外，我们在 Greenberger-Horne-Zeilinger 状态的验证和 Bell 状态的验证之间建立了密切的联系。凭借这种联系，我们构建了用于验证 Greenberger-Horne-Zeilinger 状态和检测真正多方纠缠的最佳协议。