This work develops an effective channel model for distributed passive-user Loop-Back quantum key distribution. In the intended key-establishment setting, the two passive users \(B_1\) and \(B_2\) are the legitimate lightweight parties that establish a shared secret key by using Alice as an active quantum preparation-and-measurement infrastructure. A single active station prepares and measures BB84 states, while two remote users apply only local polarization rotations. We show that the passive-user pair can be externally encapsulated as an effective Loop-Back node whose observable action is a balanced mixture of conjugate-basis dephasings, equivalently represented as an anisotropic Pauli channel with identity, \(X\), and \(Z\) components and no \(Y\) component. This structure differs from isotropic depolarization and recovers the ideal conclusive-event probability \(P_{\mathrm{conc}}=1/4\). The model also clarifies why non-orthogonal intermediate states are necessary for passive-user security. This channel-level description characterizes Alice's observable statistics and provides a compact basis for subsequent analysis of passive-user Loop-Back QKD under realistic optical and adversarial conditions.

翻译：本文针对分布式被动用户环回量子密钥分发建立了一种有效的信道模型。在目标密钥建立场景中，两个被动用户\(B_1\)和\(B_2\)作为合法的轻量级参与方，通过利用主动量子制备与测量设施Alice来建立共享密钥。单个主动站负责制备并测量BB84态，而两个远程用户仅执行本地偏振旋转操作。研究证明，被动用户对可被外部等效封装为一个环回节点，其可观测动作表现为共轭基退相位的均衡混合，等价于一种各向异性的泡利信道，包含恒等算符、\(X\)和\(Z\)分量，且不含\(Y\)分量。该结构区别于各向同性退极化，并恢复了理想情况下必然事件概率\(P_{\mathrm{conc}}=1/4\)。该模型还阐明了为何被动用户安全性要求引入非正交中间态。这种信道级描述刻画了Alice的可观测统计特性，并为进一步分析现实光学条件及对抗条件下被动用户环回量子密钥分发提供了紧凑基础。