# Loop-Back Quantum Key Distribution (QKD) for Secure and Scalable Multi-Node Quantum Networks
## 2 Loop-Back QKD in a Linear Topology
### Single-Pulse Loop-Back QKD  
In the IR attack, [PERSON] deploys a measurement device similar to [PERSON]'s to measure the quantum states sent by [PERSON] over the channel. After obtaining the measurement results, [PERSON] generates a pulse and sends it to [PERSON]'s station. This type of attack is referred to as a MitM attack. The Loop-Back QKD protocol was specifically designed to detect MitM attacks. It builds on the BB84 protocol, where [PERSON] takes on the role of [PERSON], thereby eliminating the need to reveal the measurement bases. Specifically, while [PERSON] does not measure the quantum state received from [PERSON], he polarizes it in one of the bases \((X,Z)\) and then returns it to [PERSON]. In this context, polarizing refers to preparing the quantum state using optical polarizers aligned with a specific basis \((X,Z)\), rather than applying a quantum gate or performing a measurement. This approach ensures that the state remains within a well-defined basis, analogous to how BB84 encodes information using polarizers. Upon receiving the returned state, [PERSON] measures it in the same basis she originally used to send the state. If the measurement result corresponds to a state orthogonal to the one sent, [PERSON] can deduce the basis used by [PERSON]. [PERSON] and [PERSON] use the identification of bases to encode the shared bits: cases where [PERSON] selects the \(Z\)-basis represent 0, while those where he chooses the \(X\)-basis represent 1. In the protocol, [PERSON] prepares and sends a quantum state \(\ket{\psi}\) to [PERSON] using the \(X\) basis. [PERSON] then randomly selects a measurement basis, either \(X\) or \(Z\), and applies the corresponding polarization. Figure 1 illustrates in the Bloch sphere, the case where [PERSON] sends \(\ket{\psi}=\ket{0_{X}}\). If [PERSON] measures in the \(Z\) basis, the resulting state is an equal superposition of \(\ket{0_{Z}}\) and \(\ket{1_{Z}}\), leading to a probability of 0.5 for [PERSON] to obtain the orthogonal state \(\ket{1_{X}}=\ket{\
eg\psi}\) upon measuring in the \(X\) basis. If [PERSON] measures in the \(X\) basis, the probability of obtaining \(\ket{1_{X}}\) is zero. Since [PERSON] performs her measurement in the same basis she used to prepare \(\ket{\psi}\), whenever she obtains the orthogonal state \(\ket{\
eg\psi}\), she can conclusively infer [PERSON]'s choice of basis. Conversely, Figure 2 depicts, in the Bloch sphere, the scenario where [PERSON] sends \(\ket{\psi}=\ket{0_{X}}\). If [PERSON] measures in the \(X\) basis, the probability of obtaining \(\ket{0_{X}}\) is unity. [PERSON] then measures in the \(X\) basis, and, if she obtains \(\ket{0_{X}}=\ket{\psi}\), this outcome could have originated from [PERSON] measuring in either the \(X\) or the \(Z\) basis. Thus, in this case, [PERSON] cannot infer [PERSON]'s choice of basis. In the previous explanation, we adopted a quantum mechanical approach to describe state transformations and measurement probabilities. However, in the following sections, we will minimize the use of this formalism to facilitate the exposition of QKD protocols while preserving the essential quantum principles. The probabilities associated with the \(0_{X}\) state are provided in Table 1, which allows us to deduce that the protocol's efficiency (\(\eta\)) in an ideal channel reaches \(25\%\). To provide a clearer understanding of the notation used in this work, we define the symbols in Table 2.  
The protocol can be summarized in the following steps:  
1. [PERSON] prepares a quantum state \(\psi\) in either the \(X\) or \(Z\) basis, such that \(\psi\in\{0_{Z},1_{Z},0_{X},1_{X}\}\), and sends \(\psi\) to [PERSON]. 2. [PERSON], without measuring the state, applies a polarization operation using either the \(X\) or \(Z\) basis and returns the state to [PERSON].