## 3 Methodology
### BB84 Protocol  
In our experimental methodology, we proceed with the understanding that the security of the BB84 protocol has been carefully tested and verified in prior research papers [32; 52; 53]. As a result, we focus on the practical implementation and performance aspects of BB84 without directly considering the presence of an eavesdropper ([PERSON]) in our specific experiments. Our objective is to assess the applicability and efficiency of the protocol within the confines of a controlled and without any eavesdropper. Since QKD works with the no-cloning theorem, which states that it is impossible to create perfect copies of an unknown quantum state [17], it will be impossible for [PERSON] to make a perfect copy of a transmitting qubit over the quantum channel. Any act on the qubit will ultimately cause some errors, which can be detected by both [PERSON] and [PERSON]. As previously discussed in the literature review, consider a quantum state \(|\psi\rangle\) expressed as a linear combination of basis states:  
\[|\psi\rangle=\alpha|0\rangle+\beta|1\rangle \tag{4}\]  
Suppose there exists a unitary operator \(U\) that can clone any quantum state:  
\[U(|\psi\rangle\otimes|0\rangle)=|\psi\rangle\otimes|\psi\rangle \tag{5}\]  
To test this assumption, substitute \(|\psi\rangle=\alpha|0\rangle+\beta|1\rangle\) into the cloning operation:  
\[U((\alpha|0\rangle+\beta|1\rangle)\otimes|0\rangle)=(\alpha|0\rangle+\beta|1 \rangle)\otimes(\alpha|0\rangle+\beta|1\rangle) \tag{6}\]  
Expanding the right-hand side:  
\[(\alpha|0\rangle+\beta|1\rangle)\otimes(\alpha|0\rangle+\beta|1 \rangle)=\alpha^{2}|0\rangle\otimes|0\rangle+\alpha\beta|0\rangle\otimes|1 \rangle+\beta\alpha|1\rangle\otimes|0\rangle+\beta^{2}|1\rangle\otimes|1\rangle \tag{7}\]  
Next, considering the linearity property of unitary operators and assuming perfect cloning by \(U\):  
\[U((\alpha|0\rangle+\beta|1\rangle)\otimes|0\rangle)=\alpha U(|0 \rangle\otimes|0\rangle)+\beta U(|1\rangle\otimes|0\rangle) \tag{8}\]  
Given \(U\) performs perfect cloning:  
\[U(|0\rangle\otimes|0\rangle)=|0\rangle\otimes|0\rangle,\quad U(|1 \rangle\otimes|0\rangle)=|1\rangle\otimes|1\rangle \tag{9}\]  
Substituting these results:  
\[\alpha U(|0\rangle\otimes|0\rangle)+\beta U(|1\rangle\otimes|0 \rangle)=\alpha(|0\rangle\otimes|0\rangle)+\beta(|1\rangle\otimes|1\rangle) \tag{10}\]  
Comparison of this with the expanded form reveals:  
\[\alpha(|0\rangle\otimes|0\rangle)+\beta(|1\rangle\otimes|1\rangle)\
eq\alpha ^{2}|0\rangle\otimes|0\rangle+\alpha\beta|0\rangle\otimes|1\rangle+\beta\alpha |1\rangle\otimes|0\rangle+\beta^{2}|1\rangle\otimes|1\rangle \tag{11}\]This difference shows that no unitary operator \(U\) can clone an arbitrary quantum state. As a consequence, the QKD quantum channel will be considered secure against eavesdropping attacks. The QKD protocol involves a series of steps to secure quantum communication:  
1. **[PERSON]'s initialization**: [PERSON] begins the quantum key distribution process by choosing a set of random bits and corresponding random bases. She keeps this information private to avoid unauthorized access. 2. **Encoding qubits**: [PERSON] encodes each bit into a string of qubits using the selected bases. This encoding process is necessary to ensure trustworthy transmission. She sends the encrypted data to [PERSON] in the form of a qubit string, which is [PERSON]'s result. 3. **[PERSON]'s measurement**: When [PERSON] receives [PERSON]'s encoded qubit string, he randomly measures each qubit using his own set of randomly selected bases. By keeping the measurement results private, [PERSON] protects the confidentiality of the communication and the process's integrity. 4. **Basis disclosure**: Following the measurement step, [PERSON] and [PERSON] reveal the bases used for each qubit. This disclosure allows both parties to match their measurement bases which helps in the creation of a shared secret key. Incompatible bases and their bit values will be removed. 5.