## 3 The ack-QKD Protocol with Non-Orthogonal and Parallel States  
In the ack-QKD protocol, one classical bit is encoded using two quantum states. Such encoding is done by means of non-orthogonal or parallel states (see Figures 3 and 4). In quantum physics, if \(X\)=\(\{\left|0_{X}\right\rangle,\left|1_{X}\right\rangle\}\) and \(Z\)=\(\{\left|0_{Z}\right\rangle,\left|1_{Z}\right\rangle\}\) are orthonormal bases, then the magnitude of each basis vector is the unity, and any vector in such a space can be written as a linear combination of such a basis. For example, we can rewrite \(\left|0_{X}\right\rangle\) as \(\frac{1}{\sqrt{2}}\left|0_{Z}\right\rangle+\frac{1}{\sqrt{2}}\left|1_{Z}\right\rangle\). Two qubits, \(\left|0_{X}\right\rangle\) and \(\left|0_{Z}\right\rangle\), are non-orthogonal if the inner product between them is different from zero, symbolically \(\left\langle 0_{X}|0_{Z}\right\rangle\
eq 0\). In this regard, \begin{table}
\begin{tabular}{l l l} \hline \hline
**Alice Sends** & **Bob Obtains a (\(2M\))** & **Secret Bit** \\ \hline \(\{\left|+_{X}\right\rangle,\left|+_{Z}\right\rangle\}\) & \(\{\left|+_{X}\right\rangle,\left|+_{X}\right\rangle\}\) & 1 \\ \(\{\left|+_{X}\right\rangle,\left|-_{Z}\right\rangle\}\) & \(\{\left|+_{X}\right\rangle,\left|+_{X}\right\rangle\}\) & 1 \\ \(\{\left|-_{X}\right\rangle,\left|+_{Z}\right\rangle\}\) & \(\{\left|-_{X}\right\rangle,\left|-_{X}\right\rangle\}\) & 0 \\ \(\{\left|-_{X}\right\rangle,\left|-_{Z}\right\rangle\}\) & \(\{\left|-_{X}\right\rangle,\left|-_{X}\right\rangle\}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An event in which [PERSON] measures with the \(X\) basis the two states sent by [PERSON] and obtains a (\(2M\)) result. \(\left\langle 0_{X}|0_{Z}\right\rangle=\frac{1}{\sqrt{2}}(1)+\frac{1}{\sqrt{2}}(0)\) and \(\left\langle 0_{X}|0_{Z}\right\rangle=\frac{1}{\sqrt{2}}\). The inner product of \(orthogonal\) qubits is zero, e.g., \(\left\langle 0_{X}|1_{X}\right\rangle=0\), and identical (or parallel) qubits produce the unity under the inner product, thus \(\left\langle 0_{X}|0_{X}\right\rangle=1\). In the ack-QKD protocol, [PERSON] chooses randomly between sending a pair of parallel or non-orthogonal states. At the other side, [PERSON] measures the two consecutive pulses with the same basis measurement, \(X\) or \(Z\) (see Figures 3 and 4). In this context, the pair of quantum states sent by [PERSON] is called bi-qubit. Parallel bi-qubits define the parallel quantum flow, and non-orthogonal bi-qubits define the non-orthogonal quantum flow. Now, we summarize the ack-QKD protocol with non-orthogonal and parallel states. The ack-QKD protocol was introduced in [12]. In this reference, the non-orthogonal states are called protocol states, while parallel states are named decoy states:  
(1) [PERSON] chooses randomly between a non-orthogonal bi-qubit and a parallel bi-qubit. If she chooses a non-orthogonal bi-qubit, she must pick up randomly one of the following states:\(\{(\left|0_{X}\right\rangle,\left|0_{Z}\right\rangle)\), (\(\left|0_{X}\right\rangle,\left|1_{Z}\right\rangle)\), (\(\left|1_{X}\right\rangle\), \(\left|0_{Z}\right\rangle)\)), (\(\left|1_{X}\right\rangle\), \(\left|1_{Z}\right\rangle)\}\); where the order between states \(X\) or \(Z\) is also chosen randomly. If she chooses a parallel bi-qubit, she must select at random a bi-qubit from the set: \(\{(\left|0_{X}\right\rangle\), \(\left|0_{X}\right\rangle)\), (\(\left|1_{X}\right\rangle\), \(\left|1_{X}\right\rangle\)), (\(\left|0_{Z}\right\rangle\), \(\left|0_{Z}\right\rangle\)), (\(\left|1_{Z}\right\rangle\)))\(\}\). Then, she prepares the bi-qubit and sends it to [PERSON]. (2) [PERSON] selects randomly the basis, \(X\) or \(Z\), to measure the incoming bi-qubit. (3) [PERSON] announces over the public channel his basis measurement, and he also reveals weather the outcome he obtained is a double-detected event (\(2M\) or \(2 nM\)), a single-detected event (\(S\)-1 or \(S\)-2) or a lost bi-qubit (\(2L\)) (See discussion below).