## 3 The ack-QKD Protocol with Non-Orthogonal and Parallel States
(4) In analyzing such results, [PERSON] tells [PERSON] which cases must be discarded. **Figure 4.** The parallel bi-qubits include the following states: \(\{(\left|0_{X}\right\rangle,\left|0_{X}\right\rangle),(\left|1_{X}\right\rangle,\left|1_{X}\right\rangle),\)\((\left|0_{Z}\right\rangle,\left|0_{Z}\right\rangle),(\left|1_{Z}\right\rangle,\left|1_{Z}\right\rangle)\}\). [PERSON] randomly interleaves non-orthogonal and parallel states to construct an interleaved quantum flow. [PERSON] can verify the matching cases of the non-orthogonal and the parallel states. However, [PERSON] cannot distinguish between them. Table 2 shows a running example of the ack-QKD protocol. Each time [PERSON] measures two consecutive states, one of the following detection events can be obtained:  
(i) The states produce a double detection event. We use the symbol, \((+,+)\), to denote the photonic gain of the double detection event. If the detection events are registered in the same detector, then we have a double-matching (\(2M\)) detection event. Otherwise, if the measurement of the states yields opposite result, then we have a double non-matching (\(2 nM\)) detection event. While (\(2M\)) non-orthogonal results are useful for distilling secret bits, the (\(2 nM\)) results are useless and must be discarded. In the (\(2M\)) detection event, we say that the second measurement is the acknowledgment (the ack) of the first measurement. In the example of Figure 3 (upper-right), [PERSON] sends first the qubit, \(\left|0_{X}\right\rangle\), and then the qubit, \(\left|0_{Z}\right\rangle\). Since [PERSON] uses the \(X\) basis to measure both qubits, he measures the qubit, \(\left|0_{X}\right\rangle\), as \(\left|0_{X}\right\rangle\), but he measures the qubit, \(\left|0_{Z}\right\rangle\), as \(\left|0_{X}\right\rangle\) or \(\left|1_{X}\right\rangle\) with the same 50% probability. In the case that [PERSON]'s measurement produces \(\left|0_{X}\right\rangle\), we say that this measurement is the ack of the first \(\left|0_{X}\right\rangle\) state. Conversely, if [PERSON] obtains \(\left|1_{X}\right\rangle\), we say that \(\left|1_{X}\right\rangle\) is the negative acknowledgment (the nack) of \(\left|0_{X}\right\rangle\). In a channel with losses, there are two more possible outcomes: (ii) The single detection event, which occurs when [PERSON] obtains only one detection event, because the other state is lost. We use the symbol, \((\pm,\mp)\), to denote the single detection event. To be more specific, [PERSON] will use the symbol, (\(S\)-\(i\)), to represent the single detection event, where \(i\) can be one or two, depending of the state-number that gives a click after he applies the basis measurement, \(X\) or \(Z\), to the two consecutive incoming states. Thus, the number, \(i\), will be announced publicly by [PERSON]. (iii) The two pulses are lost. This case is represented with the symbol, \((-,-)\), or alternatively as \(2L\). In the ack-QKD protocol, [PERSON] and [PERSON] use two consecutive non-orthogonal states to distill one secret bit. [PERSON] announces publicly the basis measurement, \(X\) or \(Z\), and the sifting instances he obtained: (\(2M\)), (\(2 nM\)), (\(S\)-1), (\(S\)-2) and (\(2L\)). In addition, [PERSON] use the bits obtained from the single detection events, (\(S\)-1) and (\(S\)-2), to verify the single photonic gain of the quantum channel.