## 2 BB84, SARG04 and ack-QKD  
Consider a BB84-based protocol that encodes a classical bit using one of the four non-orthogonal quantum states, \(\ket{+_{X}},\ket{-_{X}},\ket{+_{Z}}\) and \(\ket{-_{Z}}\) (See Figure 1). In the SARG04 protocol [9], [PERSON] prepares one of the four BB84 quantum states to be sent to [PERSON], that is, she prepares a state that belongs to two conjugate bases (\(X\) and \(Z\)). In SARG04, the classical bits are encoded as follows: zero is coded with \(\ket{+_{Z}}\) and \(\ket{-_{Z}}\) and one is coded with \(\ket{+_{X}}\) and \(\ket{-_{X}}\) (see Figure 2), where the qubits are represented as black dots in the bi-dimensional Bloch sphere (the non-orthogonal states are right-angled; the orthogonal states are drawn diametrically opposed, and the parallel states occupy the same place in the sphere). The basis measurements, \(X\) and \(Z\), are depicted as horizontal and vertical lines, respectively. By contrast, in the \(BB84\) protocol, the bit, 0, is encoded with \(\ket{+_{Z}}\) and \(\ket{-_{X}}\) while the bit, 1, is encoded with \(\ket{-_{Z}}\) and \(\ket{+_{X}}\). In the sifting phase of the SARG04 protocol, [PERSON] does not reveal the basis she used (this would reveal the bit). Instead, she announces the sifting set to which the state belongs according to the following four sifting sets: \(S_{(+,+)}=\left\{\ket{+_{X}},\ket{+_{Z}}\right\}\), \(S_{(+,-)}=\left\{\ket{+_{X}},\ket{-_{Z}}\right\}\), \(S_{(-,+)}=\left\{\ket{-_{X}},\ket{+_{Z}}\right\}\) and \(S_{(-,-)}=\left\{\ket{-_{X}},\ket{-_{Z}}\right\}\). For example, if [PERSON] sends \(\ket{+_{X}}\), she announces the set, \(S_{(+,+)}\). [PERSON] measured in the \(X\) basis measurement, and he gets the result \(\ket{+_{X}}\); but, since this result is possible for both states in the set, \(S_{(+,+)}\), he has to discard the bit, 1, from \(\ket{+_{X}}\). If [PERSON] has measured with the \(Z\) basis measurement and gotten \(\ket{+_{Z}}\), again, he cannot discriminate the state sent by [PERSON]. However, if he has measured in the \(Z\) basis and got \(\ket{-_{Z}}\), then he knows that [PERSON] has sent \(\ket{+_{X}}\) and adds a zero to his key. In order for the eavesdropper to get the same secret bit as [PERSON], [PERSON] would need to perform a measurement using the conjugate bases, \(X\) and \(Z\), thus requiring multi-photonic pulses with at least three photons. In the ack-QKD protocol, [PERSON] encodes a classical bit as in the BB84: zero is encoded with \(\left|+_{Z}\right\rangle\) and \(\left|-_{X}\right\rangle\) and one is encoded with \(\left|-_{Z}\right\rangle\) and \(\left|+_{X}\right\rangle\). Furthermore, as in the SARG04 protocol, the  
Figure 1: Simplified representation of the two-dimensional Bloch sphere where the quantum states and measurement bases are shown. The \(BB84\)qubits are the non-orthogonal states: \(\{\left|0_{Z}\right\rangle,\left|1_{Z}\right\rangle,\left|0_{X}\right\rangle, \left|1_{X}\right\rangle\}\). The measurement bases, \(Z\) and \(X\), are shown as vertical and horizontal lines, respectively. If [PERSON] uses basis \(X\) (\(Z\)) to measure [PERSON]’s state, \(\left|i_{X}\right\rangle\) (\(\left|i_{Z}\right\rangle\)), he effectively gets bit \(i\) (\(i=0,1\)); otherwise, if he applies basis \(X\) (\(Z\)) to measure \(\left|i_{Z}\right\rangle\) (\(\left|i_{X}\right\rangle\)), the probability to get \(i\) is \(\frac{1}{2}\). For example, if [PERSON] measures the \(\left|0_{X}\right\rangle\) state with the \(Z\) basis, he could obtain \(\left|0_{Z}\right\rangle\) or \(\left|1_{Z}\right\rangle\) with equal probability. Figure 2: The non-orthogonal states used in the \(SARG04\)protocol encodes the bit, 0, with the states, \(\left|+_{Z}\right\rangle\) and \(\left|-_{Z}\right\rangle\), and the bit, 1, is encoded with \(\left|+_{X}\right\rangle\) and \(\left|-_{X}\right\rangle\).