## 2 Two-Way Protocols  
The SDC protocol in some sense is closer to the earlier instances of two-way protocols, namely the Ping-Pong [3] and more specifically its improved version [11] which provides for a higher protocol capacity in terms of the number of classical bits encoded in each EM run. It is however more resource demanding as it requires the use of a quantum memory on [PERSON]'s side. Our description of SDC here follows closely that of [5]. Very simply, the SDC protocol sees [PERSON] sending half a Bell pair to [PERSON] (while storing the other half in a quantum memory) for her to encode by virtue of a randomly chosen unitary transformation from the set containing the identity operator and the three Pauli operators, \(\{I,\sigma_{x},\sigma_{y},\sigma_{z}\}\). She would subsequently submit the qubit back to [PERSON] for measurements. Alternatively, she could measure the received qubit in the \(Z\) basis and prepare another in the \(X\) basis to be sent to [PERSON]. The former process, done with probability \(c\approx 1\), corresponds to her actions in EM, while the latter, done with probability \(1-c\), corresponds to CM. Despite the use of \(4\) unitaries imply a larger alphabet used, particularly elements of \(\mathbb{Z}_{4}\)\(0,1,2,3\) mapped to \(I,\sigma_{x},\sigma_{y},\sigma_{z}\) respectively as opposed to bits for encoding, [PERSON] could in fact assign logical bits (in pairs) \(00,10,11\) and \(01\) to the unitaries. However the mapping \(f:\mathbb{Z}_{4}\rightarrow\mathbb{Z}_{2}^{2}\) to bits should be done only at the end of the protocol to avoid any possibility of [PERSON] capitalizing on the correlation of bits given any pair [12]. [PERSON], on the other hand would, with a probability \(c\) make a Bell measurement and with probability \(1-c\) measure his stored and received qubit in the \(Z\) and \(X\) basis respectively. A protocol run which sees Bell measurements by [PERSON] coinciding with [PERSON]'s unitary transformations would allow [PERSON] to distinguish between [PERSON]'s unitaries perfectly (given that the Pauli matrices would shift between orthogonal Bell states). Assigning logical symbols \(0,1,2\) and \(3\) to the Bell states \(|\psi^{+}\rangle,|\psi^{-}\rangle,|\phi^{+}\rangle\) and \(|\phi^{-}\rangle\) respectively, these instances referred to as EM provide for sharing of a raw key between [PERSON] and [PERSON]. Again, it is possible for [PERSON] to use bits instead when [PERSON] submits the mapping information to [PERSON] at the end of the protocol runs. The instances when [PERSON]'s and [PERSON]'s measurements in the \(Z\) and \(X\) bases coincides would allow for a meaningful CM. Hence a successful EM happens only with probability \(c^{2}\) and CM happens with probability \((1-c)^{2}\). A schematic diagram for SDC is illustrated in Figure 1 below. The case for a two-way QKD without using entanglement, namely the LM05, sees a qubit prepared by [PERSON] in a particular bases randomly chosen from either the \(X\) or \(Z\) with probabilities \(p_{X}\) and \(p_{Z}\) respectively to be sent to [PERSON] for her encoding. In the original LM05 protocol, apart from \(p_{X}=p_{Z}\), only two unitary transformations were considered, namely the passive \(I\) and \(i\sigma_{y}\). In [5], a generalization was made to include another two unitary transformation, namely the \(\sigma_{x}\) and \(\sigma_{z}\), and the possibility of having different probabilities \(p_{X}\) and \(p_{Z}\). In this protocol, upon executing her unitary transformation, [PERSON] would resubmit the qubit to [PERSON] who would measure in the same basis he prepared in. Defining the eigenstates of \(\sigma_{z}\) and \(\sigma_{x}\) as \(|z\pm\rangle\) and \(|x\pm\rangle\) respectively, bit values \(0\) may be assigned to the states \(|z+\rangle\) or \(|x+\rangle\) and \(1\) to the states \(|z-\rangle\) or \(|x-\rangle\). [PERSON] then adds (modulo 2) the bit value corresponding to  
Figure 1: Schematic diagram for SDC. Arrows represent the distribution of qubits in the channels between [PERSON] and [PERSON].