## 2 Two-Way Protocols
### Measurements and Entropic Uncertainty Relations  
Entropic uncertainty relations are used in some security proofs of particular QKD protocols given its power to describe bounds of uncertainty parties may share of a certain quantum system, say \(B\). In its simplest description, it is simply [PERSON]'s uncertainty principle in an entropic form first proven by [15]. It was later generalized [16; 17; 18] to include correlation of the system to be measured with disjoint (possibly quantum) systems \(A\) and \(E\) given by  
\[H(X|E)+H(Z|A)\geq-\log_{2}\mathcal{C}, \tag{2}\]  
where \(X\) and \(Z\) are results from POVMs \(\mathbb{X}\) and \(\mathbb{Z}\) respectively on \(B\). Writing the POVM elements of \(\mathbb{X}\) and \(\mathbb{Z}\) as \(\{E_{x}^{\mathbb{X}}\}\) and \(\{E_{z}^{\mathbb{Z}}\}\) respectively, the term \(\mathcal{C}\) is given by  
\[\mathcal{C}=\max_{x,z}\left\|\sqrt{E_{x}^{\mathbb{X}}}\sqrt{E_{z}^{\mathbb{Z}} }\right\|_{\infty}^{2}. \tag{3}\]  
Equation (2) together with the Devatak-Winter security bound [19] is employed in [5] to provide a security proof for the purified SDC and LM05 protocols. In order to make use of such an entropic relation, [5] necessarily describes the term \(\mathcal{C}\) as the (effective) overlap between [PERSON]'s measurements in a RR picture. Given the assumptions in [5], [PERSON]'s measurements in the purified SDC protocol are either the Bell measurements or a measurement that can be described as \(\sigma_{z}\otimes\sigma_{x}\) where the overlap between the POVM elements as defined in Equation (3) is \(1/4\). This maximal overlap is achieved when considering the overlap between the POVM elements in EM and that of in CM. If we let the measurements by [PERSON] and [PERSON] to result in the strings \(S_{A}^{\text{EM}}\) and \(S_{B}^{\text{EM}}\) respectively from EM and \(S_{A}^{\text{CM}}\) and \(S_{B}^{\text{CM}}\) respectively from CM, and [PERSON]'s system as \(E\) we can bound [PERSON]'s information based on Equation (2) as  
\[H(S_{B}^{\text{EM}}|E)+H(S_{B}^{\text{CM}}|S_{A}^{\text{CM}})\geq 2. \tag{4}\]  
The key rate of the SDC, \(R_{SDC}\), is given by the Devetak-Winter rate [19]  
\[R_{SDC}\geq H(S_{B}^{\text{EM}}|E)-H(S_{B}^{\text{EM}}|S_{A}^{\text{EM}}). \tag{5}\]  
With \(H(S_{B}^{\text{CM}}|S_{A}^{\text{CM}})\) upper bounded by \(h_{4}(\overline{q}_{\text{CM}})\), \(H(S_{B}^{\text{EM}}|S_{A}^{\text{EM}})\) by \(h_{4}(\overline{q}_{\text{EM}})\) where \(h_{4}\) is the 4-ary Shannon entropy and \(\overline{q}_{\text{CM}}\) and \(\overline{q}_{\text{EM}}\) are errors in [PERSON]'s and [PERSON]'s strings in CM and EM respectively, applying the bound of Equation (4) we arrive at  
\[R_{SDC}\geq 2-h_{4}(\overline{q}_{\text{CM}})-h_{4}(\overline{q}_{\text{EM} }). \tag{6}\]  
It is instructive to note that the errors \(\overline{q}_{\text{CM}}\) and \(\overline{q}_{\text{EM}}\) affects a two-bit message transmission resulting from a use of two noisy channels and thus comes as a triple; namely error exclusively in either one channel, say \(q_{1}^{a}\) and \(q_{2}^{a}\) as well as errors in both channels, \(q_{3}^{a}\) for \(a=\text{CM},\text{EM}\). Hence,  
\[h_{4}(\overline{q}_{a})=-\sum_{i}^{3}q_{i}^{a}\log_{2}q_{i}^{a}-(1-\sum_{i}^{ 3}q_{i}^{a})\log_{2}\Bigg{(}1-\sum_{i}^{3}q_{i}^{a}\Bigg{)}. \tag{7}\]  
It is certainly possible to consider the entropic bounds dictated by the overlap of [PERSON]'s measurements instead of [PERSON]'s as shown above. The key rate would remain the same nonetheless. A similar line of argument can be applied to the LM05 protocol; though the measurements made for decoding purposes by [PERSON] would be \(\mathcal{Z}=\sigma_{z}\otimes\sigma_{z}\) or \(\mathcal{X}=\sigma_{x}\otimes\sigma_{x}\) followed by an XOR of the measurements' results to reveal the encoding done by [PERSON] in the EM.