## 2 Two-Way Protocols
### Smooth Entropies and Finite Keys  
The smooth entropy is defined based on the conditional entropy. More rigorously, following the definition given in [20] for a bipartite state \(\rho_{BE}\) on \(B\) and \(E\), the entropy of \(B\) given \(E\) is defined as  
\[H_{min}(B|E)=\max_{\sigma_{E}}\sup\left\{\lambda\in\mathbb{R}:2^{-\lambda}I_{B} \otimes\sigma_{E}\geq\rho_{BE}\right\} \tag{9}\]  
where the maximum is taken overall states \(\sigma_{E}\) in \(E\) and \(I_{B}\) is the identity on \(B\). The \(\epsilon\)-smooth min-entropy for \(\epsilon\geq 0\) is then defined as  
\[H_{min}^{\epsilon}(B|E)=\max_{\rho}H_{min}(B|E). \tag{10}\]  
with maximization over all bipartite states \(\rho\) on \(B\) and \(E\) with a purified distance [21] to \(\rho_{BE}\) not exceeding \(\epsilon\). The smooth max-entropy is defined as the dual of the smooth min-entropy with regards to any purification of \(\rho_{BE}\).  
For the tripartite state \(\rho_{ABE}\) and POVMs \(\mathbb{X}\) and \(\mathbb{Z}\) respectively on \(B\) (resulting in bit strings \(X\) and \(Z\)), from [20], the smooth min-entropy of \(X\) conditioned on \(E\), \(H_{min}^{\epsilon}(X|E)\), gives the number of bits contained in \(X\) that are \(\epsilon\)-close [22] to a uniform distribution and independent of \(E\). The smooth max-entropy of \(Z\) conditioned on \(A\), \(H_{max}^{\epsilon}(Z|A)\), gives the number of bits needed to reconstruct \(Z\) from \(A\) up to a probability of failure \(\epsilon\) and the generalized uncertainty relation involving smooth entropies is given as [20]  
\[H_{min}^{\epsilon}(X|E)+H_{max}^{\epsilon}(Z|A)\geq-\log_{2}\mathcal{C}. \tag{11}\]  
In identifying the measurements and results in both the SDC and LM05 with Equation (11), similar to the above, \(H_{min}^{\epsilon}(X|E)\) will be identified with [PERSON]'s correlation with [PERSON]'s string (in EM) while \(H_{max}^{\epsilon}(Z|A)\) is to be identified with [PERSON]'s and [PERSON]'s in CM. There are two points worth mentioning with regards to the measurements made in CM. The first is in particular reference to LM05's measurements \(\mathcal{Z}_{C}\) and \(\mathcal{X}_{C}\) where a passive operation is noted on the second qubit (received in the backward path). This does not necessarily require [PERSON] to _not_ measure the qubit in the backward path; rather he could just ignore the measurement made and consider only the result of his first measurement [23]. In other words, in the cases [PERSON] note a particular round of the protocol is a CM, [PERSON] would ignore the result of his measurement on the second qubit. If it was the EM, then both the measurement results would be XOR-ed for decoding purposes.  
The second point is that as the \(\sigma_{z}\otimes\sigma_{x}\) in SDC and \(\mathcal{Z}_{C}\) and \(\mathcal{X}_{C}\) in LM05 happens only in CM, one can see the protocol as analogous to the asymmetrical prepare and measure protocol of BB84 where the measurements in EM is seen as measurements in the preferred basis in the asymmetrical BB84. In the case for the latter, in [9], where measurements are made in the \(X\) and \(Z\), a _gedankenexperiment_ was considered where all measurements were done in the \(Z\) basis to establish an uncertainty relation. Following [9], in the case for the LM05, we can use the bits derived in the CM to provide for an estimation of the errors in the application of the uncertainty relation using a similar _gedankenexperiment_ where all rounds are CM; and since [PERSON] can choose to measure for EM, security follows from the notion that the better [PERSON] could estimate [PERSON]'s bits in CM, the worse would [PERSON]'s estimation of [PERSON]'s bits in EM.