## 3 Efficiency and Secure Key Rates  
In what follows, we shall assume implementations of the two-way protocols using depolarizing channels. The depolarizing channel, \(\mathcal{D}\) is described by the parameter \(q\), such that \(0\leq q\leq 1\), which affects a quantum state, \(\rho\) independent of the basis as such  
\[\rho\mapsto\mathcal{D}(\rho)=(1-q)\rho+qI/2. \tag{12}\]  
For two-way protocols, the use of depolarizing channels can be categorized as either independent or correlated channels. Such a correlation can be understood in terms of the errors estimated in the forward and backward paths in CM against errors in EM. Given bit wise errors \(e_{1}\) and \(e_{2}\) in the forward and backward paths respectively, we say the channels are independent provided the errors in EM is given by \(e_{m}=e_{1}(1-e_{2})+e_{2}(1-e_{1})\). Otherwise they are correlated [24]. While the case for independent channels are unique by definition, the cases for correlated channels can be infinitely many. However, we shall only consider, as in [5] correlated channels where \(e_{1}=e_{2}=e_{m}\).  
To ascertain a finite key rate for the two-way protocols given such implementation, we need to determine the distribution of finite number of bits between the EM and the CM. Choices of the value of \(c\) for both protocols would determine this and ultimately how much of a key rate one can have. Hence, in order to determine the optimal value for \(c\), assuming one has a value of \(k\) for bits in CM and \(n\) in EM (more precisely the cases where [PERSON]'s encoding measurements coincide with [PERSON]'s decoding measurements), the total number qubits, \(M(n,k)\) sent [25] before \(n\) bits are derived from EM and \(k\) for CM estimation can be made is given by probability value \(c=(\sqrt{1+k/n})^{-1}\). This is immediately derived from modelling the protocol based on [9].  
Subsequently, the already mentioned _gedankenexperiment_ will provide for an uncertainty relation of Equation (11) and the smooth-max entropy term for an \(n\)-bit string, \(Z\) given \(C\), \(H^{\epsilon}_{max}(Z|C)\) would be upper bounded by \(nh_{2}(Q+\mu(n,k))\) where  
\[\mu(n,k)=\sqrt{\frac{n+k}{nk}\frac{k+1}{k}\ln\frac{2}{\epsilon_{S}}}, \tag{13}\]  
with the strings \(Z\) and \(C\) derived from CM differing at \(Q\) bits. The parameter \(\epsilon_{S}>0\) is a security parameter as defined in [9].  
Strings shared between [PERSON] and [PERSON] in the EM, which one party's may be characterized as having errors \(e\) relative to the other need be subjected to an error correction procedure. This can conventionally be understood as the amount of pre shared secret bits invested in the communications for error correction purposes [26].  
The Leftover Hashing Lemma [27] then provides for the length of the secret key as  
\[L=\left\lfloor H^{\epsilon}_{min}(X|E)+2\log 2\epsilon\right\rfloor, \tag{14}\]  
which is \(2\epsilon\) close to a bit string which is uniform and independent of [PERSON]'s knowledge.  
The efficiency of the protocol, \(\mathcal{E}\) can be defined based on the amount of resources in terms of number of qubits required,  
\[\mathcal{E}=L/M(n,k). \tag{15}\]It should be noted that for SDC, a factor of 2 should be further multiplied to the denominator reflecting the use of entangled pairs. As in [9], we do not include any classical bits necessary in the protocol unlike [28].