Tisch
Spare potatoes
46}$$where $p_{1}^{(s)}$ denotes the probability that the source is in state 1 and $p_{2}^{(s)}$ denotes the probability that the source is in state 2. Thus, the correlation term $R_{AB}^{(2)}$ is the same for both quantum protocols.
3.2. Secret keys rate {#se0050}
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The secret key generation rate of the protocol based on a BB84 qubit source can be expressed as$$r_{\text{Alice}}^{(2)} = \chi H(e_{\alpha}\text{,}e_{\beta}\text{,}R_{AB}^{(2)}\text{,}p_{2}^{(s)})\text{.}$$
Using the expression for $R_{AB}^{(2)}$ from [Eq. (44)](#eq0090){ref-type="disp-formula"} and the condition of $|\phi_{A,B}^{(2)}| = e_{\alpha}\text{,}e_{\beta}$ that $p_{2}^{(s)}$ must be the solution of $g_{2}(\eta_{s}) = g_{2}(1)$ to maximize $R_{AB}^{(2)}$, the maximal value of $R_{AB}^{(2)}$ is given by $R_{AB}^{(2)\max} = H(e_{\alpha}\text{,}e_{\beta}\text{,}H(g_{2}(\eta_{s})\text{,}p_{2}^{(s)}))$. Thus, the maximum secret key generation rate for Alice can be written as$$r_{\text{Alice}}^{(2)\max} = \chi H(e_{\alpha}\text{,}e_{\beta}\text{,}H(g_{2}(\eta_{s})\text{,}p_{2}^{(s)}))\text{.}$$
3.2. Secret keys rate {#se0050}
---------------------
The secret key generation rate of the protocol based on a BB84 qubit source can be expressed as$$r_{\text{Alice}}^{(2)} = \chi H(e_{\alpha}\text{,}e_{\beta}\text{,}R_{AB}^{(2)}\text{,}p_{2}^{(s)})\text{.}$$
Using the expression for $R_{AB}^{(2)}$ from [Eq. (44)](#eq0090){ref-type="disp-formula"} and the condition of $|\phi_{A,B}^{(2)}| = e_{\alpha}\text{,}e_{\beta}$ that $p_{2}^{(s)}$ must be the solution of $g_{2}(\eta_{s}) = g_{2}(1)$ to maximize $R_{AB}^{(2)}$, the maximal value of $R_{AB}^{(2)}$ is given by $R_{AB}^{(2)\max} = H(e_{\alpha}\text{,}e_{\beta}\text{,}H(g_{2}(\eta_{s})\text{,}p_{2}^{(s)}))$. Thus, the maximum secret key generation rate for Alice can be written as$$r_{\text{Alice}}^{(2)\max} = \chi H(e_{\alpha}\text{,}e_{\beta}\text{,}H(g_{2}(\eta_{s})\text{,}p_{2}^{(s)}))\text{.}$$