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So first, let's define a bit your notations. I guess (correct me if I'm wrong) that you consider Bob honest, and that what you denote by $\Psi_{a_k,b_k}$ is the BB84 qubit in basis $\{0,1\}$ if $b_k = 0$, and in basis $\{+,-\}$ if $b_k = 1$, whose "value" bit is $a_k$, i.e.: $$\Psi_{a_k,b_k} = H^{b_k}X^{a_k}|0\rangle$$ Then, Bob will measure in basis $b'_k$...


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