I've make progress in obtaining the values for the mutual information using the following:
$I(\rho_A:\rho_B) = S(\rho_A) +S(\rho_B) - S(\rho_{AB}) = 1 + 1 - 0 = 2.$
I would like to compute the expectation but I'm facing a problem in the case of $\langle\psi |\mathcal{O}_A|\psi \rangle$ since...