Non Coherent BPSK with no Channel State Information at the receiver
In any PSK modulation, the information is only in the phase.
An unknown $h$ randomly rotates your phase. You know nothing about the phase that $x$ had when you observe $y$. Hence, you have zero mutual information.
One could write that down as a very short formal proof based on conditional entropies, but I don't think that'll help you; the more intuitive understanding &quot;without knowing $h$, the output phase is fully random&quot; is sufficient as proof.