Non Coherent BPSK with no Channel State Information at the receiver

Question

Where or what is the proof for the failure of BPSK in fading channel when we have no knowledge of the fading coefficient $h$ at the receiver in the equation $y=hx+w$, where $x$ is BPSK symbol equally likely to be $A$ or $-A$, $h\sim \mathcal{CN}(0,1)$, and $w\sim \mathcal{CN}(0,1)$.

Answer 1

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 "without knowing $h$, the output phase is fully random" is sufficient as proof.