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 $w\sim \mathcal{CN}(0,1)$.


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.

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