U=2*U1 ;

X1=2*(U1+U2+ …+U20) ;

X=(X1-mean(X1))/std(X1);

Y=0.1*X+W1;

V=0.9*Y+0.1*W2;

where Ui(0,1) is uniform, (i=1,2, … ,20), W1~N(0,0.99), and W2~N(0,1).

I got the following numerical results for this setting using 10^7 samples.

H(U,V)= 81.4570,

H(Y,U)= 11.8206,

H(X,V)= 69.5232,

H(X,U)= 11.9757,

H(Y,V)= 7.4803,

[I have received several emails inquiring about the status of the conjecture, so I have added a comment at the top of the post to advertise the status. I will update it if the status changes.]

]]>I( b(X^n) ; Y^n ), where (X^n,Y^n) are n i.i.d. pairs of correlated Bernoulli(1/2) random variables. In this case, the data processing inequality looking at the channel X^n –> Y^n (which is what you are driving at with your channel capacity statement) simply gives:

I( b(X^n) ; Y^n ) <= n*( 1-H(\alpha) ),

which is off by a factor of n. Clearly this is no good, since we can trivially upper bound:

I( b(X^n) ; Y^n ) <= 1.

Appealing to a so-called 'strong data processing' inequality improves things, but still only gives:

I( b(X^n) ; Y^n ) <= (1-2\alpha)^2,

which is still weaker than the conjecture.

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