A simple proof of the Ahlswede - Csiszár one-bit theorem (Corresp.)

It is proved that if
(X,Y)
are two finite alphabet correlated sources with
p(x,y)>0
for all
(x,y) \in ({\cal X} \times {\cal Y})
, and if a function
F(X,Y)
is
\alpha
-sensitive, then the rate
R
of transmission from
X
to
Y
necessary to compute
F(X,Y)
reliably must be greater than
H(X|Y)
. The same result holds if the function is highly sensitive and for every
x_{1} \neq x_{2} \in {\cal X}
, then the number of elements
y \in {\cal Y}
with
p(x_{l},y) \cdot p(x_{2}, y)>0
is different from one.