The Courtade-Kumar conjecture asserts that a single coordinate is a most informative Boolean function under a binary
symmetric channel (Kumar and Courtade 2013). For 
and a Boolean function 
,
let 
have the uniform distribution on 
and put 
with independent signs 
, also independent
of 
,
and 
for 
.
The assertion is
 |
where 
is mutual information, 
is binary Shannon
entropy, 
,
and 
is measured in bits. Equality holds for 
.
