How to quantize n outputs of a binary symmetric channel to n − 1 bits?

Suppose that Y
ⁿ
is obtained by observing a uniform Bernoulli random vector X
ⁿ
through a binary symmetric channel with crossover probability α. The “most informative Boolean function” conjecture postulates that the maximal mutual information between Y
ⁿ
and any Boolean function b(X
ⁿ
) is attained by a dictator function. In this paper, we consider the “complementary” case in which the Boolean function is replaced by f : {0, 1}
ⁿ
→ {0, 1}
^n-1
, namely, an n - 1 bit quantizer, and show that I(f(X
ⁿ
); Y
ⁿ
) ≤ (n - 1)·(1 - h(α)) for any such f. Thus, in this case, the optimal function is of the form f (x
ⁿ
) = (x
₁
,..., x
_n-1
).