# How to Quantize $n$ Outputs of a Binary Symmetric Channel to $n-1$ Bits?

## TL;DR

This paper investigates the maximum mutual information achievable by an (n-1)-bit quantizer of a binary symmetric channel output, establishing that the optimal quantizer simply outputs the first n-1 bits of the input.

## Contribution

It proves an upper bound on mutual information for (n-1)-bit quantizers, showing the optimal quantizer is a simple truncation of the input vector.

## Key findings

- Maximum mutual information is bounded by (n-1) times (1 - h(α)).
- The optimal quantizer is the truncation of the first n-1 bits.
- The result extends understanding of information preservation in binary symmetric channels.

## Abstract

Suppose that $Y^n$ is obtained by observing a uniform Bernoulli random vector $X^n$ through a binary symmetric channel with crossover probability $\alpha$. The "most informative Boolean function" conjecture postulates that the maximal mutual information between $Y^n$ and any Boolean function $\mathrm{b}(X^n)$ is attained by a dictator function. In this paper, we consider the "complementary" case in which the Boolean function is replaced by $f:\left\{0,1\right\}^n\to\left\{0,1\right\}^{n-1}$, namely, an $n-1$ bit quantizer, and show that $I(f(X^n);Y^n)\leq (n-1)\cdot\left(1-h(\alpha)\right)$ for any such $f$. Thus, in this case, the optimal function is of the form $f(x^n)=(x_1,\ldots,x_{n-1})$.