The maximum mutual information between the output of a binary symmetric channel and a Boolean function of its input

Abstract

We prove the Courtade-Kumar conjecture, which states that the mutual information between any Boolean function of an $n$-dimensional vector of independent and identically distributed inputs to a memoryless binary symmetric channel and the corresponding vector of outputs is upper-bounded by $1-H(p)$, where $H(p)$ represents the binary entropy function. That is, let $\mathbf{X}=[X_1...X_n]$ be a vector of independent and identically distributed Bernoulli($1/2$) random variables, which are the input to a memoryless binary symmetric channel, with the error probability equal to $0\le p\le 1/2$, and $\mathbf{Y}=[Y_1...Y_n]$ the corresponding output. Let $f:\{0,1{\}}^n\to \{0,1\}$ be an $n$-dimensional Boolean function. Then, $\mathrm{MI}(f(\mathbf{X}),\mathbf{Y})\le 1-H(p)$. We provide the proof for the most general case of the conjecture, that is for any $n$-dimensional Boolean function $f$ and for any value of the error probability of the binary symmetric channel, $0\le p\le 1/2$. Our proof employs only basic concepts from information theory, probability theory and transformations of random variables and vectors.

This paper has been withdrawn due to a critical error in the way the equation of the mutual information that involved conditional entropies was applied

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• On the entropy of a noisy function
• The "Most informative boolean function" conjecture holds for high noise