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Common information revisited

One of the main notions of information theory is the notion of mutual information in two messages (two random variables in Shannon information theory or two binary strings in algorithmic information theory). The mutual information in $x$ and $y$ measures how much the transmission of $x$ can be simplified if both the sender and the recipient know $y$ in advance. Gács and Körner gave an example where mutual information cannot be presented as common information (a third message easily extractable from both $x$ and $y$). Then this question was studied in the framework of algorithmic information theory by An. Muchnik and A. Romashchenko who found many other examples of this type. K. Makarychev and Yu. Makarychev found a new proof of Gács--Körner results by means of conditionally independent random variables. The question about the difference between mutual and common information can be studied quantitatively: for a given $x$ and $y$ we look for three messages $a$, $b$, $c$ such that $a$ and $c$ are enough to reconstruct $x$, while $b$ and $c$ are enough to reconstruct $y$. In this paper: We state and prove (using hypercontractivity of product spaces) a quantitative version of Gács--Körn