# Direct Calculation of Mutual Information of Distant Regions

## TL;DR

This paper derives a direct formula for calculating the mutual information between distant regions in a free scalar field, simplifying numerical computations by avoiding separate entropy calculations.

## Contribution

It provides an explicit expression for the mutual information coefficient, enabling direct and efficient numerical evaluation for arbitrary regions.

## Key findings

- Derived a direct expression for $C^{(n)}_{AB}$ applicable to any regions.
- Enabled direct computation of mutual information without separate entropy calculations.
- Facilitated numerical analysis of mutual information in quantum field theory.

## Abstract

We consider the (Renyi) mutual information, $I^{(n)}(A,B) = S^{(n)}_A+S^{(n)}_{B} - S^{(n)}_{A \cup B}$, of distant compact spatial regions A and B in the vacuum state of a free scalar field. The distance r between A and B is much greater than their sizes $R_{A,B}$. It is known that $I^{(n)}(A,B) \sim C^{(n)}_{AB} \left<0| \phi(r)\phi(0) |0\right>^2$ . We obtain the direct expression of $C^{(n)}_{AB}$ for arbitrary regions A and B. We perform the analytical continuation of $n$ and obtain the mutual information. The direct expression is useful for the numerical computation. By using the direct expression, we can compute directly $I(A,B)$ without computing $S_A, S_B$ and $S_{A \cup B}$ respectively, so it reduces significantly the amount of computation.