Direct Calculation of Mutual Information of Distant Regions

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 $I^{(n)}(A,B)\sim C_{AB}^{(n)}{<0\mathrm{\mid }\varphi (r)\varphi (0)\mathrm{\mid }0>}^2$ . We obtain the direct expression of $C_{AB}^{(n)}$ 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.