Direct Calculation of Mutual Information of Distant Regions

Abstract

We consider the (Renyi) mutual information, $I^{(n)}(A,B)=S_A^{(n)}+S_B^{(n)}-S_{A\cup B}^{(n)}$, 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 $I^{(n)}(A,B)\sim C_{AB}^{(n)}{⟨0|\varphi (r)\varphi (0)|0⟩}^2$ . 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.

17 pages; v2, added references, minor corrections in eq.(3.20). arXiv admin note: text overlap with arXiv:1408.0637