Exact Mutual Information Difference: Scalar vs. Maxwell Fields

TL;DR

This work derives the exact difference between Rényi mutual informations for free Maxwell and scalar fields in $d=4$ by reducing the 4D Maxwell problem to a 2D scalar on a half-line with Dirichlet boundary conditions, then relating it to a chiral current on the full line. A key result is the exact identity $2 I_n^{(S)}(\eta) - I_n^{(M)}(\eta) = 2 I_n^{(C)}(\eta)$, enabling closed-form access to the Maxwell–scalar MI difference through known 2D results, together with a precise long-distance/OPE analysis. The authors reveal that the long-distance series converges only for integer $n>1$, while for $n=1$ and non-integer $n$ it is asymptotic, hinting at possible non-perturbative contributions. They provide detailed short- and long-distance expansions, verify consistency with existing results, and discuss implications for higher-dimensional entanglement and potential extensions to other free fields and information-theoretic quantities.

Abstract

We compute, for any Rényi index $n$, the exact difference between the mutual Rényi informations of a pair of free massless scalars and that of a Maxwell field in $d=4$ dimensions. Using the standard dimensional reduction method in polar coordinates, the problem is mapped to that of a single scalar field in $d=2$ with Dirichlet boundary conditions, which in turn can be conveniently related to the algebra of a chiral current on the full line. This latter identification, which maps algebras on an interval to two-interval algebras, yields exact results that clarify the structure of the long-distance OPE perturbative expansion of the mutual information. We find that this series has a finite radius of convergence only for integer $n>1$, while it becomes only asymptotical for $n=1$ and general non-integer values of $n$.

Figures (6)

• Figure 1: Geometric setup of some regions employed on the main text, as well as the identification between their corresponding algebras.
• Figure 2: In circles we show the numerical calculation of the $n$-th RMIs for the scalar on the half-line performed on a lattice for different values of the cross-ratio $\eta$ and for $n=0.5,1,2$ (from top to bottom). In solid lines we show the corresponding analytical results.
• Figure 3: Comparison between the analytic expression for the RMI \ref{['eq:RMI_result']} with $n=2$ and it's long distance expansion for different orders $k$ of the expansion. The same behavior is observed for other integer values of $n\neq1$.
• Figure 4: On the left we show the comparison between the analytic expression for the MI \ref{['eq:MI']} and it's long distance expansion up to the $k$-th order, for different values of $k$. We can see that the convergence is spoiled with increasing $k$. The same behavior is observed for the RMIs \ref{['eq:RMI_result']} with $n\not\in\mathbb{Z}$, as we show on the right panel for $n=2.5$
• Figure 5: Plot of the absolute value of the $k$-th coefficient in the long distance expansion of $I^{(\text{C})}_{n}(\eta)$ for some values of $k$ between 5 and 40 (from bottom to top), as a function of $n$. Note the logarithmic scale for the vertical axes.