In this work, we study the universal behaviors in the mutual information of two disjoint spheres in a conformal field theory (CFT). By using the operator product expansion of the spherical twist operator in terms of the conformal family, we show that the large distance expansion of the mutual information can be cast in terms of the conformal blocks. We develop the 1/n prescription to compute the coefficients before the conformal blocks. For a single conformal family, the leading nonvanishing contribution to the mutual information comes from the bilinear operators. We show that the coefficients of these operators take universal forms and such universal behavior persists in the bilinear operators with derivatives as well. Consequently the first few leading order contributions to the mutual information in CFT take universal forms. To illustrate our framework, we discuss the free scalars and free fermions in various dimensions. For the free scalars, we compute the mutual information to the next-to-leading order and find good agreement with the improved numerical lattice result. For the free fermion, we compute the leading order result, which is of universal form, and find the good match with the numerical study. Our formalism could be applied to any CFT potentially.
We consider the mutual Rényi information of disjoint compact spatial regions A and B in the ground state of a d + 1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R A , B . We show that in general , where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where α = d − 1, we show that is proportional to the capacitance of a thin conducting slab in the shape of A in d + 1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S d − 1 or an ellipsoid. For spherical regions in d = 2 and 3 we obtain explicit results for C ( n ) for all n and hence for the leading term in the mutual information by taking n → 1. We also compute a universal logarithmic correction to the area law for the Rényi entropies of a single spherical region for a scalar field theory with a small mass.
A bstract The operator product expansion (OPE) of twist operators in the replica trick framework enables a long-distance expansion of the mutual information (MI) in conformal field theories (CFTs). In this expansion, the terms are labeled by primary operators, as contributions from descendant operators can be resummed into conformal blocks. However, for the MI, the expansion involves primaries from the multi-replica theory, which includes far more operators than those in the original theory. In this work, we develop a method to resum this series, yielding an expansion in terms of the primaries of the original theory, specifically restricted to the two-copy sector. This is achieved by expressing the twist operators in a non-local manner across different replicas and using a modular flow representation to obtain the n → 1 limit of the Rényi index. We explicitly compute the resulting “enhanced conformal blocks,” which, surprisingly, provide excellent approximations to the MI of generalized free fields across the full range of cross ratios. Remarkably, this approximation appears to be exact in the limit of large spacetime dimensions.
We obtain an exact expression for the Rényi mutual information (RMI) between two spherical regions for a free, massless Dirac field in
d spacetime dimensions and for a massless Rarita-Schwinger field in
d=4. We identify each angular momentum mode in the spherical dimensional reduction of the parent theory with a massive Dirac fermion in two dimensional anti-de Sitter space (AdS). The RMI for each mode can be computed analytically for any Rényi index
n in terms of solutions of a Painlevé differential equation. We explore the long and short distance asymptotics of the result, where the sum over modes can be computed explicitly. As a result we are able to find subleading contributions to the long distance expansion in
d dimensions for arbitrary
n, as well as to extract the area and logarithmic coefficients for the case
n=1 in the short distance regime.
The mutual information of measurement of a local operator between two spatial regions characterizes the reduction in the uncertainty of measurement outcomes in one region given the outcome in the other. We study the Rényi-Shannon mutual information of measurement in conformal field theories (CFTs). Explicit calculations for scalar primaries and descendants in large
N CFTs are performed in the two extreme limits of the intervals being adjacent or largely separated. The results depend on only the cross ratio and the scaling dimension
Δ of the operator being measured, with striking similarities to the mutual information of entanglement. We find a surprising quadratic dependence as
(Δ−1/2)2 for adjacent intervals. This dependence superficially seems to violate the information processing inequality, which we discuss and resolve. We also offer a holographic interpretation of the mutual information of measurement.
We present several new results for the
N-partite information,
IN, of spatial regions in the ground state of
d-dimensional conformal field theories. First, we show that
IN can be written in terms of a single
N-point function of twist operators. Using this, we argue that in the limit in which all mutual separations are much greater than the regions sizes, the
N-partite information scales as
IN∼r−2NΔ, where
r is the typical distance between pairs of regions and
Δ is the lowest primary scaling dimension. In the case of spherical entangling surfaces, we obtain a completely explicit formula for the
I4 in terms of 2-, 3- and 4-point functions of the lowest-dimensional primary. Then, we consider a three-dimensional scalar field in the lattice. We verify the predicted long-distance scaling and provide strong evidence that
IN is always positive for general regions and arbitrary
N for that theory. For the
I4, we find excellent numerical agreement between our general formula and the lattice result for disk regions. We also perform lattice calculations of the mutual information for more general regions and general separations both for a free scalar and a free fermion, and conjecture that, normalized by the corresponding disk entanglement entropy coefficients, the scalar result is always greater than the fermion one. Finally, we verify explicitly the equality between the
N-partite information of bulk and boundary fields in holographic theories for spherical entangling surfaces in general dimensions.