Correlation functions of von Neumann entropy

In this note, we study two-point correlation functions of modular Hamiltonians. We show that in general quantum systems, these correlators obey properties similar to those of von Neumann entropy and capacity of entanglement, both of which are special cases of these correlators. Then we specialize to two spacelike-separated spherical subregions in conformal field theories. We present direct computations of the vacuum two-point function that confirm its equivalence to the stress-tensor conformal block. We explore the two-point function in various kinematic regimes, including imaginary time separation between subsystems. The material presented in this note may be useful for further studying modular Hamiltonian correlators in generic quantum systems and in conformal field theories, including those with holographic duals.