Renyi information from entropic effects in one higher dimension

TL;DR

This paper introduces a novel approach to compute Rènyi entropies and information in free field theories by mapping them to thermodynamic free energy changes in higher dimensions, enabling new calculations and insights.

Contribution

It establishes a mapping between Rènyi entropy and thermodynamic free energy in higher dimensions, facilitating the use of Casimir effect tools for entropy calculations.

Findings

Computed Rènyi information between two disks at any separation.

Represented Rènyi information as a sum over closed-loop polymers.

Connected entropic effects to Rènyi information and discussed extensions beyond free theories.

Abstract

Computing entanglement entropy and its cousins is often challenging even in the simplest continuum and lattice models, partly because such entropies depend nontrivially on all geometric characteristics of the entangling region. Quantum information measures between two or more regions are even more complicated, but contain more, and universal, information. In this paper, we focus on R\'{e}nyi entropy and information of the order $n=2$. For a free field theory, we show that these quantities are mapped to the change of the thermodynamic free energy by introducing boundaries subject to Dirichlet and Neumann boundary conditions in one higher dimension. This mapping allows us to exploit the powerful tools available in the context of thermal Casimir effect, specifically a multipole expansion suited for computing the R\'{e}nyi information between arbitrarily-shaped regions. In particular, we compute the R\'{e}nyi information between two disk-shaped regions at an arbitrary separation distance. We provide an alternative representation of the R\'{e}nyi information as a sum over closed-loop polymers, which establishes a connection to purely entropic effects, and proves useful in deriving information inequalities. Finally, we discuss extensions of our results beyond free field theories.