An Alternative Method for Extracting the von Neumann Entropy from Renyi Entropies

TL;DR

This paper introduces a novel method to compute von Neumann entropy from Renyi entropies using a generating function approach, avoiding direct analytic continuation, and demonstrates its effectiveness through analytical and numerical examples in quantum systems.

Contribution

The paper presents a new generating function technique to extract von Neumann entropy from Renyi entropies without direct analytic continuation, applicable to quantum entanglement calculations.

Findings

Successfully derived the von Neumann entropy for two intervals in CFT using the new method.

Validated the approach with numerical calculations for various cross ratios and finite temperature cases.

Reproduced known results in conformal field theory, confirming the method's accuracy.

Abstract

An alternative method is presented for extracting the von Neumann entropy $-\operatorname{Tr} (\rho \ln \rho)$ from $\operatorname{Tr} (\rho^n)$ for integer $n$ in a quantum system with density matrix $\rho$. Instead of relying on direct analytic continuation in $n$, the method uses a generating function $-\operatorname{Tr} \{ \rho \ln [(1-z \rho) / (1-z)] \}$ of an auxiliary complex variable $z$. The generating function has a Taylor series that is absolutely convergent within $|z|<1$, and may be analytically continued in $z$ to $z = -\infty$ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in $n$. Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.