Efficient Calculation of the Maximal R\'{e}nyi Divergence for a Matrix Product State via Generalized Eigenvalue Density Matrix Renormalization Group

TL;DR

This paper introduces an efficient method to compute the maximal Rényi divergence for matrix product states using a generalized eigenvalue density matrix renormalization group algorithm, revealing different correlation trends than von Neumann mutual information.

Contribution

The authors develop a generalized eigenvalue DMRG algorithm to compute the maximal Rényi divergence for 1D quantum states represented as matrix product states, enabling efficient analysis.

Findings

Maximal Rényi divergence can be computed via a generalized eigenvalue problem.

The method is benchmarked on the XXZ chain, demonstrating efficiency.

Rényi divergence shows different correlation trends than von Neumann mutual information.

Abstract

The study of quantum and classical correlations between subsystems is fundamental to understanding many-body physics. In quantum information theory, the quantum mutual information, $I(A;B)$, is a measure of correlation between the subsystems $A,B$ in a quantum state, and is defined by the means of the von Neumann entropy: $I\left(A;B\right)=S\left(\rho_{A}\right)+S\left(\rho_{B}\right)-S\left(\rho_{AB}\right)$. However, such a computation requires an exponential amount of resources. This is a defining feature of quantum systems, the infamous ``curse of dimensionality'' . Other measures, which are based on R\'{e}nyi divergences instead of von Neumann entropy, were suggested as alternatives in a recent paper showing them to possess important theoretical features, and making them leading candidates as mutual information measures. In this work, we concentrate on the maximal R\'{e}nyi divergence. This measure can be shown to be the solution of a generalized eigenvalue problem. To calculate it efficiently for a 1D state represented as a matrix product state, we develop a generalized eigenvalue version of the density matrix renormalization group algorithm. We benchmark our method for the paradigmatic XXZ chain, and show that the maximal R\'enyi divergence may exhibit different trends than the von Neumann mutual information.