Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems

TL;DR

This paper introduces a numerical algorithm to accurately compute the equilibrium-reduced density matrix and effective Hamiltonian for strongly coupled open quantum systems, enabling better analysis of quantum phase transitions and entanglement.

Contribution

It generalizes typicality algorithms with trace estimators and Krylov methods to strongly coupled systems, providing a new computational tool with validated accuracy.

Findings

Algorithm accurately approximates reduced density matrices.

Numerical experiments confirm effectiveness for quantum phase transition studies.

Potential applications include entanglement entropy analysis.

Abstract

We describe a numerical algorithm for approximating the equilibrium-reduced density matrix and the effective (mean force) Hamiltonian for a set of system spins coupled strongly to a set of bath spins when the total system (system+bath) is held in canonical thermal equilibrium by weak coupling with a "super-bath". Our approach is a generalization of now standard typicality algorithms for computing the quantum expectation value of observables of bare quantum systems via trace estimators and Krylov subspace methods. In particular, our algorithm makes use of the fact that the reduced system density, when the bath is measured in a given random state, tends to concentrate about the corresponding thermodynamic averaged reduced system density. Theoretical error analysis and numerical experiments are given to validate the accuracy of our algorithm. Further numerical experiments demonstrate the potential of our approach for applications including the study of quantum phase transitions and entanglement entropy for long-range interaction systems.