Uhlmann fidelities from tensor networks

TL;DR

This paper presents an efficient tensor network-based method for computing subsystem Uhlmann fidelities to analyze similarities between quantum many-body states locally, with applications in quenches, criticality, and simulation convergence.

Contribution

It introduces a practical approach to compute subsystem fidelities in tensor network states, enhancing analysis of local quantum state similarities.

Findings

Subsystem fidelities can be efficiently computed using tensor networks.

The method is applicable to Matrix Product States and Tree Tensor Networks.

Demonstrated usefulness in studying local quenches, criticality, and convergence.

Abstract

Given two states $|\psi\rangle$ and $|\phi\rangle$ of a quantum many-body system, one may use the overlap or fidelity $|\langle\psi|\phi\rangle|$ to quantify how similar they are. To further resolve the similarity of $|\psi\rangle$ and $|\phi\rangle$ in space, one can consider their reduced density matrices $\rho$ and $\sigma$ on various regions of the system, and compute the Uhlmann fidelity $F(\rho, \sigma) = \operatorname{Tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}$. In this paper, we show how computing such subsystem fidelities can be done efficiently in many cases when the two states are represented as tensor networks. Formulated using Uhlmann's theorem, such subsystem fidelities appear as natural quantities to extract for certain subsystems for Matrix Product States and Tree Tensor Networks, and evaluating them is algorithmically simple and computationally affordable. We demonstrate the usefulness of evaluating subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.