Quantum Fisher information from tensor network integration of Lyapunov equation

TL;DR

This paper introduces a new tensor network-based numerical method to efficiently compute Quantum Fisher Information for many-body quantum systems, overcoming traditional computational challenges.

Contribution

The authors develop a novel approach combining Lyapunov integrals and tensor networks to calculate QFI without eigendecomposition, enabling analysis of larger quantum systems.

Findings

Method effectively computes QFI for mixed states in many-body systems.

Application demonstrated on thermal states of the transverse-field Ising model.

Method shows advantages in quantum metrology scenarios.

Abstract

The Quantum Fisher Information (QFI) is a geometric measure of state deformation calculated along the trajectory parameterizing an ensemble of quantum states. It serves as a key concept in quantum metrology, where it is linked to the fundamental limit on the precision of the parameter that we estimate. However, the QFI is notoriously difficult to calculate due to its non-linear mathematical form. For mixed states, standard numerical procedures based on eigendecomposition quickly become impractical with increasing system size. To overcome this limitation, we introduce a novel numerical approach based on Lyapunov integrals that combines the concept of symmetric logarithmic derivative and tensor networks. Importantly, this approach requires only the elementary matrix product states algorithm for time-evolution, opening a perspective for broad usage and application to many-body systems. We discuss the advantages and limitations of our methodology through an illustrative example in quantum metrology, where the thermal state of the transverse-field Ising model is used to measure magnetic field amplitude.