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  • Open Access

Quantum Fisher information from tensor-network integration of the Lyapunov equation

Gabriela Wójtowicz1,*, Susana F. Huelga1, Marek M. Rams2,3, and Martin B. Plenio1

  • 1Institut für Theoretische Physik und IQST, Albert-Einstein-Allee 11, Universität Ulm, D-89081 Ulm, Germany
  • 2Institute of Theoretical Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
  • 3Mark Kac Complex Systems Research Center, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland

  • *Contact author: gabriela.wojtowicz@uni-ulm.de
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Phys. Rev. A 112, 052454 – Published 26 November, 2025

DOI: https://doi.org/10.1103/xjln-7ddy

Abstract

The quantum Fisher information (QFI) is a geometric measure of state deformation calculated along the trajectory parametrizing 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 nonlinear 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 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.

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Phys. Rev. A 112, 052454– Published 26 November, 2025
  • Received 16 June 2025
  • Accepted 29 October 2025
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