An Investigation on Lie Algebraic Aspect of Quantum Fisher Information Applied to Unitary and Non-Unitary Evolution
Tatiana Iakovleva
In this thesis, I focus on the quantum Fisher information (QFI) - a central quantity in parameter estimation theory for quantum systems, which provides a lower bound on the precision one can achieve. In the first part of the thesis, I explore a Lie algebraic approach to evaluate the quantum Fisher information for unitary dynamics. I show how the Lie algebra associated with the unitary evolution, together with the initial state, allows one to compute various quantities of interest, circumventing the computation of the time-evolved state. In particular, I apply this shortcut approach to calculate the QFI and use it for the QFI evaluation in various prototype models, both finiteand infinite-dimensional. In the second part of the thesis, I shift focus to non-unitary dynamics, such as those described by a quantum master equation. In this setting, I theoretically study the description of open system dynamics in the Liouville space of vectorized density matrices and investigate the so-called dissipative quantum Fisher information (DQFI) within that framework. Specifically, I demonstrate how the metric defined in Liouville space differs from its counterpart in Hilbert space. I establish a map between the Hilbert space QFI and the Liouville dissipative quantum Fisher information for qubit and qudit systems, expressed via the quantum state’s purity. Furthermore, I discuss the physical meaning of the DQFI by studying its behavior under completely positive trace-preserving (CPTP) maps. The present results demonstrate that the Lie algebraic approach provides a computationally efficient framework for the QFI evaluation. This approach allows to avoid the explicit solution of quantum dynamics, and remains well-defined in cases when conventional approaches like WeiNorman expansion exhibit singularities.
•Feb 28, 2026•0 citations