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An Investigation on Lie Algebraic Aspect of Quantum Fisher Information Applied to Unitary and Non-Unitary Evolution

Tatiana Iakovleva

Published Feb 28, 2026• 0 citations

Abstract

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.

DOI: 10.15102/0002001112

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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.

Open quantum system dynamics traditionally described by a quantum master equation can be effectively reformulated in the Liouville space of vectorised density matrices. Despite an extensive use of the Liouville space framework to describe quantum states evolution, its application to quantum metrology is not well explored. Several papers have studied a Liouville-space generalisation of the quantum Fisher information, called the dissipative quantum Fisher information (DQFI). We perform a comparative study between the quantum Fisher information (QFI) in Hilbert space and DQFI, deriving an explicit relation between the two quantities for qubit, qudit, and harmonic oscillator systems. The derived relations are nontrivial, and depend on the state's purity. We show that for a single qubit, the QFI can be efficiently recovered from the DQFI via a simple mapping, while in higher dimensions the expression becomes convoluted and does not convert to a compact form. We find examples where the DQFI is neither an upper nor lower bound for the Hilbert space QFI, and suggest that the informational content of the DQFI itself cannot be straightforwardly interpreted. Our results clarify the limitations of the Liouville space based approach and structural differences with the QFI in the context of quantum parameter estimation problem.

The dissipative quantum Fisher information (DQFI) for a dynamic map with a general parameter in an open quantum system is investigated, which can be regarded as an analog of the quantum Fisher information (QFI) in the Liouville space. We first derive a general dissipative generator in the Liouville space, and based on its decomposition form, find the DQFI stems from two parts. One is the dependence of eigenvalues of the Liouvillian supermatrix on the estimated parameter, which shows a linear dependence on time. The other is the variation of the eigenvectors with the estimated parameter. The relationship between this part and time presents rich characteristics, including harmonic oscillation, pure exponential gain and attenuation, as well as exponential gain and attenuation of oscillatory type, which depend specifically on the properties of the Liouville spectrum. This is in contrast to that of the conventional generator, where only oscillatory dependencies are seen. Further, we illustrate the theory through a toy model: a two-level system with spin-flip noise. Especially, by using the DQFI, we demonstrated that the exceptional estimation precision cannot be obtained at the Liouvillian exceptional point.

The dissipative quantum Fisher information (DQFI) for a dynamic map with a general parameter in an open quantum system is investigated, which can be regarded as an analog of the quantum Fisher information in the Liouville space. We first derive a general dissipative generator in the Liouville space, and based on its decomposition form, find the DQFI stems from two parts. One is the dependence of eigenvalues of the Liouvillian supermatrix on the estimated parameter, which shows a linear dependence on time. The other is the variation of the eigenvectors with the estimated parameter. The relationship between this part and time presents rich characteristics, including harmonic oscillation, pure exponential gain and attenuation, as well as exponential gain and attenuation of oscillatory type, which depend specifically on the properties of the Liouville spectrum. This is in contrast to that of the conventional generator, where only oscillatory dependencies are seen. Particularly, we find that the nonunitary parameter encoding process induced by the dissipative generator can be transformed into two unitary encoding processes plus the contribution of the commutator between the two. Further, we illustrate the theory through a toy model: a two-level system with spin-flip noise. Especially, by using the DQFI, we demonstrated that the exceptional estimation precision cannot be obtained for this model at the Liouvillian exceptional point.

Abstract Quantum Fisher information (QFI) is a key concept of quantum metrology, which is widely used in quantum parameter estimation. We introduce a general approach to analytically get the QFI of pure states for unitary parametrization processes governed by arbitrary finite dimensional Lie algebras. We show the universality of this approach by comparing the result of algebra with the previous work and applying it to and algebras. For case, there are some interesting findings. The phase of the initial coherent state has an optimal value for maximum QFI, and the QFI can vanish and revive during the time evolution.

Lie algebraic techniques are powerful and widely used tools for studying dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie algebra with finite dimension, the unitary evolution can be expressed as a finite product of exponentials using the Wei-Norman expansion. The system is then exactly described by a finite set of scalar differential equations, even if the Hilbert space is infinite. However, the differential equations provided by the Wei-Norman expansion are nonlinear and often have singularities that prevent both analytic and numerical evaluation. We derive a new Lie algebra expansion for the quantum Fisher information, which results in linear differential equations. Together with existing Lie algebra techniques this allows many metrology problems to be analysed entirely in the Heisenberg picture. This substantially reduces the calculations involved in many metrology problems, and provides analytical solutions for problems that cannot even be solved numerically using the Wei-Norman expansion. It also allows us to study general features of metrology problems, valid for all quantum states. We provide detailed examples of these methods applied to problems in quantum optics and nonlinear optomechanics. Published by the American Physical Society 2024

The dynamics of two variants of quantum Fisher information under decoherence is investigated from a geometrical point of view. We first derive the explicit formulas of these two quantities for a single qubit in terms of the Bloch vector. Moreover, we obtain analytical results for them under three different decoherence channels, which are expressed as affine transformation matrices. Using the hierarchy equation method, we numerically study the dynamics of both sets of information in a dissipative model and compare the numerical results with the analytical ones obtained by applying the rotating-wave approximation. We further express the two information quantities in terms of the Bloch vector for a qudit by expanding the density matrix and Hermitian operators in a common set of generators of the Lie algebra

\text{su}(d)

. By calculating the dynamical quantum Fisher information, we find that the collisional dephasing significantly diminishes the precision of the phase parameter with the Ramsey interferometry.