Abstract geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the Fisher-Rao metric with the non-Hermitian systems. By approximating the Lindblad master equation in the non-Hermitian Hamiltonian, we calculate the time evolution of the quantum geometric metric. Finally, we give an example of the quantum spin Ising model of the imaginary magnetic field, explore the energy spectrum of PT-symmetric Hamiltonian and the evolution of geometric metric, and discuss that the dissipative effect of the imaginary magnetic field can be eliminated under the condition of adding the control Hamiltonian, so as to improve the accuracy of parameter estimation.
Abstract geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the Fisher-Rao metric with the non-Hermitian systems. By approximating the Lindblad master equation in the non-Hermitian Hamiltonian, we calculate the time evolution of the quantum geometric metric. Finally, we give an example of the quantum spin Ising model of the imaginary magnetic field, explore the energy spectrum of PT-symmetric Hamiltonian and the evolution of geometric metric, and discuss that the dissipative effect of the imaginary magnetic field can be eliminated under the condition of adding the control Hamiltonian, so as to improve the accuracy of parameter estimation.
Information geometry provides a tool to systematically investigate the parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric.
Quantum Fisher information bounds the ultimate precision limit in the parameter estimation and has important applications in quantum metrology. In recent years, the theoretical and experimental studies of non-Hermitian Hamiltonians realized in quantum systems have attracted wide attention. Here, the parameter estimation based on eigenstates of non-Hermitian Hamiltonians is investigated, and the corresponding quantum Fisher information and quantum Cramér-Rao bound for the single-parameter and two-parameter estimations are given. In particular, the quantum Fisher information about estimating intrinsic momentum and external parameters in the non-reciprocal and gain-and-loss Su-Schrieffer-Heeger models, and non-Hermitian versions of the quantum Ising chain, Chern-insulator model and two-level system are calculated and analyzed. For these non-Hermitian models, the results show that in the case of single-parameter estimation in these non-Hermitian models, the quantum Fisher information increases significantly in the gapless regime and near the exceptional points, which can improve the accuracy limit of parameter estimation. For the two-parameter estimation, the determinant of the quantum Fisher information matrix also increases obviously near the gapless and exceptional points. In addition, a higher overall accuracy can be achieved in the topological regime than in the trivial regime, and the topological bound in two-parameter estimation can be determined by the Chern number.
We discuss an information-geometric framework for characterizing quantum states in non-equilibrium dynamics. Using the transverse-field Ising chain model as a laboratory, we investigate the quantum Fisher information metric with particular emphasis on the mixed metric component ght and related geometric observables, where h is a controllable post-quench Hamiltonian parameter and t is real time. The framework treats h and t as coordinates of an extended information manifold (h,t). The geometric observables characterize the speed of evolution on the manifold and the alignment between the temporal and parameter-deformation directions. Correlations between these geometric quantities and two widely used measures of non-equilibrium dynamics, the entanglement entropy and the Loschmidt echo, are analyzed.
Information geometry usually refers to a class of geometries characterized from metrics and corresponding connections related to the notion of statistical information during the process of parameter estimation on classical or quantum systems.
We are aware of the fact that in parameter estimation we can't achieve perfectness. There is a lower bound, which is not zero, for the variance - covariance of the estimated parameter compared to the real parameter, the Cramer - Rao bound. In classical parameter estimation, this bound is given by the reciprocal of Fisher information and is unique in the sense that the Fisher is the unique monotone metric under stochastic transformations. In quantum parameter estimation there is a class of monotone metrics and the lowest possible bound, which determines the best quality of estimation, is given by that of the Symmetric Logarithmic Derivative (SLD). The term symmetric determines the derivative type, since there are more than one types due to the non-commutativity of operators, in general, in quantum mechanics.
In our first published paper in the framework of this Ph.D. thesis, we are working beyond the class on monotone metrics. We are using the phase space to define a new statistical metric. This is achieved by introducing logarithmic derivatives through the Husimi Q distribution and consequently the new metric. As it turns out the related bound of Cramer - Rao type, can be greater even of this of the Symmetric Logarithmic Derivative.
In our second paper, we refer to a particular bi-parameter family of generalized entropies used for the classification of complex systems. By defining a divergence for the discrete distribution, we consequently define a metric and the corresponding Levi - Civita connection, which depend on the two parameters say c and d. So, we are having geometrical objects with statistical content like the Cramer - Rao bound and the scalar curvature, which is characteristic (and different) for every class of complex systems.
In our third paper, we are treating the above parameters c and d as coordinates of a two dimensional manifold. By calculating the Fisher metric and the corresponding scalar curvature we observe particular values of c and d for which we are making the conjecture that they related with particular properties of complex systems.
Finally, we are highlighting a correlation of the curvature of a particular manifold and quantum entanglement. In particular, we are using the entropy of entanglement to define a two dimensional manifold, which we embed into three dimensional real space. We find that in the region which this surface has maximum curvature we have product quantum states and only there.
Abstract One of the key features of information geometry in the classical setting is the existence of a metric structure and a family of connections on the space of probability distributions. The uniqueness of the Fisher–Rao metric and the duality of these connections is at the heart of classical information geometry. However, these features do not carry over straightforwardly to quantum systems, where a Hermitian inner product structure on the Hilbert space induces a metric on the complex projective space of pure states, the Fubini–Study tensor, which is preserved under unitary evolution. In this work, we explore how the modification of the Hermitian tensor structure on the projective space may affect the geometry of pure quantum states, and whether such generalisations can be used to define dual connections with a direct correspondence to classical probability distribution functions, modified by the presence of a non-trivial phase. We show that it is indeed possible to construct a family of connections that are dual to each other in a generalised sense with respect to the real-valued sector of the Fubini–Study tensor. Using this biorthogonal formalism, we systematically classify the four types of tensors that can arise when the dynamics of a quantum system are governed by a non-Hermitian Hamiltonian, identifying both the complex-valued metric and the Berry curvature. Finally, we elucidate the role of the metric in a quantum natural gradient descent optimisation problem, generalised to the non-Hermitian case for a suitable choice of cost function.
The ability to control matter at the quantum level is a central requirement for the continued advance of quantum technologies. Recent experimental progress has enabled continuous monitoring of quantum systems in time, giving rise to dynamics conditioned on measurement outcomes. Such measurement-conditioned evolution is naturally described by non-Hermitian Hamiltonians, whose complex eigenvalues encode gain and loss of population associated with the measurement record. Non-Hermitian quantum dynamics exhibit geometric features without Hermitian counterparts, including non-orthogonal eigenspaces, exceptional points, and complex geometric phases. This thesis investigates geometric structures underlying non-Hermitian quantum dynamics from two complementary perspectives: the geometry of quantum states under measurement-conditioned evolution, and the geometry of Hamiltonians and parameter spaces relevant for control in experimentally accessible platforms. First, measurement-conditioned non-Hermitian dynamics are formulated within quantum information geometry. The normalized evolution preserves the rank of the density operator, confining the dynamics to fixed-rank manifolds with a natural geometric structure. This framework enables optimization of the success probability of prescribed non-Hermitian trajectories and, via a purification-bundle formulation, leads to a non-Hermitian quantum speed limit expressed in terms of the Bures geometry. Explicit shortest geodesics and their associated non-Hermitian generators are derived, including results on the existence of time-independent generators in finite dimensions. The thesis then addresses control of non-Hermitian dynamics from a complementary geometric viewpoint focused on Hamiltonians and parameter spaces. A complex mixing angle is introduced for the detuned parity--time-symmetric dimer, providing a transparent mapping between parameter space and eigenspaces and enabling implementable counter-diabatic control on a superconducting transmon qubit. This approach is validated experimentally through accurate eigenstate tracking and exceptional-point encircling, and is subsequently generalized to arbitrary two-level systems, bipartite lattice models, and multilevel transmons. Finally, the geometric origin of complex Berry phases is identified, with predictions indicating their observability on the transmon platform and their role in purity stabilization and noise-assisted chiral transfer. Overall, the thesis establishes geometry as a unifying and operational language for the analysis and control of non-Hermitian quantum systems, addressing both foundational theory and experimentally relevant protocols.