Information-Geometric Representation of Upper Bounds on Fluctuation Growth in Open Quantum Systems

Abstract

Building on our earlier results for closed quantum systems, we investigate the dynamical evolution of quantum fluctuations and their connections to Fisher information and the quantum operations describing open-system dynamics. Starting from a general expression for the time derivative of an observable's uncertainty, we derive an alternative upper bound on fluctuation growth in an open quantum system using the Cauchy-Schwarz inequality. The resulting bound separates contributions from the evolution of the quantum state and the explicit time dependence of the observable, reducing to a known result for closed-system dynamics. We then establish a connection between fluctuation dynamics and quantum Fisher information by decomposing the density operator and the symmetric logarithmic derivative (SLD) into coherent and incoherent contributions, which leads to a bound relating the evolution of observable uncertainty to the coherent and incoherent components of the Fisher information, providing an information-geometric interpretation of fluctuation growth. To connect fluctuation dynamics with quantum operations governing open-system evolutions, we derive the time evolution of the eigenvalues of the density operator in terms of Kraus operators. Using the infinitesimal-time relation between Kraus and Lindblad operators, we obtain a corresponding expression for the eigenvalue dynamics in terms of Lindblad operators. This formulation directly relates changes in the density operator eigenvalues to the quantum processes acting on the system. We then study the results using some examples. Our results provide a framework connecting observable fluctuation growth, Fisher information, and the Kraus and Lindblad descriptions of quantum evolution, offering a route toward understanding how specific quantum channels influence the information and fluctuations available in an evolving quantum system.