Quantum fisher metric and uncertainty relations

Shunlong Luo

Z. Hradil, Jaroslav Řeháček

Mondal, Debasis

We provide an experimentally measurable local gauge $U(1)$ invariant Fubini-Study (FS) metric for mixed states. Like the FS metric for pure states, it also captures only the quantum part of the uncertainty in the evolution Hamiltonian. We show that this satisfies the quantum Cramer-Rao bound and thus arrive at a more general and measurable bound. Upon imposing the monotonicity condition, it reduces to the square-root derivative quantum Fisher Information. We show that on the Fisher information metric space dynamical phase is zero. A relation between square root derivative and logarithmic derivative is formulated such that both give the same Fisher information. We generalize the Fubini-Study metric for mixed states further and arrive at a set of Fubini-Study metric---called $α$ metric. This newly defined $α$ metric also satisfies the Cramer-Rao bound. Again by imposing the monotonicity condition on this metric, we derive the monotone $α$ metric. It reduces to the Fisher information metric for $α=1$.

Laetitia P. Bettmann, John Goold

Recent advancements have revealed new links between information geometry and classical stochastic thermodynamics, particularly through the Fisher information (FI) with respect to time. Recognizing the nonuniqueness of the quantum Fisher metric in Hilbert space, we exploit the fact that any quantum Fisher information (QFI) can be decomposed into a metric-independent incoherent part and a metric-dependent coherent contribution. We demonstrate that the incoherent component of any QFI can be directly linked to entropic acceleration, and for GKSL dynamics with local detailed balance, to the rate of change of generalized thermodynamic forces and entropic flow, paralleling the classical results. Furthermore, we tighten a classical uncertainty relation between the geometric uncertainty of a path in state space and the time-averaged rate of information change and demonstrate that it also holds for quantum systems. We generalize a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a nonnegative quantum contribution that arises from the geometric action due to coherent dynamics. Finally, we apply an information-geometric analysis to the recently proposed quantum-thermodynamic Mpemba effect, demonstrating this framework's ability to capture thermodynamic phenomena.

Paolo Gibilisco, Tommaso Isola

Tomohiro Shitara, Masahito Ueda

The quantum Fisher information represents a continuous family of metrics on the space of quantum states and places the fundamental limit on the accuracy of quantum state estimation. We show that the entire family of quantum Fisher information can be determined from linear-response theory through generalized covariances. We derive the generalized fluctuation-dissipation theorem that relates linear-response functions to generalized covariances and hence allows us to determine the quantum Fisher information from linear-response functions, which are experimentally measurable quantities. As an application, we examine the skew information, which is a quantum Fisher information, of a harmonic oscillator in thermal equilibrium, and show that the equality of the skew-information-based uncertainty relation holds.