Kubo–Mori–Bogoliubov Fisher information on the quantum Gaussian model and violation of the scale invariance

The classical Gaussian model, a parametric family of the Gaussian distribution, is known to be a space of constant negative curvature if one regards the Fisher information on the model as a Riemannian metric. Constant curvature reflects the scale invariance of the classical Gaussian model, which is well known in information geometry. However, it is shown that if the Kubo–Mori–Bogoliubov Fisher information on the quantum Gaussian model is adopted as a Riemannian metric, then this scale invariance on the Gaussian model is broken due to the quantum effect. In the present study, the connection between the geometry of the classical Gaussian model and its quantum counterpart is clarified using the Taylor expansion with respect to the Planck constant. It is further shown that such a method is not applicable to a finite-dimensional system such as the spin system.