Fluctuations, uncertainty relations, and the geometry of quantum state manifolds

The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher-order geometric tensors (affine connection, Riemann curvature tensor), can be obtained in terms of gauge-invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many-operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the $\text{SU}(1,1)$ group. In our examples nontrivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the $\text{SU}(1,1)$ group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states nontrivial geometry results.