Quantum metric statistics for random-matrix families

The quantum metric tensor G _ij for parameterised families of quantum states, in particular the trace G = trG _ij, depends on the symmetry of the system (e.g. time-reversal), and the dimension N of the underlying matrices. Modelling the families by the stationary Gaussian ensembles of random-matrix, theory, we calculate the probability distribution of G, exactly for N = 2, and approximately for N = 3 and N → ∞. Codimension arguments establish the scalings of the distributions near the singularities at G → ∞ and G = 0, near which asymptotics gives the explicit analytic behaviour. Numerical simulations support the theory. (paper)