Quantum metric statistics for random-matrix families
Michael Berry, Pragya Shukla
Abstract
Abstract The quantum metric tensor G ij for parameterised families of quantum states, in particular the trace G = tr G 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.
Topics & Concepts
Random matrixMathematicsTRACE (psycholinguistics)Dimension (graph theory)QuantumGaussianMatrix (chemical analysis)Symmetry (geometry)Metric (unit)Probability distributionStatistical physicsMathematical physicsPure mathematicsStatisticsQuantum mechanicsPhysicsGeometryPhilosophyEconomicsMaterials scienceLinguisticsComposite materialEigenvalues and eigenvectorsOperations managementQuantum many-body systemsQuantum Information and CryptographyQuantum Mechanics and Applications