Related papers: Statistical Geometry in Quantum Mechanics
The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing…
This article is an expository account aimed at viewing entanglement in finite-dimensional quantum many-body systems as a phenomenon of global geometry. While the mathematics of general quantum states has been studied extensively, this…
Statistical nuclear spectroscopy (also called spectral distribution method), introduced by J.B. French in late 60's and developed in detail in the later years by his group and many other groups, is based on the Gaussian forms for the state…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new…
`Hypergeometric states', which are a one-parameter generalization of binomial states of the single-mode quantized radiation field, are introduced and their nonclassical properties are investigated. Their limits to the binomial states and to…
A powerful tool for studying geometrical problems in Hilbert space is developed. In particular, we study the quantum pure state tomography problem in finite dimensions from the point of view of dynamical systems and bifurcations theory.…
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The…
Statistical depth is the act of gauging how representative a point is compared to a reference probability measure. The depth allows introducing rankings and orderings to data living in multivariate, or function spaces. Though widely applied…
This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory including some recent results. It is based on the algebraic approach where the starting point is a star-algebra and on the geometric…
We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically…
We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential…
We demonstrate how Quantum Cognition Machine Learning (QCML) encodes data as quantum geometry. In QCML, features of the data are represented by learned Hermitian matrices, and data points are mapped to states in Hilbert space. The quantum…
We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical…
We explore the interplay between symmetry and randomness in quantum information. Adopting a geometric approach, we consider states as $H$-equivalent if related by a symmetry transformation characterized by the group $H$. We then introduce…
A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in $\bbR^n$, or $\bbS^n$, $n\in \bbN$, which are distributed according to the configurational microcanonical…
We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery $\Gamma$-calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide…
We apply De Haro's Geometric View of Theories to one of the simplest quantum systems: a spinless particle on a line and on a circle. The classical phase space M = T*Q is taken as the base of a trivial Hilbert bundle E ~ M x H, and the…
Quantum statistical mechanics is formulated as an integral over classical phase space. Some details of the commutation function for averages are discussed, as is the factorization of the symmetrization function used for the grand potential…
Stochastic models share many characteristics with generic parametric models. In some ways they can be regarded as a special case. But for stochastic models there is a notion of weak distribution or generalised random variable, and the same…