Statistical properties of Pauli matrices going through noisy channels

International audienceWe study the statistical properties of the triplet $(\sigma_x,\sigma_y,\sigma_z)$ of Pauli matrices going through a sequence of noisy channels, modeled by the repetition of a general, trace-preserving, completely positive map. We show a non-commutative central limit theorem for the distribution of this triplet, which shows up a 3-dimensional Brownian motion in the limit with a non-trivial covariance matrix. We also prove a large deviation principle associated to this convergence, with an explicit rate function depending on the stationary state of the noisy channel

A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry

We present here a review of the fundamental topics of Hartree-Fock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer approximation, we arrive to the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. All Hartree-Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. Then, the discretization of the one-electron orbitals space is reviewed and the Roothaan-Hall formalism introduced. This leads to a exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing in algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (specially those related to the design of linear-scaling methods) are commented and linked to the issues discussed. The whole work is intentionally introductory and rather self-contained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.Comment: 64 pages, 3 figures, tMPH2e.cls style file, doublesp, mathbbol and subeqn package

Frontiers of open quantum system dynamics

We briefly examine recent developments in the field of open quantum system theory, devoted to the introduction of a satisfactory notion of memory for a quantum dynamics. In particular, we will consider a possible formalization of the notion of non-Markovian dynamics, as well as the construction of quantum evolution equations featuring a memory kernel. Connections will be drawn to the corresponding notions in the framework of classical stochastic processes, thus pointing to the key differences between a quantum and classical formalization of the notion of memory effects.Comment: 15 pages, contribution to "Quantum Physics and Geometry", Lecture Notes of the Unione Matematica Italiana 25,E. Ballico et al. (eds.

The elusive Heisenberg limit in quantum enhanced metrology

We provide efficient and intuitive tools for deriving bounds on achievable precision in quantum enhanced metrology based on the geometry of quantum channels and semi-definite programming. We show that when decoherence is taken into account, the maximal possible quantum enhancement amounts generically to a constant factor rather than quadratic improvement. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: dephasing,depolarization, spontaneous emission and photon loss.Comment: 10 pages, 4 figures, presentation imporved, implementation of the semi-definite program finding the precision bounds adde

Wavelet methods

This overview article motivates the use of wavelets in statistics, and introduces the basic mathematics behind the construction of wavelets. Topics covered include the continuous and discrete wavelet transforms, multiresolution analysis and the non-decimated wavelet transform. We describe the basic mechanics of nonparametric function estimation via wavelets, emphasising the concepts of sparsity and thresholding. A simple proof of the mean-square consistency of the wavelet estimator is also included. The article ends with two special topics: function estimation with Unbalanced Haar wavelets, and variance stabilisation via the Haar-Fisz transformation. Wavelets aremathematical functions which, when plotted, resemble “little waves”: that is, they are compactly or almost-compactly supported, and they integrate to zero. This is in contrast to “big waves” – sines and cosines in Fourier analysis, which also oscillate, but the amplitude of their oscillation never changes. Wavelets are useful for decomposing data into “wavelet coefficients”, which can then be processed in a way which depends on the aim of the analysis. One possibly advantageous feature of this decomposition is that in some set-ups, the decomposition will be sparse, i.e. most of the coefficients will be close to zero, with only a few coefficients carrying most of the information about the data. One can imagine obvious uses of this fact, e.g. in image compression. The decomposition is particularly informative, fast and easy to invert if it is performed using wavelets at a range of scales and locations. The role of scale is similar to the role of frequency in Fourier analysis. However, the concept of location is unique to wavelets: as mentioned above, they are localised around a particular point of the domain, unlike Fourier functions. This article provides a self-contained introduction to the applications of wavelets in statistics and attempts to justify the extreme popularity which they have enjoyed in the literature over the past 15 years