33 research outputs found
Statistical properties of Pauli matrices going through noisy channels
International audienceWe study the statistical properties of the triplet 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 âlittle wavesâ: that is, they are compactly or almost-compactly supported, and they integrate to zero. This is in contrast to âbig wavesâ â sines and cosines in Fourier analysis, which also oscillate, but the amplitude of their oscillation never changes. Wavelets are useful for decomposing data into âwavelet coefficientsâ, 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