Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states

We discuss an alternative to relative entropy as a measure of distance between mixed quantum states. The proposed quantity is an extension to the realm of quantum theory of the Jensen-Shannon divergence (JSD) between probability distributions. The JSD has several interesting properties. It arises in information theory and, unlike the Kullback-Leibler divergence, it is symmetric, always well defined and bounded. We show that the quantum JSD (QJSD) shares with the relative entropy most of the physically relevant properties, in particular those required for a "good" quantum distinguishability measure. We relate it to other known quantum distances and we suggest possible applications in the field of the quantum information theory.Comment: 14 pages, corrected equation 1

Wootters' distance revisited: a new distinguishability criterium

The notion of distinguishability between quantum states has shown to be fundamental in the frame of quantum information theory. In this paper we present a new distinguishability criterium by using a information theoretic quantity: the Jensen-Shannon divergence (JSD). This quantity has several interesting properties, both from a conceptual and a formal point of view. Previous to define this distinguishability criterium, we review some of the most frequently used distances defined over quantum mechanics' Hilbert space. In this point our main claim is that the JSD can be taken as a unifying distance between quantum states.Comment: 15 pages, 3 figures, changed content, added reference for last sectio

On kernel engineering via Paley–Wiener

A radial basis function approximation takes the form $$s(x)=\sum_{k=1}^na_k\phi(x-b_k),\quad x\in {\mathbb{R}}^d,$$ where the coefficients a 1,…,a n are real numbers, the centres b 1,…,b n are distinct points in ℝ d , and the function φ:ℝ d →ℝ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which φ is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ψ=μ φ is a function of compact support, and when φ is polyharmonic. The novelty of this construction is its use of the Paley–Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ψ, so providing a new form of kernel engineering

On the Schoenberg Transformations in Data Analysis: Theory and Illustrations

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and visualized on artificial data sets by classical multidimensional scaling. A simple distance-based discriminant algorithm illustrates the theory, intimately connected to the Gaussian kernels of Machine Learning

Interplay of size and Landau quantizations in the de Haas-van Alphen oscillations of metallic nanowires

We examine the interplay between size quantization and Landau quantization in the De Haas-Van Alphen oscillations of clean, metallic nanowires in a longitudinal magnetic field for `hard' boundary conditions, i.e. those of an infinite round well, as opposed to the `soft' parabolically confined boundary conditions previously treated in Alexandrov and Kabanov (Phys. Rev. Lett. {\bf 95}, 076601 (2005) (AK)). We find that there exist {\em two} fundamental frequencies as opposed to the one found in bulk systems and the three frequencies found by AK with soft boundary counditions. In addition, we find that the additional `magic resonances' of AK may be also observed in the infinite well case, though they are now damped. We also compare the numerically generated energy spectrum of the infinite well potential with that of our analytic approximation, and compare calculations of the oscillatory portions of the thermodynamic quantities for both models.Comment: Title changed, paper streamlined on suggestion of referrees, typos corrected, numerical error in figs 2 and 3 corrected and final result simplified -- two not three frequencies (as in the previous version) are observed. Abstract altered accordingly. Submitted to Physical Review

Adult asthma associated with roadway density and housing in rural Appalachia: the Mountain Air Project (MAP).

BACKGROUND: Appalachian Kentucky is a rural area with a high prevalence of asthma among adults. The relative contribution of environmental exposures in the etiology of adult asthma in these populations has been understudied. OBJECTIVE: This manuscript describes the aims, study design, methods, and characteristics of participants for the Mountain Air Project (MAP), and focuses on associations between small area environmental exposures, including roadways and mining operations, and lifetime and current asthma in adults. METHODS: A cohort of residents, aged 21 and older, in two Kentucky counties, was enrolled in a community-based, cross-sectional study. Stratified cluster sampling was used to select small geographic areas denoted as 14-digit USGS hydrologic units (HUCs). Households were enumerated within selected HUCs. Community health workers collected in-person interviews. The proximity of nearby active and inactive coal mining operations, density of oil and gas operations, and density of roadways were characterized for all HUCs. Poisson regression analyses were used to estimate adjusted prevalence ratios. RESULTS: From 1,459 eligible households contacted, 1,190 individuals were recruited, and 972 persons completed the interviews. The prevalence of lifetime asthma was 22.8%; current asthma was 16.3%. Adjusting for covariates, roadway density was positively associated with current asthma in the second (aPR = 1.61; 95% CI 1.04-2.48) and third tertiles (aPR = 2.00; 95% CI 1.32-3.03). Increased risk of current asthma was associated with residence in public, multi-unit housing (aPR = 2.01; 95% CI 1.27-3.18) compared to a residence in a single-family home. There were no notable associations between proximity to coal mining and oil and gas operations and asthma prevalence. CONCLUSIONS: This study suggests that residents in rural areas with higher roadway density and those residing in public housing units may be at increased risk for current asthma after accounting for other known risk factors. Confirming the role of traffic-related particulates in producing high asthma risk among adults in this study contributes to the understanding of the multiple environmental exposures that influence respiratory health in the Appalachia region

On spherical averages of radial basis functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log  ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration

Restoring Ureagenesis in Hepatocytes by CRISPR/Cas9-mediated Genomic Addition to Arginase-deficient Induced Pluripotent Stem Cells.

Urea cycle disorders are incurable enzymopathies that affect nitrogen metabolism and typically lead to hyperammonemia. Arginase deficiency results from a mutation in Arg1, the enzyme regulating the final step of ureagenesis and typically results in developmental disabilities, seizures, spastic diplegia, and sometimes death. Current medical treatments for urea cycle disorders are only marginally effective, and for proximal disorders, liver transplantation is effective but limited by graft availability. Advances in human induced pluripotent stem cell research has allowed for the genetic modification of stem cells for potential cellular replacement therapies. In this study, we demonstrate a universally-applicable CRISPR/Cas9-based strategy utilizing exon 1 of the hypoxanthine-guanine phosphoribosyltransferase locus to genetically modify and restore arginase activity, and thus ureagenesis, in genetically distinct patient-specific human induced pluripotent stem cells and hepatocyte-like derivatives. Successful strategies restoring gene function in patient-specific human induced pluripotent stem cells may advance applications of genetically modified cell therapy to treat urea cycle and other inborn errors of metabolism