Fundamental Limits on the Speed of Evolution of Quantum States

This paper reports on some new inequalities of Margolus-Levitin-Mandelstam-Tamm-type involving the speed of quantum evolution between two orthogonal pure states. The clear determinant of the qualitative behavior of this time scale is the statistics of the energy spectrum. An often-overlooked correspondence between the real-time behavior of a quantum system and the statistical mechanics of a transformed (imaginary-time) thermodynamic system appears promising as a source of qualitative insights into the quantum dynamics.Comment: 6 pages, 1 eps figur

On the use of non-canonical quantum statistics

We develop a method using a coarse graining of the energy fluctuations of an equilibrium quantum system which produces simple parameterizations for the behaviour of the system. As an application, we use these methods to gain more understanding on the standard Boltzmann-Gibbs statistics and on the recently developed Tsallis statistics. We conclude on a discussion of the role of entropy and the maximum entropy principle in thermodynamics.Comment: 29 pages, uses iopart.cls, major revisions of text for better readability, added a discussion about essentially microcanonical ensemble

Three-body bound states on a lattice

Journal ArticleThe theory of three-body bound states for particles moving on a lattice and interacting with attractive two-body pointlike potentials is presented. The applications are to bosons, fermions (no three-body bound states are found), and magnons. When a three-body bound state forms in three dimensions, it does so discontinuously. Thus there is a maximum size for the three-body bound state, of approximately two lattice constants. Some of the various analyses are relevant to magnetism and superconductivity

The Big Data Newsvendor: Practical Insights from Machine Learning

We investigate the data-driven newsvendor problem when one has n observations of p features related to the demand as well as historical demand data. Rather than a two-step process of first estimating a demand distribution then optimizing for the optimal order quantity, we propose solving the “Big Data” newsvendor problem via single step machine learning algorithms. Specifically, we propose algorithms based on the Empirical Risk Minimization (ERM) principle, with and without regularization, and an algorithm based on Kernel-weights Optimization (KO). The ERM approaches, equivalent to high-dimensional quantile regression, can be solved by convex optimization problems and the KO approach by a sorting algorithm. We analytically justify the use of features by showing that their omission yields inconsistent decisions. We then derive finite-sample performance bounds on the out-of-sample costs of the feature-based algorithms, which quantify the effects of dimensionality and cost parameters. Our bounds, based on algorithmic stability theory, generalize known analyses for the newsvendor problem without feature information. Finally, we apply the feature-based algorithms for nurse staffing in a hospital emergency room using a data set from a large UK teaching hospital and find that (i) the best ERM and KO algorithms beat the best practice benchmark by 23% and 24% respectively in the out-of-sample cost, and (ii) the best KO algorithm is faster than the best ERM algorithm by three orders of magnitude and the best practice benchmark by two orders of magnitude

Detection of a Moving Rigid Solid in a Perfect Fluid

In this paper, we consider a moving rigid solid immersed in a potential fluid. The fluid-solid system fills the whole two dimensional space and the fluid is assumed to be at rest at infinity. Our aim is to study the inverse problem, initially introduced in [3], that consists in recovering the position and the velocity of the solid assuming that the potential function is known at a given time. We show that this problem is in general ill-posed by providing counterexamples for which the same potential corresponds to different positions and velocities of a same solid. However, it is also possible to find solids having a specific shape, like ellipses for instance, for which the problem of detection admits a unique solution. Using complex analysis, we prove that the well-posedness of the inverse problem is equivalent to the solvability of an infinite set of nonlinear equations. This result allows us to show that when the solid enjoys some symmetry properties, it can be partially detected. Besides, for any solid, the velocity can always be recovered when both the potential function and the position are supposed to be known. Finally, we prove that by performing continuous measurements of the fluid potential over a time interval, we can always track the position of the solid.Comment: 19 pages, 14 figure

Power of unentangled measurements on two antiparallel spins

We consider a pair of antiparallel spins polarized in a random direction to encode quantum information. We wish to extract as much information as possible on the polarization direction attainable by an unentangled measurement, i.e., by a measurement, whose outcomes are associated with product states. We develop analytically the upper bound 0.7935 bits to the Shannon mutual information obtainable by an unentangled measurement, which is definitely less than the value 0.8664 bits attained by an entangled measurement. This proves our main result, that not every ensemble of product states can be optimally distinguished by an unentangled measurement, if the measure of distinguishability is defined in the sense of Shannon. We also present results from numerical calculations and discuss briefly the case of parallel spins.Comment: Latex file, 18 pages, 1 figure; published versio

Density of states of a two-dimensional electron gas in a non-quantizing magnetic field

We study local density of electron states of a two-dimentional conductor with a smooth disorder potential in a non-quantizing magnetic field, which does not cause the standart de Haas-van Alphen oscillations. It is found, that despite the influence of such ``classical'' magnetic field on the average electron density of states (DOS) is negligibly small, it does produce a significant effect on the DOS correlations. The corresponding correlation function exhibits oscillations with the characteristic period of cyclotron quantum $\hbar\omega_c$.Comment: 7 pages, including 3 figure