R\'{e}nyi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells

• URL: http://arxiv.org/abs/2003.04351v1
• Date: Mon, 9 Mar 2020 18:34:00 GMT
• Title: R\'{e}nyi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• Authors: O. Olendski
• Abstract summary: Dirichlet and Neumann boundary conditions (BCs) of 1D quantum well are studied. For either BC the dependencies of the R'enyi position components on the parameter $alpha$ are the same for all orbitals. The gap between the thresholds $alpha_TH$ of the two BCs causes different behavior of the R'enyi uncertainty relations as functions of $alpha$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A comparative analysis of the Dirichlet and Neumann boundary conditions (BCs) of the one-dimensional (1D) quantum well extracts similarities and differences of the R\'{e}nyi $R(\alpha)$ as well as Tsallis $T(\alpha)$ entropies between these two geometries. It is shown, in particular, that for either BC the dependencies of the R\'{e}nyi position components on the parameter $\alpha$ are the same for all orbitals but the lowest Neumann one for which the corresponding functional $R$ is not influenced by the variation of $\alpha$. Lower limit $\alpha_{TH}$ of the semi infinite range of the dimensionless R\'{e}nyi/Tsallis coefficient where {\em momentum} entropies exist crucially depends on the {\em position} BC and is equal to one quarter for the Dirichlet requirement and one half for the Neumann one. At $\alpha$ approaching this critical value, the corresponding momentum functionals do diverge. The gap between the thresholds $\alpha_{TH}$ of the two BCs causes different behavior of the R\'{e}nyi uncertainty relations as functions of $\alpha$. For both configurations, the lowest-energy level at $\alpha=1/2$ does saturate either type of the entropic inequality thus confirming an earlier surmise about it. It is also conjectured that the threshold $\alpha_{TH}$ of one half is characteristic of any 1D non-Dirichlet system. Other properties are discussed and analyzed from the mathematical and physical points of view.

Related papers

• On stability of k-local quantum phases of matter [0.4999814847776097]
  We analyze the stability of the energy gap to Euclids for Hamiltonians corresponding to general quantum low-density parity-check codes. We discuss implications for the third law of thermodynamics, as $k$-local Hamiltonians can have extensive zero-temperature entropy.
  arXiv  Detail & Related papers  (2024-05-29T18:00:20Z)
• On the $O(\frac{\sqrt{d}}{T^{1/4}})$ Convergence Rate of RMSProp and Its Momentum Extension Measured by $\ell_1$ Norm [59.65871549878937]
  This paper considers the RMSProp and its momentum extension and establishes the convergence rate of $frac1Tsum_k=1T. Our convergence rate matches the lower bound with respect to all the coefficients except the dimension $d$. Our convergence rate can be considered to be analogous to the $frac1Tsum_k=1T.
  arXiv  Detail & Related papers  (2024-02-01T07:21:32Z)
• Inner Structure of Many-Body Localization Transition and Fulfillment of Harris Criterion [6.83731714529242]
  Two independent order parameters stemming purely from the half-chain von Neumann entanglement entropy $S_textrmvN$ are introduced to probe its eigenstate transition. From symmetry-endowed entropy decomposition, they are probability distribution deviation $|d(p_n)|$ and von Neumann entropy $S_textrmvNn(D_n!=!!mboxmax)$ of the maximum-dimensional symmetry subdivision.
  arXiv  Detail & Related papers  (2024-01-20T22:13:59Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Quantum-information theory of magnetic field influence on circular dots with different boundary conditions [0.0]
  2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC) Physical interpretation is based on the different roles of the two BCs and their interplay with the field: Dirichlet (Neumann) surface is a repulsive (attractive) interface.
  arXiv  Detail & Related papers  (2023-06-28T11:33:11Z)
• Bounds on Renyi entropy growth in many-body quantum systems [0.0]
  We prove rigorous bounds on the growth of $alpha$-Renyi entropies $S_alpha(t)$. For completely non-local Hamiltonians, we show that the instantaneous growth rates $|S'_alpha(t)|$ can be exponentially larger than $|S'_alpha(t)|$.
  arXiv  Detail & Related papers  (2022-12-14T19:00:01Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots [0.0]
  $mathttd$-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) This paves the way to an efficient computation in either space of Shannon, R'enyi and Tsallis entropies, Onicescu energies and Fisher informations.
  arXiv  Detail & Related papers  (2021-03-24T11:08:31Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.