Bounds on Renyi entropy growth in many-body quantum systems

• URL: http://arxiv.org/abs/2212.07444v1
• Date: Wed, 14 Dec 2022 19:00:01 GMT
• Title: Bounds on Renyi entropy growth in many-body quantum systems
• Authors: Zhengyan Darius Shi
• Abstract summary: 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 $|S'_alpha(t)|$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove rigorous bounds on the growth of $\alpha$-Renyi entropies $S_{\alpha}(t)$ (the Von Neumann entropy being the special case $\alpha = 1$) associated with any subsystem $A$ of a general lattice quantum many-body system with finite onsite Hilbert space dimension. For completely non-local Hamiltonians, we show that the instantaneous growth rates $|S'_{\alpha}(t)|$ (with $\alpha \neq 1$) can be exponentially larger than $|S'_1(t)|$ as a function of the subsystem size $|A|$. For $D$-dimensional systems with geometric locality, we prove bounds on $|S'_{\alpha}(t)|$ that depend on the decay rate of interactions with distance. When $\alpha = 1$, the bound is $|A|$-independent for all power-law decaying interactions $V(r) \sim r^{-w}$ with $w > 2D+1$. But for $\alpha > 1$, the bound is $|A|$-independent only when the interactions are finite-range or decay faster than $V(r) \sim e^{- c\, r^D}$ for some $c$ depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on $k$-local systems with or without geometric locality. A central theme of this work is that the value of $\alpha$ strongly influences the interplay between locality and entanglement growth. In other words, the Von Neumann entropy and the $\alpha$-Renyi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for non-local dynamics.

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)
• 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)
• Entanglement in XYZ model on a spin-star system: Anisotropy vs. field-induced dynamics [0.0]
  We show that for vanishing $xy$-anisotropy $gamma$, bipartite entanglement on the peripheral spins exhibits a logarithmic growth with $n_p$. When the system is taken out of equilibrium by the introduction of a magnetic field of constant strength on all spins, the time-averaged bipartite entanglement on the periphery exhibits a logarithmic growth with $n_p$ irrespective of the value of $gamma$.
  arXiv  Detail & Related papers  (2023-07-29T10:13:39Z)
• Average R\'{e}nyi Entropy of a Subsystem in Random Pure State [0.0]
  We plot the $ln m$-dependence of the quantum information derived from $widetildeS_alpha (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $alpha$, and eventually disappears in the limit of $alpha rightarrow infty$.
  arXiv  Detail & Related papers  (2023-01-22T08:08:51Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• Scattering data and bound states of a squeezed double-layer structure [77.34726150561087]
  A structure composed of two parallel homogeneous layers is studied in the limit as their widths $l_j$ and $l_j$, and the distance between them $r$ shrinks to zero simultaneously. The existence of non-trivial bound states is proven in the squeezing limit, including the particular example of the squeezed potential in the form of the derivative of Dirac's delta function. The scenario how a single bound state survives in the squeezed system from a finite number of bound states in the finite system is described in detail.
  arXiv  Detail & Related papers  (2020-11-23T14:40:27Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• R\'{e}nyi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells [0.0]
  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 $alpha$.
  arXiv  Detail & Related papers  (2020-03-09T18:34:00Z)
• On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions [62.01594253618911]
  We exploit the finite noise structure of finite sums to derive a matching $O(n2)$-upper bound under the global oracle model. Following a similar approach, we propose a novel adaptation of SVRG which is both emphcompatible with oracles, and achieves complexity bounds of $tildeO(n2+nsqrtL/mu)log (1/epsilon)$ and $O(nsqrtL/epsilon)$, for $mu>0$ and $mu=0$
  arXiv  Detail & Related papers  (2020-02-09T03:39:46Z)
• 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)
• Multifractality meets entanglement: relation for non-ergodic extended states [0.0]
  We show that entanglement entropy takes an ergodic value even though the wave function is highly non-ergodic. We also show that their fluctuations have ergodic behavior in narrower vicinity of the ergodic state, $D=1$.
  arXiv  Detail & Related papers  (2020-01-09T19:00:02Z)

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.