Holographic R\'enyi entropy in AdS3_3/LCFT2_2 correspondence

The recent study in AdS$_3$/CFT$_2$ correspondence shows that the tree level contribution and 1-loop correction of holographic R\'enyi entanglement entropy (HRE) exactly match the direct CFT computation in the large central charge limit. This allows the R\'enyi entanglement entropy to be a new window to study the AdS/CFT correspondence. In this paper we generalize the study of R\'enyi entanglement entropy in pure AdS$_3$ gravity to the massive gravity theories at the critical points. For the cosmological topological massive gravity (CTMG), the dual conformal field theory (CFT) could be a chiral conformal field theory or a logarithmic conformal field theory (LCFT), depending on the asymptotic boundary conditions imposed. In both cases, by studying the short interval expansion of the R\'enyi entanglement entropy of two disjoint intervals with small cross ratio $x$, we find that the classical and 1-loop HRE are in exact match with the CFT results, up to order $x^6$. To this order, the difference between the massless graviton and logarithmic mode can be seen clearly. Moreover, for the cosmological new massive gravity (CNMG) at critical point, which could be dual to a logarithmic CFT as well, we find the similar agreement in the CNMG/LCFT correspondence. Furthermore we read the 2-loop correction of graviton and logarithmic mode to HRE from CFT computation. It has distinct feature from the one in pure AdS$_3$ gravity.Comment: 28 pages. Typos corrected, published versio

Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT

We investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval R\'enyi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of R\'enyi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central charge expansion. We then evaluate the relative entropy of the reduced density matrices to measure the difference between the heavy primary state and thermal state of canonical ensemble, and find that the aforementioned nontrivial eighth order results make the relative entropy unsuppressed in the large central charge limit. By using Pinsker's and Fannes-Audenaert inequalities, we can exploit the results of relative entropy to yield the lower and upper bounds on trace distance of the excited-state and thermal-state reduced density matrices. Our results are consistent with subsystem weak ETH, which requires the above trace distance is of power-law suppression by the large central charge. However, we are unable to pin down the exponent of power-law suppression. As a byproduct we also calculate the relative entropy to measure the difference between the reduced density matrices of two different heavy primary states.Comment: 28 pages, 4 figures;v2 change author list;v3 related subtleties about weak ETH clarified; v4 minor correction to match JHEP versio

Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length $\ell$ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the R\'enyi entropy, entanglement entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm. We adopt the method of operator product expansion of twist operators, and calculate the short interval expansion of these quantities up to order of $\ell^9$ for the contributions from the vacuum conformal family. The formal forms of these dissimilarity measures and the derived Fisher information metric from contributions of general operators are also given. As an application of the results, we use these dissimilarity measures to compare the excited and thermal states, and examine the eigenstate thermalization hypothesis (ETH) by showing how they behave in high temperature limit. This would help to understand how ETH in 2D CFT can be defined more precisely. We discuss the possibility that all the dissimilarity measures considered here vanish when comparing the reduced density matrices of an excited state and a generalized Gibbs ensemble thermal state. We also discuss ETH for a microcanonical ensemble thermal state in a 2D large central charge CFT, and find that it is approximately satisfied for a small subsystem and violated for a large subsystem.Comment: V1, 34 pages, 5 figures, see collection of complete results in the attached Mathematica notebook; V2, 38 pages, 5 figures, published versio

Handwritten Chinese Character Recognition Using Deep Neural Networks

Chinese script is utilized by one-sixth of the world’s population. Unlike Latin, consisting only of twenty-six characters, the Chinese script comprises approximate ninety thousand characters. Due to the tremendous variation among these calligraphically rather complex characters, it is worthwhile for researchers to design and build fast and efficient technologies to classify handwritten Chinese characters. Recently, deep neural networks gained tremendous success in object recognition and different Latin based character recognition which led us to the idea to conduct several research attempts in this area by considering convolutional neural networks to recognize this complex script. To validate the efficiency and the robustness of the proposed Chinese character recognition system the CASIA HWDB1.1 database was considered. This dataset is a benchmark used by the research community to assess different handwriting recognizers. The dataset - built by 300 different writers - is considered a large one containing 1.1 million images split into 3755 most commonly used Chinese characters. Our various experiments culminated at 95% accuracy, which is a very promising result and comparable with the current state-of-the-art results