Journal of High Energy Physics

, 2017:73 | Cite as

Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

  • Song He
  • Feng-Li Lin
  • Jia-ju ZhangEmail author
Open Access
Regular Article - Theoretical Physics


We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length ℓ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the Rényi 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 ℓ9 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.


AdS-CFT Correspondence Conformal Field Theory Field Theories in Lower Dimensions 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Supplementary material

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ESM 1 (NB 559 kb)


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© The Author(s) 2017

Authors and Affiliations

  1. 1.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)GolmGermany
  2. 2.CAS Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingChina
  3. 3.Department of PhysicsNational Taiwan Normal UniversityTaipeiTaiwan
  4. 4.Dipartimento di FisicaUniversità degli Studi di Milano-BicoccaMilanoItaly
  5. 5.INFN, Sezione di Milano-BicoccaMilanoItaly

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