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Open AccessFeature PaperArticle

Generic Entanglement Entropy for Quantum States with Symmetry

by Yoshifumi Nakata 1,2,*,† and Mio Murao 3,†
Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan
Japan Science and Technology Agency (JST), Precursory Research for Embryonic Science and Technology (PRESTO), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Author to whom correspondence should be addressed.
Both authors contributed to the formulation of the problem. The main analysis was done by Y.N. in collaboration with M.M.
Entropy 2020, 22(6), 684;
Received: 1 June 2020 / Revised: 16 June 2020 / Accepted: 17 June 2020 / Published: 19 June 2020
(This article belongs to the Special Issue Quantum Probability, Statistics and Control)
When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry. View Full-Text
Keywords: entanglement entropy; symmetry; random states entanglement entropy; symmetry; random states
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Nakata, Y.; Murao, M. Generic Entanglement Entropy for Quantum States with Symmetry. Entropy 2020, 22, 684.

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