TY - JOUR

T1 - Volume-Law Entanglement Entropy of Typical Pure Quantum States

AU - Bianchi, Eugenio

AU - Hackl, Lucas

AU - Kieburg, Mario

AU - Rigol, Marcos

AU - Vidmar, Lev

N1 - Funding Information:
We would like to thank Pietro Donà, Peter Forrester, Patrycja Łydżba, Lorenzo Piroli, and Nicholas Witte for inspiring discussions. E.B. acknowledges support from the National Science Foundation, Grant No. PHY-1806428, and from the John Templeton Foundation via the ID 61466 grant, as part of the “Quantum Information Structure of Spacetime (QISS)” project (qiss.fr). L.H. gratefully acknowledges support from the Alexander von Humboldt Foundation. M.K. acknowledges support from the Australian Research Council (ARC) under Grant No. DP210102887. M.R. acknowledges support from the National Science Foundation under Grant No. 2012145. L.V. acknowledges support from the Slovenian Research Agency (ARRS), Research core fundings Grants No. P1-0044 and No. J1-1696. L.H. and M.K. are also grateful to the MATRIX Institute in Creswick for hosting the online research programme and workshop “Structured Random Matrices Downunder” (26 July–13 August 2021).
Publisher Copyright:
© 2022 authors. Published by the American Physical Society.

PY - 2022/7

Y1 - 2022/7

N2 - The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has recently been conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that, while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.

AB - The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has recently been conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that, while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.

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U2 - 10.1103/PRXQuantum.3.030201

DO - 10.1103/PRXQuantum.3.030201

M3 - Article

AN - SCOPUS:85136523265

SN - 2691-3399

VL - 3

JO - PRX Quantum

JF - PRX Quantum

IS - 3

M1 - 030201

ER -