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arXiv:2301.07778 · Physical review. B./Physical review. B

Symmetry classification of typical quantum entanglement

Yuhan Liu, Jonah Kudler-Flam, Kohei Kawabata

Published Aug 4, 2023• 5 citations
arXivHTMLPDF

Abstract

Entanglement entropy of typical quantum states, also known as the Page curve, plays an important role in quantum many-body systems and quantum gravity. However, little has hitherto been understood about the role of symmetry in quantum entanglement. Here, we establish the comprehensive classification of typical quantum entanglement for free fermions, or equivalently the quadratic Sachdev-Ye-Kitaev model with symmetry, on the basis of the tenfold fundamental symmetry classes of time reversal, charge conjugation, and chiral transformation. Through both analytical and numerical calculations of random matrix theory, we show that the volume-law contribution to average entanglement entropy is robust and remains unaffected by symmetry. Conversely, we uncover that the constant terms of the average and variance of entanglement entropy yield tenfold universal values unique to each symmetry class. These constant terms originate from the combination of a global scaling of the entanglement spectrum due to time-reversal symmetry and a singular peak at the center of the entanglement spectrum due to chiral or particle-hole symmetry. Our work elucidates the interplay of symmetry and entanglement in quantum physics and provides a characterization of symmetry-enriched quantum chaos.

DOI
10.1103/physrevb.108.085109

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Entanglement entropy of typical quantum states, also known as the Page curve, plays an important role in quantum many-body systems and quantum gravity. However, little has hitherto been understood about the role of symmetry in quantum entanglement. Here, we establish the comprehensive classification of typical quantum entanglement for free fermions, or equivalently the quadratic Sachdev-Ye-Kitaev model with symmetry, on the basis of the tenfold fundamental symmetry classes of time reversal, charge conjugation, and chiral transformation. Through both analytical and numerical calculations of random matrix theory, we show that the volume-law contribution to average entanglement entropy is robust and remains unaffected by symmetry. Conversely, we uncover that the constant terms of the average and variance of entanglement entropy yield tenfold universal values unique to each symmetry class. These constant terms originate from the combination of a global scaling of the entanglement spectrum due to time-reversal symmetry and a singular peak at the center of the entanglement spectrum due to chiral or particle-hole symmetry. Our work elucidates the interplay of symmetry and entanglement in quantum physics and provides a characterization of symmetry-enriched quantum chaos.
We study bipartite entanglement statistics in one-dimensional anyon chains, whose Hilbert spaces are constrained by fusion rules of unitary pre-modular categories. Our setup generalizes previous frameworks on symmetry-resolved entanglement entropy for non-abelian Lie group symmetries to the setting of quantum groups. We derive analytical expressions for the average anyonic entanglement entropy and its variance. Surprisingly, despite the constrained Hilbert space structure, the large LLL expansion has no universal O(L)O(\sqrt{L})O(L​) or O(1)O(1)O(1) symmetry-type corrections except for a subleading topological correction term that produces a Page curve asymmetry. We further show that the variance decays exponentially with system size, establishing the typicality. Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions, supporting the use of eigenstate entanglement as a diagnostic of quantum chaos. Our results establish the anyonic Page curve as an appropriate chaotic benchmark in topological many-body systems and connect anyonic entanglement to Page-type universality in quantum many-body physics.
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.
Given a statistical ensemble of quantum states, the corresponding Page curve quantifies the average entanglement entropy associated with each possible spatial bipartition of the system. In this work, we study a natural extension in the presence of a conservation law and introduce the symmetry-resolved Page curves, characterizing average bipartite symmetry-resolved entanglement entropies. We derive explicit analytic formulas for two important statistical ensembles with a U(1)U(1)U(1)-symmetry: Haar-random pure states and random fermionic Gaussian states. In the former case, the symmetry-resolved Page curves can be obtained in an elementary way from the knowledge of the standard one. This is not true for random fermionic Gaussian states. In this case, we derive an analytic result in the thermodynamic limit based on a combination of techniques from random-matrix and large-deviation theories. We test our predictions against numerical calculations and discuss the subleading finite-size corrections.
We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global SU(2)\mathrm{SU}(2)SU(2) symmetry, constrained to fixed total spin JJJ and magnetization Jz=0J_z = 0Jz​=0. Focusing on spin-12\tfrac1221​ lattices and subsystem fractions f<12f < \frac{1}{2}f<21​, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume VVV. In addition to the expected leading volume law term, we prove the existence of a 12log⁡V\frac{1}{2}\log V21​logV finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the O(1)O(1)O(1) contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how SU(2)\mathrm{SU}(2)SU(2) symmetry shapes average entanglement.
We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.
We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2) S U ( 2 ) symmetry.