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arXiv:1904.08370 · Physical review. D/Physical review. D.

Typical entanglement entropy in the presence of a center: Page curve and its variance

Eugenio Bianchi A, Pietro Donà

Published Nov 18, 2019• 68 citations
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Abstract

In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability $P({S}_{A})\mathrm{d}{S}_{A}$ that a subsystem has entanglement entropy ${S}_{A}$. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, $\mathrm{\ensuremath{\Delta}}{S}_{A}/⟨{S}_{A}⟩\ensuremath{\ll}1$, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.

DOI
10.1103/physrevd.100.105010

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Typical entropy of a subsystem: Page curve and its variance.

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In a quantum system in a pure state, a subsystem generally has a non-zero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? The exact formula for the average $\langle{S_{A}}\rangle$ was conjectured by Page in 1995 and later proved. Here we compute the exact formula for the average entropy and variance $(\Delta S_A)^2 = \langle{S_A}^2\rangle - \langle{S_A}\rangle{}^2$ both for random pure states and for energy eigenstates of a non-interacting system. We show that for large systems the variance is small, $\Delta S_A/\langle{S_{A}}\rangle\ll 1$, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution $P(S_A) \mathrm{d}S_A$, and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems in an energy eigenstate. In particular, for a system of non-interacting spins in a magnetic field, we show how the thermal entropy arises as the typical entanglement entropy of a subsystem.

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In a seminal paper [Phys. Rev. Lett. 71, 1291 (1993)], Page conjectured that the entanglement entropy of typical pure states is $S_{\rm typ}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}$, where ${\cal D}_{\rm A}$ and ${\cal D}$ are the Hilbert space dimensions of the subsystem and the system, respectively, and typicality is with respect to the Haar measure. Typical pure states are hence (nearly) maximally entangled. We prove that $S^+_1=\ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}$ is an upper bound for the average entanglement entropy of all many-body fermionic eigenstates of a class of translationally invariant quadratic Hamiltonians in one dimension, and that the variance vanishes with increasing system size. Consequently, whenever the size of the subsystem is not a vanishing fraction of the size of the system, typical eigenstates of such Hamiltonians have a volume-law entanglement but do not become maximally entangled as ${\cal D}\rightarrow\infty$, i.e., they are qualitatively different from typical pure states. We also prove that, in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal.

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In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is ${S}_{\text{ave}}\ensuremath{\simeq}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(1/2){\mathcal{D}}_{\mathrm{A}}^{2}/\mathcal{D}$ for $1\ensuremath{\ll}{\mathcal{D}}_{\mathrm{A}}\ensuremath{\le}\sqrt{\mathcal{D}}$, where ${\mathcal{D}}_{\mathrm{A}}$ and $\mathcal{D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $⟨S⟩$ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}\ensuremath{\le}⟨S⟩\ensuremath{\le}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}[1/(2\mathrm{ln}2)](\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}$. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then $⟨S⟩<\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}$ in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

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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.

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On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State

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The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.

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In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability P(SA)dSAP({S}_{A})\mathrm{d}{S}_{A}P(SA​)dSA​ that a subsystem has entanglement entropy SA{S}_{A}SA​. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, \ensuremathΔSA/⟨SA⟩\ensuremath≪1\mathrm{\ensuremath{\Delta}}{S}_{A}/⟨{S}_{A}⟩\ensuremath{\ll}1\ensuremathΔSA​/⟨SA​⟩\ensuremath≪1, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.
In a quantum system in a pure state, a subsystem generally has a non-zero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? The exact formula for the average ⟨SA⟩\langle{S_{A}}\rangle⟨SA​⟩ was conjectured by Page in 1995 and later proved. Here we compute the exact formula for the average entropy and variance (ΔSA)2=⟨SA2⟩−⟨SA⟩2(\Delta S_A)^2 = \langle{S_A}^2\rangle - \langle{S_A}\rangle{}^2(ΔSA​)2=⟨SA​2⟩−⟨SA​⟩2 both for random pure states and for energy eigenstates of a non-interacting system. We show that for large systems the variance is small, ΔSA/⟨SA⟩≪1\Delta S_A/\langle{S_{A}}\rangle\ll 1ΔSA​/⟨SA​⟩≪1, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution P(SA)dSAP(S_A) \mathrm{d}S_AP(SA​)dSA​, and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems in an energy eigenstate. In particular, for a system of non-interacting spins in a magnetic field, we show how the thermal entropy arises as the typical entanglement entropy of a subsystem.
In a seminal paper [Phys. Rev. Lett. 71, 1291 (1993)], Page conjectured that the entanglement entropy of typical pure states is Styp≃ln⁡DA−(1/2)DA2/DS_{\rm typ}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}Styp​≃lnDA​−(1/2)DA2​/D, where DA{\cal D}_{\rm A}DA​ and D{\cal D}D are the Hilbert space dimensions of the subsystem and the system, respectively, and typicality is with respect to the Haar measure. Typical pure states are hence (nearly) maximally entangled. We prove that S1+=ln⁡DA−[1/(2ln⁡2)](ln⁡DA)2/ln⁡DS^+_1=\ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}S1+​=lnDA​−[1/(2ln2)](lnDA​)2/lnD is an upper bound for the average entanglement entropy of all many-body fermionic eigenstates of a class of translationally invariant quadratic Hamiltonians in one dimension, and that the variance vanishes with increasing system size. Consequently, whenever the size of the subsystem is not a vanishing fraction of the size of the system, typical eigenstates of such Hamiltonians have a volume-law entanglement but do not become maximally entangled as D→∞{\cal D}\rightarrow\inftyD→∞, i.e., they are qualitatively different from typical pure states. We also prove that, in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal.
In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is ${S}_{\text{ave}}\ensuremath{\simeq}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(1/2){\mathcal{D}}_{\mathrm{A}}^{2}/\mathcal{D}$ for $1\ensuremath{\ll}{\mathcal{D}}_{\mathrm{A}}\ensuremath{\le}\sqrt{\mathcal{D}}$, where DA{\mathcal{D}}_{\mathrm{A}}DA​ and D\mathcal{D}D are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy ⟨S⟩⟨S⟩⟨S⟩ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}\ensuremath{\le}⟨S⟩\ensuremath{\le}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}[1/(2\mathrm{ln}2)](\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}$. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then ⟨S⟩<lnDA⟨S⟩<\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}⟨S⟩<lnDA​ in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.
In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of NA{N}_{A}NA​ out of NNN degrees of freedom is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=(N\ensuremath{-}\frac{1}{2})\mathrm{\ensuremath{\Psi}}(2N)+(\frac{1}{4}\ensuremath{-}{N}_{A})\mathrm{\ensuremath{\Psi}}(N)+(\frac{1}{2}+{N}_{A}\ensuremath{-}N)\mathrm{\ensuremath{\Psi}}(2N\ensuremath{-}2{N}_{A})\ensuremath{-}\frac{1}{4}\mathrm{\ensuremath{\Psi}}(N\ensuremath{-}{N}_{A})\ensuremath{-}{N}_{A}$, where $\mathrm{\ensuremath{\Psi}}$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=N(log2\ensuremath{-}1)f+N(f\ensuremath{-}1)log(1\ensuremath{-}f)+\frac{1}{2}f+\frac{1}{4}log(1\ensuremath{-}f)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}O(1/N)$, where $f={N}_{A}/N\ensuremath{\le}1/2$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by \L{}yd\ifmmode \dot{z}\else \.{z}\fi{}ba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant ${lim}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}{(\mathrm{\ensuremath{\Delta}}{S}_{A})}_{\mathrm{G}}^{2}=\frac{1}{2}[f+{f}^{2}+log(1\ensuremath{-}f)]$.
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.
The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.