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.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.

Apr 17, 20193 citations

Quantum Entropy

Dominik Janzing

Compendium of Quantum PhysicsJan 1, 20090 citations

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

Eugenio Bianchi A, Pietro Donà

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.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.

arXiv:1904.08370Physical review. D/Physical review. D.Nov 18, 201968 citations

State Ensembles and Quantum Entropy

Subhash Kak

arXiv:1509.03220International Journal of Theoretical PhysicsJan 30, 201611 citations

Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

Lev Vidmar, Lucas Hackl, Eugenio Bianchi A +1 more

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.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.

arXiv:1703.02979Physical Review LettersJul 11, 2017134 citations

\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.

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.

{\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.