We investigate bipartite entanglement in spin- $1∕2$ systems on a generic lattice. For states that are an equal superposition of elements of a group $G$ of spin flips acting on the fully polarized state ${\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems $A$ and $B$ . These states are stabilized by the group $G$ . A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus $\mathfrak{g}$ . For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem $A$ and is equal to the perimeter (up to an additive constant) when $A$ is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.We investigate bipartite entanglement in spin-$1∕2$ systems on a generic lattice. For states that are an equal superposition of elements of a group $G$ of spin flips acting on the fully polarized state ${\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems $A$ and $B$. These states are stabilized by the group $G$. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus $\mathfrak{g}$. For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem $A$ and is equal to the perimeter (up to an additive constant) when $A$ is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.

arXiv:quant-ph/0409073Physical Review AFeb 22, 2005296 citations

Multipartite entanglement in quantum spin chains (4 pages)

Dagmar Bruß, Nilanjana Datta, Chiara Macchiavello +2 more

Physical Review AJan 1, 20053 citations

An area law for 2d frustration-free spin systems

Anurag Anshu, Itai Arad, David Gosset

We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error є by a degree O(√nlog(є−1)) multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error є by a degree O(√nlog(є−1)) multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.

arXiv:2103.02492ACM Symposium on Theory of Computing (STOC)Jun 9, 202240 citations

Conditions for entanglement in spin systems and for multipartite entanglement

Arkaitz Zubiaga

Jan 1, 20120 citations

Short-time versus long-time dynamics of entanglement in quantum lattice models

Razmik G. Unanyan, Dominik Muth, Michael Fleischhauer

We study the short-time evolution of the bipartite entanglement in quantum lattice systems with local interactions in terms of the purity of the reduced density matrix. A lower bound for the purity is derived in terms of the eigenvalue spread of the interaction Hamiltonian between the partitions. Starting from an initially separable state the purity decreases as $1\ensuremath{-}(t/\ensuremath{\tau}){}^{2}$ (i.e., quadratically in time, with a characteristic timescale $\ensuremath{\tau}$ that is inversely proportional to the boundary size of the subsystem, that is, as an area law). For larger times an exponential lower bound is derived corresponding to the well-known linear-in-time bound of the entanglement entropy. The validity of the derived lower bound is illustrated by comparison to the exact dynamics of a one-dimensional spin lattice system as well as a pair of coupled spin ladders obtained from numerical simulations.We study the short-time evolution of the bipartite entanglement in quantum lattice systems with local interactions in terms of the purity of the reduced density matrix. A lower bound for the purity is derived in terms of the eigenvalue spread of the interaction Hamiltonian between the partitions. Starting from an initially separable state the purity decreases as $1\ensuremath{-}(t/\ensuremath{\tau}){}^{2}$ (i.e., quadratically in time, with a characteristic timescale $\ensuremath{\tau}$ that is inversely proportional to the boundary size of the subsystem, that is, as an area law). For larger times an exponential lower bound is derived corresponding to the well-known linear-in-time bound of the entanglement entropy. The validity of the derived lower bound is illustrated by comparison to the exact dynamics of a one-dimensional spin lattice system as well as a pair of coupled spin ladders obtained from numerical simulations.

arXiv:0910.3868Physical Review AFeb 24, 20108 citations

Random matrix prediction of average entanglement entropy in non-Abelian symmetry sectors

Anwesha Chakraborty, Lucas Hackl, Mario Kieburg

We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global $\mathrm{SU}(2)$ symmetry, constrained to fixed total spin $J$ and magnetization $J_z = 0$ . Focusing on spin- $\tfrac12$ lattices and subsystem fractions $f < \frac{1}{2}$ , we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume $V$ . In addition to the expected leading volume law term, we prove the existence of a $\frac{1}{2}\log V$ finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the $O(1)$ 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 $\mathrm{SU}(2)$ symmetry shapes average entanglement.We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global $\mathrm{SU}(2)$ symmetry, constrained to fixed total spin $J$ and magnetization $J_z = 0$. Focusing on spin-$\tfrac12$ lattices and subsystem fractions $f < \frac{1}{2}$, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume $V$. In addition to the expected leading volume law term, we prove the existence of a $\frac{1}{2}\log V$ finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the $O(1)$ 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 $\mathrm{SU}(2)$ symmetry shapes average entanglement.

quant-pharXiv:2512.22942Dec 30, 20250 citations

We investigate bipartite entanglement in spin-

1∕2

systems on a generic lattice. For states that are an equal superposition of elements of a group

G

of spin flips acting on the fully polarized state ${\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems

A

and

B

. These states are stabilized by the group

G

. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus

\mathfrak{g}

. For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem

A

and is equal to the perimeter (up to an additive constant) when

A

is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.

We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global

\mathrm{SU}(2)

symmetry, constrained to fixed total spin

J

and magnetization

J_z = 0

. Focusing on spin-

\tfrac12

lattices and subsystem fractions

f < \frac{1}{2}

, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume

V

. In addition to the expected leading volume law term, we prove the existence of a

\frac{1}{2}\log V

finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the

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

\mathrm{SU}(2)

symmetry shapes average entanglement.