Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0409073 · Physical Review A

Bipartite entanglement and entropic boundary law in lattice spin systems

Alioscia Hamma, Radu Ionicioiu, Paolo Zanardi A

Published Feb 22, 2005• 296 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.71.022315

Related papers

Quantum entanglement in states generated by bilocal group algebras

Alioscia Hamma, Radu Ionicioiu, Paolo Zanardi A

Given a finite group $G$ with a bilocal representation, we investigate the bipartite entanglement in the state constructed from the group algebra of $G$ acting on a separable reference state. We find an upper bound for the von Neumann entropy for a bipartition $(A,B)$ of a quantum system and conditions to saturate it. We show that these states can be interpreted as ground states of generic Hamiltonians or as the physical states in a quantum gauge theory and that under specific conditions their geometric entropy satisfies the entropic area law. If $G$ is a group of spin flips acting on a set of qubits, these states are locally equivalent to 2-colorable (i.e., bipartite) graph states and they include Greenberger-Horne-Zeilinger, cluster states, etc. Examples include an application to qudits and a calculation of the $n$-tangle for 2-colorable graph states.

arXiv:quant-ph/0504049•Physical Review A•Jul 20, 2005•24 citations

Exact results for the bipartite entanglement entropy of the AKLT spin-1 chain

Scott D Geraedts, Erik Schwartz Sørensen

We study the bipartite entanglement between a subsystem of size l and the rest of the system of total size L as it occurs in a spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) chain subject to open boundary conditions. In this case, the ground-state manifold is four-fold degenerate and there is strong dependence on the parity of the number of spins, L . We present exact analytical results for the von Neumann entanglement entropy, as a function of both the size of the subsystem, l , and the total system size, L , for all four degenerate ground states for both odd and even L . In the large l , L limits the entanglement entropy approaches ln (2) for the S z T = ±1 while it approaches twice that value, 2ln (2), for the S z T = 0 states. In all cases, it is found that this constant is approached exponentially fast defining a length scale ξ = 1/ln (3) equal to the known bulk correlation length.

arXiv:0912.4733•Journal of Physics A Mathematical and Theoretical•Apr 16, 2010•10 citations

Entanglement in a second-order topological insulator on a square lattice

Qiang Wang, Da Wang, Qiang-Hua Wang

In a d -dimensional topological insulator of order d , there are zero-energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero-energy states caused by the same physical corners. We then considered finite-size systems in which cases these corner states are coupled together by long-range hybridizations to form multipartite entangled states. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.

arXiv:1808.01085•Europhysics Letters (EPL)•Dec 28, 2018•19 citations

Comb entanglement in quantum spin chains

Jonathan P. Keating, Francesco Mezzadri, Marcel Novaes

Bipartite entanglement in the ground state of a chain of $N$ quantum spins can be quantified either by computing pairwise concurrence or by dividing the chain into two complementary subsystems. In the latter case the smaller subsystem is usually a single spin or a block of adjacent spins and the entanglement differentiates between critical and noncritical regimes. Here we extend this approach by considering a more general setting: our smaller subsystem ${S}_{A}$ consists of a comb of $L$ spins, spaced $p$ sites apart. Our results are thus not restricted to a simple area law, but contain nonlocal information, parametrized by the spacing $p$. For the $XX$ model we calculate the von Neumann entropy analytically when $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and investigate its dependence on $L$ and $p$. We find that an external magnetic field induces an unexpected length scale for entanglement in this case.

arXiv:quant-ph/0604016•Physical Review A•Jul 18, 2006•24 citations

Entanglement in low dimensional systems

Raul A. Santos

In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages

•Jul 23, 2013•0 citations

Exact description of the boundary theory of the Kitaev toric code with open boundary conditions

Yevheniia Cheipesh, Lorenzo Cevolani, Stefan K. Kehrein

In this paper, we consider the Kitaev toric code with specific open boundary conditions. Such a physical system has a highly degenerate ground state determined by the degrees of freedom localized at the boundaries. We can write down an explicit expression for the ground state of this model. Based on this, the entanglement properties of the model are studied for two types of bipartition: one, where subsystem $A$ is completely contained in $B$; and the second, where the boundary of the system is shared between $A$ and $B$. In the former configuration, the entanglement entropy is the same as for the periodic boundary condition case, which means that the bulk is completely decoupled from the boundary on distances larger than the correlation length. In the latter, deviations from the torus configuration appear due to the edge states and lead to an increase of the entropy. We then determine an effective theory for the boundary of the system. In the case where we apply a small magnetic field as a perturbation, the degrees of freedom on the boundary acquire a dispersion relation. The system can there be described by a Hamiltonian of the Ising type with a generic spin-exchange term.

arXiv:1803.10217•Physical review. B./Physical review. B•Jan 22, 2019•8 citations
We investigate bipartite entanglement in spin-1∕21∕21∕2 systems on a generic lattice. For states that are an equal superposition of elements of a group GGG of spin flips acting on the fully polarized state \ensuremath∣0⟩\ensuremath⨂n{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}\ensuremath∣0⟩\ensuremath⨂n, we find that the von Neumann entropy depends only on the boundary between the two subsystems AAA and BBB. These states are stabilized by the group GGG. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus g\mathfrak{g}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 AAA and is equal to the perimeter (up to an additive constant) when AAA 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.
Given a finite group GGG with a bilocal representation, we investigate the bipartite entanglement in the state constructed from the group algebra of GGG acting on a separable reference state. We find an upper bound for the von Neumann entropy for a bipartition (A,B)(A,B)(A,B) of a quantum system and conditions to saturate it. We show that these states can be interpreted as ground states of generic Hamiltonians or as the physical states in a quantum gauge theory and that under specific conditions their geometric entropy satisfies the entropic area law. If GGG is a group of spin flips acting on a set of qubits, these states are locally equivalent to 2-colorable (i.e., bipartite) graph states and they include Greenberger-Horne-Zeilinger, cluster states, etc. Examples include an application to qudits and a calculation of the nnn-tangle for 2-colorable graph states.
We study the bipartite entanglement between a subsystem of size l and the rest of the system of total size L as it occurs in a spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) chain subject to open boundary conditions. In this case, the ground-state manifold is four-fold degenerate and there is strong dependence on the parity of the number of spins, L . We present exact analytical results for the von Neumann entanglement entropy, as a function of both the size of the subsystem, l , and the total system size, L , for all four degenerate ground states for both odd and even L . In the large l , L limits the entanglement entropy approaches ln (2) for the S z T = ±1 while it approaches twice that value, 2ln (2), for the S z T = 0 states. In all cases, it is found that this constant is approached exponentially fast defining a length scale ξ = 1/ln (3) equal to the known bulk correlation length.
In a d -dimensional topological insulator of order d , there are zero-energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero-energy states caused by the same physical corners. We then considered finite-size systems in which cases these corner states are coupled together by long-range hybridizations to form multipartite entangled states. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.
Bipartite entanglement in the ground state of a chain of NNN quantum spins can be quantified either by computing pairwise concurrence or by dividing the chain into two complementary subsystems. In the latter case the smaller subsystem is usually a single spin or a block of adjacent spins and the entanglement differentiates between critical and noncritical regimes. Here we extend this approach by considering a more general setting: our smaller subsystem SA{S}_{A}SA​ consists of a comb of LLL spins, spaced ppp sites apart. Our results are thus not restricted to a simple area law, but contain nonlocal information, parametrized by the spacing ppp. For the XXXXXX model we calculate the von Neumann entropy analytically when $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and investigate its dependence on LLL and ppp. We find that an external magnetic field induces an unexpected length scale for entanglement in this case.
In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages
In this paper, we consider the Kitaev toric code with specific open boundary conditions. Such a physical system has a highly degenerate ground state determined by the degrees of freedom localized at the boundaries. We can write down an explicit expression for the ground state of this model. Based on this, the entanglement properties of the model are studied for two types of bipartition: one, where subsystem AAA is completely contained in BBB; and the second, where the boundary of the system is shared between AAA and BBB. In the former configuration, the entanglement entropy is the same as for the periodic boundary condition case, which means that the bulk is completely decoupled from the boundary on distances larger than the correlation length. In the latter, deviations from the torus configuration appear due to the edge states and lead to an increase of the entropy. We then determine an effective theory for the boundary of the system. In the case where we apply a small magnetic field as a perturbation, the degrees of freedom on the boundary acquire a dispersion relation. The system can there be described by a Hamiltonian of the Ising type with a generic spin-exchange term.