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:1110.3300 · Journal of Physics A Mathematical and Theoretical

Entanglement of disjoint blocks in the one-dimensional spin-1 VBS

Raul A Santos, V. E. Korepin

Published Mar 13, 2012• 13 citations
arXivHTMLPDF

Abstract

Starting with the valence-bond solid (VBS) ground state of the 1D AKLT Hamiltonian, we make a partition of the system in two subsystems A and B , where A is a block of L consecutive spins and B is its complement. In that setting we compute the partial transpose density matrix with respect to A , . We obtain the spectrum of the transposed density matrix of the VBS pure system. Subsequently, we define two disjoint blocks, A and B , containing L A and L B spins, respectively, separated by L sites. Tracing away the spins which do not belong to A ∪ B , we find an expression for the reduced density matrix of the A and B blocks, ρ( A , B ). With this expression (in the thermodynamic limit), we compute the entanglement spectrum and other several entanglement measures, such as the purity P = tr(ρ( A , B ) 2 ), the negativity and the mutual entropy.

DOI
10.1088/1751-8113/45/12/125307

Related papers

ENTANGLEMENT IN VALENCE-BOND-SOLID STATES

V. E. Korepin, YING XU

This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review, we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Rényi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.

arXiv:0908.2345•International Journal of Modern Physics B•Apr 30, 2010•30 citations

Entanglement of the valence-bond-solid state on an arbitrary graph

Ying Xu, V. E. Korepin

The Affleck–Kennedy–Lieb–Tasaki (AKLT) spin interacting model can be defined on an arbitrary graph. We explain the construction of the AKLT Hamiltonian. Given certain conditions, the ground state is unique and known as the valence-bond-solid (VBS) state. It can be used in measurement-based quantum computation as a resource state instead of the cluster state. We study the VBS ground state on an arbitrary connected graph. The graph is cut into two disconnected parts: the block and the environment. We study the entanglement between these two parts and prove that many eigenvalues of the density matrix of the block are zero. We describe a subspace of eigenvectors of the density matrix corresponding to non-zero eigenvalues. The subspace is the degenerate ground states of some Hamiltonian which we call the block Hamiltonian.

arXiv:0805.3542•Journal of Physics A Mathematical and Theoretical•Nov 3, 2008•6 citations

Entanglement and Density Matrix of a Block of Spins in AKLT Model

Ying Xu, Hosho Katsura, Takaaki Hirano +1 more

We study a 1-dimensional AKLT spin chain, consisting of spins S in the bulk and S/2 at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension (S + 1) 2. This subspace is described by nonzero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to ln (S + 1) 2.

•Nov 20, 2012•27 citations

Entanglement spectra ofq-deformed higher spin VBS states

Raul A Santos, Francis N. C. Paraan, V. E. Korepin +1 more

We calculate the reduced density matrix of a block of integer spin- S 's in a q -deformed valence-bond-solid (VBS) state. This matrix is diagonalized exactly for an infinitely long block in an infinitely long chain. We construct an effective Hamiltonian with the same spectrum as the logarithm of the density matrix. We also derive analytic expressions for the von Neumann and Rényi entanglement entropies. For blocks of finite length, we calculate the eigenvalues of the reduced density matrix by perturbation theory and numerical diagonalization. These results enable us to describe the effects of finite-size corrections on the entanglement spectrum and entropy in this generalized VBS model.

arXiv:1201.5927•Journal of Physics A Mathematical and Theoretical•Apr 12, 2012•15 citations

Entanglement in a Valence-Bond Solid State

Heng Fan, V. E. Korepin, Vwani Roychowdhury

We study entanglement in a valence-bond solid state, which describes the ground state of an Affleck-Kennedy-Lieb-Tasaki quantum spin chain, consisting of bulk spin-1's and two spin-1/2's at the ends. We characterize entanglement between various subsystems of the ground state by mostly calculating the entropy of one of the subsystems; when appropriate, we evaluate concurrences as well. We show that the reduced density matrix of a continuous block of bulk spins is independent of the size of the chain and the location of the block relative to the ends. Moreover, we show that the entanglement of the block with the rest of the sites approaches a constant value exponentially fast, as the size of the block increases. We also calculate the entanglement of (i) any two bulk spins with the rest, and (ii) the end spin-1/2's (together and separately) with the rest of the ground state.

arXiv:quant-ph/0406067•Physical Review Letters•Nov 22, 2004•134 citations

Entanglement spectra of the two-dimensional Affleck-Kennedy-Lieb-Tasaki model: Correspondence between the valence-bond-solid state and conformal field theory

Jie Lou, Shu Tanaka, Hosho Katsura +1 more

We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than $\mathrm{ln}2$. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.

arXiv:1107.3888•Physical Review B•Dec 27, 2011•47 citations
Starting with the valence-bond solid (VBS) ground state of the 1D AKLT Hamiltonian, we make a partition of the system in two subsystems A and B , where A is a block of L consecutive spins and B is its complement. In that setting we compute the partial transpose density matrix with respect to A , . We obtain the spectrum of the transposed density matrix of the VBS pure system. Subsequently, we define two disjoint blocks, A and B , containing L A and L B spins, respectively, separated by L sites. Tracing away the spins which do not belong to A ∪ B , we find an expression for the reduced density matrix of the A and B blocks, ρ( A , B ). With this expression (in the thermodynamic limit), we compute the entanglement spectrum and other several entanglement measures, such as the purity P = tr(ρ( A , B ) 2 ), the negativity and the mutual entropy.
This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review, we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Rényi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.
The Affleck–Kennedy–Lieb–Tasaki (AKLT) spin interacting model can be defined on an arbitrary graph. We explain the construction of the AKLT Hamiltonian. Given certain conditions, the ground state is unique and known as the valence-bond-solid (VBS) state. It can be used in measurement-based quantum computation as a resource state instead of the cluster state. We study the VBS ground state on an arbitrary connected graph. The graph is cut into two disconnected parts: the block and the environment. We study the entanglement between these two parts and prove that many eigenvalues of the density matrix of the block are zero. We describe a subspace of eigenvectors of the density matrix corresponding to non-zero eigenvalues. The subspace is the degenerate ground states of some Hamiltonian which we call the block Hamiltonian.
We study a 1-dimensional AKLT spin chain, consisting of spins S in the bulk and S/2 at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension (S + 1) 2. This subspace is described by nonzero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to ln (S + 1) 2.
We calculate the reduced density matrix of a block of integer spin- S 's in a q -deformed valence-bond-solid (VBS) state. This matrix is diagonalized exactly for an infinitely long block in an infinitely long chain. We construct an effective Hamiltonian with the same spectrum as the logarithm of the density matrix. We also derive analytic expressions for the von Neumann and Rényi entanglement entropies. For blocks of finite length, we calculate the eigenvalues of the reduced density matrix by perturbation theory and numerical diagonalization. These results enable us to describe the effects of finite-size corrections on the entanglement spectrum and entropy in this generalized VBS model.
We study entanglement in a valence-bond solid state, which describes the ground state of an Affleck-Kennedy-Lieb-Tasaki quantum spin chain, consisting of bulk spin-1's and two spin-1/2's at the ends. We characterize entanglement between various subsystems of the ground state by mostly calculating the entropy of one of the subsystems; when appropriate, we evaluate concurrences as well. We show that the reduced density matrix of a continuous block of bulk spins is independent of the size of the chain and the location of the block relative to the ends. Moreover, we show that the entanglement of the block with the rest of the sites approaches a constant value exponentially fast, as the size of the block increases. We also calculate the entanglement of (i) any two bulk spins with the rest, and (ii) the end spin-1/2's (together and separately) with the rest of the ground state.
We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than ln2\mathrm{ln}2ln2. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.