Block spin density matrix of the inhomogeneous AKLT model

Xu, Ying; Katsura, Hosho; Hirano, Takaaki; Korepin, V. E.

doi:10.1007/s11128-008-0080-y

Cited by 5 publications

(10 citation statements)

References 83 publications

(189 reference statements)

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“…It was also proved in [82], [83] that in the large block limit N b → ∞, all eigenvalues become the same so that lim…”

Section: Examples Of the Density Matrix And Open Problemsmentioning

confidence: 90%

“…It is constructed by introducing the Schwinger boson representation [5], [28], [53], [49], [82], [83]. We define a pair of independent canonical bosonic operators a l and b l for each vertex l:…”

Section: The Generalized Aklt Modelmentioning

confidence: 99%

“…We discuss the density matrix for a single vertex block at the end of next section. It was shown for 1-dimensional models in [82], [83] that the spectrum of density matrix ρ b is closely related to the block Hamiltonian. The density matrix is a projector onto the ground space multiplied by another matrix.…”

Section: The Entanglement Between Block and Environmentmentioning

confidence: 99%

“…The density matrix of the block has been studied in [19], [49] and diagonalized directly in [82], [83] for 1-dimensional models, which illustrates the Theorem explicitly. It was shown for different 1-dimensional AKLT models that the inequality D ≤ deg.…”

Section: Examples Of the Density Matrix And Open Problemsmentioning

confidence: 99%

“…For a 1-dimensional AKLT spin chain the density matrix has a lot of zero eigenvalues [82], [83]. The eigenvectors with non-zero eigenvalues are the degenerate ground states of some Hamiltonian, which we shall call the block Hamiltonian (see (23)).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Xu¹,

Korepin²

2008

J. Phys. A: Math. Theor.

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Abstract. 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-BondSolid (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.

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“…It was also proved in [82], [83] that in the large block limit N b → ∞, all eigenvalues become the same so that lim…”

Section: Examples Of the Density Matrix And Open Problemsmentioning

confidence: 90%

Section: The Generalized Aklt Modelmentioning

confidence: 99%

Section: The Entanglement Between Block and Environmentmentioning

confidence: 99%

Section: Examples Of the Density Matrix And Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Xu¹,

Korepin²

2008

J. Phys. A: Math. Theor.

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Entanglement in valence-bond-solid states on symmetric graphs

Katsura¹,

Kawashima²,

Kirillov³

et al. 2010

J. Phys. A: Math. Theor.

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We study quantum entanglement in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model defined on two-dimensional graphs with reflection and/or inversion symmetry. The ground state of this spin model is known as the valence-bond-solid state. We investigate the properties of reduced density matrix of a subsystem which is a mirror image of the other one. Thanks to the reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is ln 2. This deviation is interpreted in terms of the hidden spin chain along the boundary between the subsystems. In some cases where graphs are on ladders, the numerical results are analytically or algebraically confirmed.

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Geometric entanglement in a one-dimensional valence-bond solid state

Orús

2008

Phys. Rev. A

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In this paper we provide the analytical derivation of the global geometric entanglement per block for the valence-bond solid ground state of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. In particular, we show that this quantity saturates exponentially fast to a constant when the sizes of the blocks are sufficiently large. Our result provides an example of an analytical calculation of the geometric entanglement for a gapped quantum many-body system in one dimension and far away from a quantum critical point.

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Block spin density matrix of the inhomogeneous AKLT model

Cited by 5 publications

References 83 publications

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

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

Entanglement in valence-bond-solid states on symmetric graphs

Geometric entanglement in a one-dimensional valence-bond solid state

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