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

AbstractWe 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 $\left(S+1\right)^{2}$ . This subspace is described by non-zero 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 $\ln\left(S+1\right)^{2}$ .

[1]  Pieter Kok.  Quantum information and computation web-corner , 2001, Quantum Inf. Comput..

[2]  P. Zanardi,et al.  Sublattice entanglement and quantum phase transitions in antiferromagnetic spin chains , 2006 .

[3]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[4]  F. Haldane.  Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model , 1983 .

[5]  F. Franchini,et al.  Ellipses of constant entropy in the XY spin chain , 2006, quant-ph/0609098.

[6]  J J García-Ripoll,et al.  Implementation of spin Hamiltonians in optical lattices. , 2004, Physical review letters.

[7]  Sergei Ketov.  Nonlinear Sigma model , 2009, Scholarpedia.

[8]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[9]  V. Korepin,et al.  Universality of entropy scaling in one dimensional gapless models. , 2003, Physical Review Letters.

[10]  Shi-Jian Gu,et al.  Entanglement and quantum phase transition in the extended Hubbard model. , 2004, Physical review letters.

[11]  M Roncaglia,et al.  Long-distance entanglement in spin systems. , 2006, Physical review letters.

[12]  Paolo Zanardi,et al.  Holonomic quantum computation , 1999 .

[13]  J. Latorre,et al.  Universality of entanglement and quantum-computation complexity , 2003, quant-ph/0311017.

[14]  M. Rasetti,et al.  Spin network quantum simulator , 2002, quant-ph/0209016.

[15]  V. Vedral,et al.  Entanglement in Many-Body Systems , 2007, quant-ph/0703044.

[16]  V. Roychowdhury,et al.  Entanglement in a valence-bond solid state. , 2004, Physical review letters.

[17]  J Eisert,et al.  Entropy, entanglement, and area: analytical results for harmonic lattice systems. , 2005, Physical review letters.

[18]  Quantum entanglement and the self-trapping transition in polaronic systems , 2004, quant-ph/0407080.

[19]  A. Auerbach.  Interacting electrons and quantum magnetism , 1994 .

[20]  E. Lieb,et al.  A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state , 1988 .

[21]  R.Ionicioiu,et al.  Ground state entanglement and geometric entropy in the Kitaev's model , 2004, quant-ph/0406202.

[22]  C. Hadley.  Single-copy entanglement in a gapped quantum spin chain. , 2008, Physical review letters.

[23]  M. Fannes,et al.  Finitely correlated states on quantum spin chains , 1992 .

[24]  D Porras,et al.  Density matrix renormalization group and periodic boundary conditions: a quantum information perspective. , 2004, Physical review letters.

[25]  Vladimir E. Korepin,et al.  Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture , 2004 .

[26]  Auerbach,et al.  Extended Heisenberg models of antiferromagnetism: Analogies to the fractional quantum Hall effect. , 1988, Physical review letters.

[27]  J I Cirac,et al.  Diverging entanglement length in gapped quantum spin systems. , 2004, Physical review letters.

[28]  Paolo Zanardi,et al.  Ground state entanglement and geometric entropy in the Kitaev model , 2005 .

[29]  J I Cirac,et al.  Entanglement versus correlations in spin systems. , 2004, Physical review letters.

[30]  E. Lieb,et al.  Valence bond ground states in isotropic quantum antiferromagnets , 1988 .

[31]  J. Eisert,et al.  Entanglement-area law for general bosonic harmonic lattice systems (14 pages) , 2005, quant-ph/0505092.

[32]  V. Roychowdhury,et al.  Boundary Effects to Entropy and Entanglement of the Spin-1 Valence-Bond Solid , 2006 .

[33]  F. Verstraete,et al.  Valence-bond states for quantum computation , 2003, quant-ph/0311130.

[34]  Roman Orus,et al.  Adiabatic quantum computation and quantum phase transitions , 2004 .

[35]  R. Laughlin.  Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations , 1983 .

[36]  V. Korepin,et al.  Entanglement in the XY spin chain , 2004 .

[37]  V. Vedral.  Quantum physics: Entanglement hits the big time , 2003, Nature.

[38]  G. Aeppli,et al.  Entangled quantum state of magnetic dipoles , 2003, Nature.

[39]  M. Hamermesh.  Group theory and its application to physical problems , 1962 .

[40]  J. Eisert,et al.  Geometric Entropy in Harmonic Lattice Systems , 2004 .

[41]  M. Hammermesh,et al.  Group theory and its applications to physical problems , 1989 .

[42]  K. Audenaert,et al.  Entanglement properties of the harmonic chain , 2002, quant-ph/0205025.

[43]  M. Nielsen,et al.  Entanglement in a simple quantum phase transition , 2002, quant-ph/0202162.

[44]  F. Arecchi,et al.  Atomic coherent states in quantum optics , 1972 .

[45]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[46]  S Lloyd,et al.  A Potentially Realizable Quantum Computer , 1993, Science.

[47]  Statistics dependence of the entanglement entropy. , 2006, Physical review letters.

[48]  S. Lloyd.  Envisioning a quantum supercomputer. , 1994, Science.

[49]  Martin B Plenio,et al.  Three-spin interactions in optical lattices and criticality in cluster Hamiltonians. , 2004, Physical review letters.

[50]  Complete analysis of two spin correlations of valence bond solid chains for all integer spins , 1991 .

[51]  F. Franchini,et al.  Renyi entropy of the XY spin chain , 2007, 0707.2534.

[52]  Kennedy,et al.  Rigorous results on valence-bond ground states in antiferromagnets. , 1987, Physical review letters.

[53]  S. Lloyd,et al.  Entanglement and off-diagonal long-range order of an η-pairing state , 2005 .

[54]  José Ignacio Latorre,et al.  Ground state entanglement in quantum spin chains , 2004, Quantum Inf. Comput..

[55]  F. Mezzadri,et al.  Random Matrix Theory and Entanglement in Quantum Spin Chains , 2004, quant-ph/0407047.

[56]  Y. Hatsugai,et al.  Entanglement entropy and the Berry phase in the solid state , 2006, cond-mat/0601237.

[57]  J. Cardy,et al.  Entanglement entropy and quantum field theory , 2004, hep-th/0405152.

[58]  Scaling of entanglement at a quantum phase transition for a two-dimensional array of quantum dots , 2004, quant-ph/0405087.

[59]  Vlatko Vedral,et al.  High-temperature macroscopic entanglement , 2004, quant-ph/0405102.

[60]  SCALING OF ENTANGLEMENT IN FINITE ARRAYS OF EXCHANGE-COUPLED QUANTUM DOTS , 2003, quant-ph/0405085.

[61]  F. Wilczek,et al.  Geometric and renormalized entropy in conformal field theory , 1994, hep-th/9403108.

[62]  H. Briegel,et al.  2D multipartite valence bond states in quantum anti-ferromagnets , 2007, 0710.2349.

[63]  Y. Hatsugai,et al.  Entanglement Entropy of One-dimensional Gapped Spin Chains(Condensed matter: structure and mechanical and thermal properties) , 2007, cond-mat/0703642.

[64]  P. Zanardi,et al.  Entanglement and Quantum Phase Transition in Low Dimensional Spin Systems , 2004, quant-ph/0407228.

[65]  Heng Fan,et al.  Boundary effects on entropy and two-site entanglement of the spin-1 valence-bond solid , 2007 .

[66]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[67]  J. Latorre,et al.  Entanglement entropy in the Lipkin-Meshkov-Glick model (4 pages) , 2004, cond-mat/0409611.

[68]  Vladimir E. Korepin,et al.  Entanglement in an SU(n) valence-bond-solid state , 2007, 0711.3882.

[69]  G. Vidal,et al.  Entanglement in quantum critical phenomena. , 2002, Physical review letters.

[70]  Hosho Katsura,et al.  Exact analysis of entanglement in gapped quantum spin chains , 2007, cond-mat/0702196.

[71]  Shi-Jian Gu,et al.  Entanglement, quantum phase transition, and scaling in the XXZ chain , 2003 .

[72]  A. Osterloh,et al.  Scaling of entanglement close to a quantum phase transition , 2002, Nature.

[73]  Entanglement and majorization in (1+1)-dimensional quantum systems , 2005 .

[74]  J. Eisert,et al.  Dynamics and manipulation of entanglement in coupled harmonic systems with many degrees of freedom , 2004, quant-ph/0402004.

[75]  S. Bose,et al.  Natural thermal and magnetic entanglement in the 1D Heisenberg model. , 2000, Physical review letters.

[76]  F. Haldane.  Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State , 1983 .

[77]  S Iblisdir,et al.  Entropy and exact matrix-product representation of the Laughlin wave function. , 2007, Physical review letters.