Block spin density matrix of the inhomogeneous AKLT model

We study the inhomogeneous generalization of a 1-dimensional AKLT spin chain model. Spins at each lattice site could be different. Under certain conditions, the ground state of this AKLT model is unique and is described by the Valence-Bond-Solid (VBS) state. We calculate the density matrix of a contiguous block of bulk spins in this ground state. The density matrix is independent of spins outside the block. It is diagonalized and shown to be a projector onto a subspace. We prove that for large block the density matrix behaves as the identity in the subspace. The von Neumann entropy coincides with Renyi entropy and is equal to the saturated value.

[1]  Bipartite entanglement and entropic boundary law in lattice spin systems (10 pages) , 2004, quant-ph/0409073.

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

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

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

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

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

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

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

[9]  Román Orús,et al.  Universal geometric entanglement close to quantum phase transitions. , 2007, Physical review letters.

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

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

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

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

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

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

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

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

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

[19]  David P. DiVincenzo,et al.  Quantum information and computation , 2000, Nature.

[20]  Vladimir E. Korepin,et al.  Entanglement and Density Matrix of a Block of Spins in AKLT Model , 2008, 0802.3221.

[21]  Quantum phase transitions in antiferromagnetic planar cubic lattices , 2004, quant-ph/0401144.

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

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

[24]  Dimitri Gioev,et al.  Entanglement entropy of fermions in any dimension and the Widom conjecture. , 2006, Physical review letters.

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

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

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

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

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

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

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

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

[33]  Michael M Wolf.  Violation of the entropic area law for fermions. , 2006, Physical review letters.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[62]  W Dür,et al.  Entanglement in spin chains and lattices with long-range Ising-type interactions. , 2005, Physical review letters.

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

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

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

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

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

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

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

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

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

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

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

[74]  Half the entanglement in critical systems is distillable from a single specimen , 2005, quant-ph/0509023.

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

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

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

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