Matrix product states represent ground states faithfully

We quantify how well matrix product states approximate exact ground states of one-dimensional quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms and justifies their use even in the case of critical systems.

[1]  Chen Ning Yang,et al.  One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System , 1966 .

[2]  Chen Ning Yang,et al.  One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I. Proof of Bethe's Hypothesis for Ground State in a Finite System , 1966 .

[3]  K. Wilson.  The renormalization group: Critical phenomena and the Kondo problem , 1975 .

[4]  R. Werner.  An application of Bell's inequalities to a quantum state extension problem , 1989 .

[5]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[6]  Andreas Schadschneider,et al.  Equivalence and solution of anisotropic spin-1 models and generalized t-J fermion models in one dimension , 1991 .

[7]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[8]  M. Fannes,et al.  Abundance of translation invariant pure states on quantum spin chains , 1992 .

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

[10]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[11]  Östlund,et al.  Thermodynamic limit of density matrix renormalization. , 1995, Physical review letters.

[12]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[13]  M. A. Martin-Delgado,et al.  Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains , 1998 .

[14]  Density-matrix spectra for integrable models , 1998, cond-mat/9810174.

[15]  Yasuhiro Hieida,et al.  Two-Dimensional Tensor Product Variational Formulation , 2001 .

[16]  D. D. Betts,et al.  Exact diagonalization and quantum Monte Carlo study of the spin-12 XXZ model on the square lattice , 2001 .

[17]  G. Vidal.  Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[18]  Barbara M Terhal,et al.  Symmetric extensions of quantum States and local hidden variable theories. , 2003, Physical review letters.

[19]  ENTANGLEMENT AND FRUSTRATION IN ORDERED SYSTEMS , 2003, quant-ph/0311051.

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

[21]  F. Verstraete,et al.  Entanglement frustration for Gaussian states on symmetric graphs. , 2003, Physical review letters.

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

[24]  M. Hastings,et al.  Locality in quantum and Markov dynamics on lattices and networks. , 2004, Physical review letters.

[25]  Mark R. Dowling,et al.  Energy as an entanglement witness for quantum many-body systems (15 pages) , 2004, quant-ph/0408086.

[26]  A. Winter,et al.  Randomizing Quantum States: Constructions and Applications , 2003, quant-ph/0307104.

[27]  Barbara M. Terhal.  Is entanglement monogamous? , 2004, IBM J. Res. Dev..

[28]  J. Ignacio Cirac,et al.  New frontiers in quantum information with atoms and ions , 2004 .

[29]  Adiabatic time evolution in spin systems , 2003, quant-ph/0309026.

[30]  F. Verstraete,et al.  Matrix product density operators: simulation of finite-temperature and dissipative systems. , 2004, Physical review letters.

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

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

[33]  U. Schollwoeck.  The density-matrix renormalization group , 2004, cond-mat/0409292.

[34]  J I Cirac,et al.  Renormalization-group transformations on quantum states. , 2004, Physical review letters.