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Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement. (English) Zbl 1312.82002

Summary: We study the projection on classical spins starting from quantum equilibria. We show Gibbsianness or quasi-locality of the resulting classical spin system for a class of gapped quantum systems at low temperatures including quantum ground states. A consequence of Gibbsianness is the validity of a large deviation principle in the quantum system which is known and here recovered in regimes of high temperature or for thermal states in one dimension. On the other hand, we give an example of a quantum ground state with strong nonlocality in the classical restriction, giving rise to what we call measurement induced entanglement and still satisfying a large deviation principle.{
©2015 American Institute of Physics}

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B30 Statistical thermodynamics
81P40 Quantum coherence, entanglement, quantum correlations
60F10 Large deviations
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[1] Affleck, I.; Kennedy, T.; Lieb, E.; Tasaki, H., Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett., 57, 799-802 (1987)
[2] Araki, H.; Matsui, T., Ground states of the XY-model, Commun. Math. Phys., 101, 213-245 (1985) · Zbl 0581.46058
[3] Borgs, C.; Koteckỳ, R.; Ueltschi, D., Low temperature phase diagrams for quantum perturbations of classical spin systems, Commun. Math. Phys., 181, 409-446 (1996) · Zbl 0858.60097
[4] Bratteli, O.; Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics 2 (1996)
[5] Calabrese, P.; Essler, F.; Fagotti, M., Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators, J. Stat. Mech., 2012, 7, P07016
[6] Cherng, R.; Demler, E., Quantum noise analysis of spin systems realized with cold atoms, New J. Phys., 9, 7 (2007)
[7] Datta, N.; Fernàndez, R.; Fröhlich, J., Low-temperature phase diagrams of quantum lattice systems: I. stability for quantum perturbations of classical systems with finitely many ground states, J. Stat. Phys., 84, 455-534 (1996) · Zbl 1081.82506
[8] Dembo, A.; Zeitouni, O., Large Deviation Techniques and Applications (1993)
[9] De Roeck, W.; Maes, C.; Netočný, K., Quantum macrostates, equivalence of ensembles and an H-theorem, J. Math. Phys., 47, 073303 (2006) · Zbl 1112.82005
[10] Van Enter, A. C. D.; Fernández, R.; Sokal, A. D., Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory, J. Stat. Phys., 72, 879-1167 (1993) · Zbl 1101.82314
[11] Ellis, R., Large Deviations and Statistical Mechanics (1985) · Zbl 0567.60031
[12] Fannes, M.; Nachtergaele, B.; Werner, R., Finitely correlated states on quantum spin chains, Commun. Math. Phys., 144, 443-490 (1992) · Zbl 0755.46039
[13] Fannes, M.; Nachtergaele, B.; Werner, R., Abundance of translation invariant pure states on quantum spin chains, Lett. Math. Phys., 25, 249-258 (1992) · Zbl 0779.46047
[14] Fernández, R., Gibbsianness and non-Gibbsianness in lattice random fields, Mathematical Statistical Physics (2006) · Zbl 1458.82009
[15] Georgii, H.-O., Gibbs Measures and Phase Transitions (1988) · Zbl 0657.60122
[16] Deift, P.; Its, A.; Krasovsky, I., Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results, Comm. Pure Appl. Math., 66, 1360-1438 · Zbl 1292.47016
[17] Kotecký, R.; Preis, D., Cluster expansion for abstract polymer models, Commun. Math. Phys., 103, 491-498 (1986) · Zbl 0593.05006
[18] Kozlov, O. K., Gibbs description of a system of random variables, Probl. Inf. Transm., 10, 258-265 (1974)
[19] Lanford, O. E. III, Entropy and equilibrium states in classical statistical mechanics, Statistical Mechanics and Mathematical Problems. Battelle Seattle Rencontres 1971, 20, 1-113 (1973)
[20] Lenci, M.; Rey-Bellet, L., Large deviations in quantum lattice systems: One-phase region, J. Stat. Phys., 119, 715-746 (2005) · Zbl 1170.82307
[21] Matsui, T., Uniqueness of the translationally invariant ground state in quantum spin systems, Commun. Math. Phys., 126, 453-467 (1990) · Zbl 0691.46055
[22] Netočný, K.; Redig, F., Large deviations for quantum spin systems, J. Stat. Phys., 117, 521-547 (2004) · Zbl 1113.82008
[23] Ogata, Y., Large deviations in quantum spin chains, Commun. Math. Phys., 296, 35-68 (2010) · Zbl 1193.82007
[24] Ogata, Y.; Rey-Bellet, L., Ruelle-Lanford functions and large deviations for asymptotically decoupled quantum systems, Reviews in Mathematical Physics, 23, 2, 211-232 (2011) · Zbl 1220.82010
[25] Pfister, C.-E., Thermodynamical aspects of classical lattice systems, In and Out of Equilibrium, 3939-3472 (2002)
[26] Popp, M.; Verstraete, F.; Cirac, J. I., Entanglement versus Correlations in Spin Systems, Phys. Rev. Lett., 92, 027901 (2004)
[27] Ruelle, D., Thermodynamic Formalism (2004)
[28] Sachdev, S., Quantum Phase Transitions (2001)
[29] Simon, B., The Statistical Mechanics of Lattice Gases (1993)
[30] Sullivan, W. G., Potentials for almost Markovian random fields, Commun. Math. Phys., 33, 61-74 (1973) · Zbl 0267.60108
[31] Ueltschi, D., Cluster expansions and correlation functions, Moscow Math. J., 4, 511-522 (2004) · Zbl 1070.82002
[32] Verstraete, F.; Martín-Delgado, M.; Cirac, J., Diverging entanglement length in gapped quantum spin systems, Phys. Rev. Lett., 92, 087201 (2004)
[33] Wahl, T.; Pérez-García, D.; Cirac, J., Matrix product states with long-range localizable entanglement, Phys. Rev. A, 86, 062314 (2012)
[34] Yarotsky, D. A., Uniqueness of the ground state in weak perturbations of non-interacting gapped quantum lattice systems, J. Stat. Phys., 118, 119-144 (2005) · Zbl 1130.82007
[35] Yarotski, D. A., Ground states in relatively bounded quantum perturbations of classical lattice systems, Commun. Math. Phys., 261, 799-819 (2005) · Zbl 1113.82015
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