@article{CramerEisert2006,
  author    = {Cramer, Marcus and Eisert, Jens},
  title     = {Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/8/5/071},
  year      = {2006},
  abstract  = {We investigate the relationship between the gap between the energy of the ground state and the first excited state and the decay of correlation functions in harmonic lattice systems. We prove that in gapped systems, the exponential decay of correlations follows for both the ground state and thermal states. Considering the converse direction, we show that an energy gap can follow from algebraic decay and always does for exponential decay. The underlying lattices are described as general graphs of not necessarily integer dimension, including translationally invariant instances of cubic lattices as special cases. Any local quadratic couplings in position and momentum coordinates are allowed for, leading to quasi-free ( Gaussian) ground states. We make use of methods of deriving bounds to matrix functions of banded matrices corresponding to local interactions on general graphs. Finally, we give an explicit entanglement-area relationship in terms of the energy gap for arbitrary, not necessarily contiguous regions on lattices characterized by general graphs},
  language  = {en}
}
@article{CramerEisertPlenioetal.2006,
  author    = {Cramer, Marcus and Eisert, Jens and Plenio, Martin B. and Dreißig, Julian},
  title     = {Entanglement-area law for general bosonic harmonic lattice systems},
  doi       = {10.1103/Physreva.73.012309},
  year      = {2006},
  abstract  = {We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic-lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in noncritical harmonic-lattice systems of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape, and states of nonzero temperature. For nearest-neighbor interactions-corresponding to the Klein-Gordon case-upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of Phys. Rev. Lett. 94, 060503 (2005). These higher-dimensional analogs of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block- banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two- point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement},
  language  = {en}
}
@article{OrusLatorreEisertetal.2006,
  author    = {Orus, Roman and Latorre, Jose Ignacio and Eisert, Jens and Cramer, Marcus},
  title     = {Half the entanglement in critical systems is distillable from a single specimen},
  doi       = {10.1103/Physreva.73.060303},
  year      = {2006},
  abstract  = {We establish a quantitative relationship between the entanglement content of a single quantum chain at a critical point and the corresponding entropy of entanglement. We find that, surprisingly, the leading critical scaling of the single-copy entanglement with respect to any bipartitioning is exactly one-half of the entropy of entanglement, in a general setting of conformal field theory and quasifree systems. Conformal symmetry imposes that the single-copy entanglement scales as E-1(rho(L))=(c/6)ln L-(c/6)(pi(2)/ln L)+O(1/L), where L is the number of constituents in a block of an infinite chain and c denotes the central charge. This shows that from a single specimen of a critical chain, already half the entanglement can be distilled compared to the rate that is asymptotically available. The result is substantiated by a quantitative analysis for all translationally invariant quantum spin chains corresponding to all isotropic quasifree fermionic models. An example of the XY spin chain shows that away from criticality the above relation is maintained only near the quantum phase transition},
  language  = {en}
}