@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}
}
@article{EisertCramerPlenio2010,
  author    = {Eisert, Jens and Cramer, Marcus and Plenio, Martin B.},
  title     = {Colloquium : area laws for the entanglement entropy},
  issn      = {0034-6861},
  doi       = {10.1103/RevModPhys.82.277},
  year      = {2010},
  abstract  = {Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such "area laws" for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.},
  language  = {en}
}
@article{EisertCramer2005,
  author    = {Eisert, Jens and Cramer, Marcus},
  title     = {Single-copy entanglement in critical quantum spin chains},
  year      = {2005},
  abstract  = {We consider the single-copy entanglement as a quantity to assess quantum correlations in the ground state in quantum many-body systems. We show for a large class of models that already on the level of single specimens of spin chains, criticality is accompanied with the possibility of distilling a maximally entangled state of arbitrary dimension from a sufficiently large block deterministically, with local operations and classical communication. These analytical results-which refine previous results on the divergence of block entropy as the rate at which maximally entangled pairs can be distilled from many identically prepared chains-are made quantitative for general isotropic translationally invariant spin chains that can be mapped onto a quasifree fermionic system, and for the anisotropic XY model. For the XX model, we provide the asymptotic scaling of similar to(1/6)log(2)(L), and contrast it with the block entropy},
  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{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{CramerEisert2010,
  author    = {Cramer, Marcus and Eisert, Jens},
  title     = {A quantum central limit theorem for non-equilibrium systems : exact local relaxation of correlated states},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/12/5/055020},
  year      = {2010},
  abstract  = {We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non- equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states-pure or mixed-which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly link our findings to the ideas of typicality and concentration of measure.},
  language  = {en}
}