TY  - JOUR
A1  - Cramer, Marcus
A1  - Eisert, Jens
T1  - A quantum central limit theorem for non-equilibrium systems : exact local relaxation of correlated states
N2  - 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.
Y1  - 2010
UR  - http://iopscience.iop.org/1367-2630
U6  - https://doi.org/10.1088/1367-2630/12/5/055020
SN  - 1367-2630
ER  - 
TY  - JOUR
A1  - Eisert, Jens
A1  - Cramer, Marcus
A1  - Plenio, Martin B.
T1  - Colloquium : area laws for the entanglement entropy
N2  - 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.
Y1  - 2010
UR  - http://rmp.aps.org/
U6  - https://doi.org/10.1103/RevModPhys.82.277
SN  - 0034-6861
ER  -