@article{EisertPlenio1999, author = {Eisert, Jens and Plenio, M. B.}, title = {A comparison of entanglement measures}, year = {1999}, abstract = {We compare the entanglement of formation with a measure defined as the modulus of the negative eigenvalue of the partial transpose. In particular we investigate whether both measures give the same ordering of density perators with respect to the amount of entanglement.}, 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} } @article{GogolinMuellerEisert2011, author = {Gogolin, Christian and M{\"u}ller, Markus P. and Eisert, Jens}, title = {Absence of thermalization in nonintegrable systems}, series = {Physical review letters}, volume = {106}, journal = {Physical review letters}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.106.040401}, pages = {4}, year = {2011}, abstract = {We establish a link between unitary relaxation dynamics after a quench in closed many-body systems and the entanglement in the energy eigenbasis. We find that even if reduced states equilibrate, they can have memory on the initial conditions even in certain models that are far from integrable. We show that in such situations the equilibrium states are still described by a maximum entropy or generalized Gibbs ensemble, regardless of whether a model is integrable or not, thereby contributing to a recent debate. In addition, we discuss individual aspects of the thermalization process, comment on the role of Anderson localization, and collect and compare different notions of integrability.}, language = {en} } @article{EisertWilkens2000, author = {Eisert, Jens and Wilkens, Martin}, title = {Catlysis of Entanglement Manipulation for Mixed States}, year = {2000}, abstract = {We consider entanglement-assisted remote quantum state manipulation of bipartite mixed states. Several aspects are addressed: we present a class of mixed states of rank two that can be transformed into another class of mixed states under entanglement-assisted local operations with classical communication, but for which such a transformation is impossible without assistance. Furthermore, we demonstrate enhancement of the efficiency of purification protocols with the help of entanglement-assisted operations. Finally, transformations from one mixed state to mixed target states which are sufficiently close to the source state are contrasted with similar transformations in the pure-state case.}, language = {en} } @article{EisertFelbingerPapadopolousetal.2000, author = {Eisert, Jens and Felbinger, Timo and Papadopolous, P. and Plenio, M. B. and Wilkens, Martin}, title = {Classical information and distillable entanglement}, year = {2000}, abstract = {We establish a quantitative connection between the amount of lost classical information about a quantum state and the concomitant loss of entanglement. Using menthods that have been developed for the optimal purification of miced states, we find a class of miced states with known distillable entanglement. These results can be used to determine the quantum capacity of a quantum channel which randomizes the order of transmitted signals.}, language = {en} } @article{WolfEisert2005, author = {Wolf, M. M. and Eisert, Jens}, title = {Classical information capacity of a class of quantum channels}, issn = {1367-2630}, year = {2005}, abstract = {We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. For this class of channels, the norm of the output is maximized for the output being a normalized projection. We prove the additivity of the minimal output Renyi entropies with entropic parameters alpha is an element of [ 0, 2], generalizing an argument by Alicki and Fannes, and present a number of examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify various instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated}, 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{EisertHyllusGuhneetal.2004, author = {Eisert, Jens and Hyllus, P. and Guhne, O. and Curty, M.}, title = {Complete hierarchies of efficient approximations to problems in entanglement theory}, year = {2004}, abstract = {We investigate several problems in entanglement theory from the perspective of convex optimization. This list of problems comprises (A) the decision whether a state is multiparty entangled, (B) the minimization of expectation values of entanglement witnesses with respect to pure product states, (C) the closely related evaluation of the geometric measure of entanglement to quantify pure multiparty entanglement, (D) the test whether states are multiparty entangled on the basis of witnesses based on second moments and on the basis of linear entropic criteria, and (E) the evaluation of instances of maximal output purities of quantum channels. We show that these problems can be formulated as certain optimization problems: as polynomially constrained problems employing polynomials of degree 3 or less. We then apply very recently established known methods from the theory of semidefinite relaxations to the formulated optimization problems. By this construction we arrive at a hierarchy of efficiently solvable approximations to the solution, approximating the exact solution as closely as desired, in a way that is asymptotically complete. For example, this results in a hierarchy of efficiently decidable sufficient criteria for multiparticle entanglement, such that every entangled state will necessarily be detected in some step of the hierarchy. Finally, we present numerical examples to demonstrate the practical accessibility of this approach}, language = {en} } @article{MuellerGrossEisert2011, author = {M{\"u}ller, Markus P. and Gross, David and Eisert, Jens}, title = {Concentration of Measure for Quantum States with a Fixed Expectation Value}, series = {Communications in mathematical physics}, volume = {303}, journal = {Communications in mathematical physics}, number = {3}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-011-1205-1}, pages = {785 -- 824}, year = {2011}, abstract = {Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors vertical bar psi >> that have a fixed expectation value < psi vertical bar H vertical bar psi > = E with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.}, language = {en} } @article{BarthelPinedaEisert2009, author = {Barthel, Thomas and Pineda, Carlos and Eisert, Jens}, title = {Contraction of fermionic operator circuits and the simulation of strongly correlated fermions}, issn = {1050-2947}, doi = {10.1103/Physreva.80.042333}, year = {2009}, abstract = {A fermionic operator circuit is a product of fermionic operators of usually different and partially overlapping support. Further elements of fermionic operator circuits (FOCs) are partial traces and partial projections. The presented framework allows for the introduction of fermionic versions of known qudit operator circuits (QUOC), important for the simulation of strongly correlated d-dimensional systems: the multiscale entanglement renormalization ansaumltze (MERA), tree tensor networks (TTN), projected entangled pair states (PEPS), or their infinite-size versions (iPEPS etc.). After the definition of a FOC, we present a method to contract it with the same computation and memory requirements as a corresponding QUOC, for which all fermionic operators are replaced by qudit operators of identical dimension. A given scheme for contracting the QUOC relates to an analogous scheme for the corresponding fermionic circuit, where additional marginal computational costs arise only from reordering of modes for operators occurring in intermediate stages of the contraction. Our result hence generalizes efficient schemes for the simulation of d- dimensional spin systems, as MERA, TTN, or PEPS to the fermionic case.}, language = {en} }