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J. Eisert; V. Eisler; Z. Zimborás
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The entanglement negativity is a versatile measure of entanglement that has numerous applications in quantum information and in condensed matter theory. It can not only efficiently be computed in the Hilbert space dimension, but for noninteracting bosonic systems, one can compute the negativity efficiently in the number of modes. However, such an efficient computation does not carry over to the fermionic realm, the ultimate reason for this being that the partial transpose of a fermionic...
Topics: Quantum Physics, Condensed Matter, Other Condensed Matter
Source: http://arxiv.org/abs/1611.08007
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M. Friesdorf; A. H. Werner; M. Goihl; J. Eisert; W. Brown
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Interacting quantum manybody systems are usually expected to thermalise, in the sense that the evolution of local expectation values approach a stationary value resembling a thermal ensemble. This intuition is notably contradicted in systems exhibiting manybody localisation, a phenomenon receiving significant recent attention. One of its most intriguing features is that, in stark contrast to the noninteracting case, entanglement of states grows without limit over time, albeit slowly. In this...
Topics: Quantum Physics, Mathematics, Mathematical Physics, Disordered Systems and Neural Networks,...
Source: http://arxiv.org/abs/1412.5605
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M. Gluza; M. Kliesch; J. Eisert; L. Aolita
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The experimental interest in realizing quantum spin1/2chains has increased uninterruptedly over the last decade. In many instances, the target quantum simulation belongs to the broader class of noninteracting fermionic models, constituting an important benchmark. In spite of this class being analytically efficiently tractable, no direct certification tool has yet been reported for it. In fact, in experiments, certification has almost exclusively relied on notions of quantum state tomography...
Topics: Condensed Matter, Quantum Physics, Other Condensed Matter
Source: http://arxiv.org/abs/1703.03152
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D. Gross; J. Eisert; N. Schuch; D. PerezGarcia
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We introduce novel schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in [Phys. Rev. Lett. 98, 220503 (2007), quantph/0609149]. Our method makes use of tools from manybody physics  matrix product states, finitely correlated states or projected entangled pairs states  to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where...
Source: http://arxiv.org/abs/0706.3401v1
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J. Eisert; K. Jacobs; P. Papadopoulos; M. B. Plenio
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We investigate the minimal resources that are required in the local implementation of nonlocal quantum gates in a distributed quantum computer. Both classical communication requirements and entanglement consumption are investigated. We present general statements on the minimal resource requirements and present optimal procedures for a number of important gates, including CNOT and Toffoli gates. We show that one bit of classical communication in each direction is both necessary and sufficient...
Source: http://arxiv.org/abs/quantph/0005101v2
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M. B. Plenio; J. Eisert; J. Dreissig; M. Cramer
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We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free KleinGordon fields. For the ground state of the ddimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of...
Source: http://arxiv.org/abs/quantph/0405142v3
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H. Wilming; M. J. Kastoryano; A. H. Werner; J. Eisert
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A cornerstone of the theory of phase transitions is the observation that manybody systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analogue of such a theory. Specifically, we consider local dissipative dynamics preparing a steadystate of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive...
Topics: Quantum Physics, Other Condensed Matter, Statistical Mechanics, Condensed Matter, Mathematics,...
Source: http://arxiv.org/abs/1602.01108
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K. Kieling; J. Eisert
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This article is a draft of a book chapter of the book entitled "Quantum Percolation and Breakdown", to appear 2008.
Source: http://arxiv.org/abs/0712.1836v1
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The density matrix renormalization group (DMRG) approach is arguably the most successful method to numerically find ground states of quantum spin chains. It amounts to iteratively locally optimizing matrixproduct states, aiming at better and better approximating the true ground state. To date, both a proof of convergence to the globally best approximation and an assessment of its complexity are lacking. Here we establish a result on the computational complexity of an approximation with...
Source: http://arxiv.org/abs/quantph/0609051v3
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D. Gross; K. Audenaert; J. Eisert
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We clarify the mathematical structure underlying unitary $t$designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any $t$th order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2designs are derived. We show how to turn mutually unbiased bases into approximate 2designs...
Source: http://arxiv.org/abs/quantph/0611002v2
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J. Eisert; T. Tyc; T. Rudolph; B. C. Sanders
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The quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system. This setting, also referred to as the question of the compatibility of local spectra, has several applications in quantum information theory. Here, we introduce the analogue of this statement for Gaussian states for any number of modes, and solve it in generality, for pure and mixed states, both concerning necessary and sufficient conditions. Formally, our...
Source: http://arxiv.org/abs/quantph/0703225v2
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J. Eisert; P. Hyllus; O. Guehne; M. Curty
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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...
Source: http://arxiv.org/abs/quantph/0407135v3
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J. Eisert; M. M. Wolf
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This article provides an elementary introduction to Gaussian channels and their capacities. We review results on the classical, quantum, and entanglement assisted capacities and discuss related entropic quantities as well as additivity issues. Some of the known results are extended. In particular, it is shown that the quantum conditional entropy is maximized by Gaussian states and that some implications for additivity problems can be extended to the Gaussian setting.
Source: http://arxiv.org/abs/quantph/0505151v1
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M. Paternostro; D. Vitali; S. Gigan; M. S. Kim; C. Brukner; J. Eisert; M. Aspelmeyer
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We describe a scheme showing signatures of macroscopic optomechanical entanglement generated by radiation pressure in a cavity system with a massive movable mirror. The system we consider reveals genuine multipartite entanglement. We highlight the way the entanglement involving the inaccessible massive object is unravelled, in our scheme, by means of fieldfield quantum correlations.
Source: http://arxiv.org/abs/quantph/0609210v1
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In this paper, the problem of finding optimal success probabilities of static linear optics quantum gates is linked to the theory of convex optimization. It is shown that by exploiting this link, upper bounds for the success probability of networks realizing singlemode gates can be derived, which hold in generality for linear optical networks followed by postselection, i.e., for networks of arbitrary size, any number of auxiliary modes, and arbitrary photon numbers. As a corollary, the...
Source: http://arxiv.org/abs/quantph/0409156v4
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K. Kieling; T. Rudolph; J. Eisert
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We apply a notion of static renormalization to the preparation of entangled states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in the context of optical architectures, where probabilistic gates are common, and cold atoms in optical lattices, where hole defects occur. We demonstrate how to efficiently construct cluster states without the need for...
Source: http://arxiv.org/abs/quantph/0611140v3
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M. Ohliger; V. Nesme; J. Eisert
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We present a novel method to perform quantum state tomography for manyparticle systems which are particularly suitable for estimating states in lattice systems such as of ultracold atoms in optical lattices. We show that the need for measuring a tomographically complete set of observables can be overcome by letting the state evolve under some suitably chosen random circuits followed by the measurement of a single observable. We generalize known results about the approximation of unitary...
Source: http://arxiv.org/abs/1204.5735v2
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H. Bernigau; M. J. Kastoryano; J. Eisert
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We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel framework for computing determinants of Toeplitz operators with smooth symbols, and for treating Toeplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the onedimensional lattice satisfies an area law, with a prefactor...
Source: http://arxiv.org/abs/1301.5646v1
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J. Goold; C. Gogolin; S. R. Clark; J. Eisert; A. Scardicchio; A. Silva
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The intriguing phenomenon of manybody localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work, we introduce the total correlations, a concept from quantum information theory capturing multipartite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pindown, probe, and better understand the MBL transition and ergodicity...
Topics: Condensed Matter, Quantum Physics, Disordered Systems and Neural Networks
Source: http://arxiv.org/abs/1504.06872
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M. M. Wolf; J. Eisert; T. S. Cubitt; J. I. Cirac
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We investigate what a snapshot of a quantum evolution  a quantum channel reflecting open system dynamics  reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of `Markovianity' is introduced. We discuss how...
Source: http://arxiv.org/abs/0711.3172v2
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K. Audenaert; J. Eisert; E. Jane; M. B. Plenio; S. Virmani; B. De Moor
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We present an analytical formula for the asymptotic relative entropy of entanglement for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. To our knowledge, this is the first case in which the value of a subadditive entanglement measure has been obtained in the asymptotic limit.
Source: http://arxiv.org/abs/quantph/0103096v2
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M. PerarnauLlobet; H. Wilming; A. Riera; R. Gallego; J. Eisert
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We derive general limitations concerning efficiency and power of heat engines strongly coupled to thermal baths. We build this framework on the insight that quantum systems strongly coupled to manybody systems will equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weakcoupling limit. Taking this observation as the starting point of our analysis, we first provide strongcoupling corrections to the second law in...
Topics: Condensed Matter, Quantum Physics, Other Condensed Matter
Source: http://arxiv.org/abs/1704.05864
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A. Steffens; P. Rebentrost; I. Marvian; J. Eisert; S. Lloyd
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We develop an efficient quantum implementation of an important signal processing algorithm for line spectral estimation: the matrix pencil method, which determines the frequencies and damping factors of signals consisting of finite sums of exponentially damped sinusoids. Our algorithm provides a quantum speedup in a natural regime where the sampling rate is much higher than the number of sinusoid components. Along the way, we develop techniques that are expected to be useful for other quantum...
Topic: Quantum Physics
Source: http://arxiv.org/abs/1609.08170
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A. H. Werner; D. Jaschke; P. Silvi; M. Kliesch; T. Calarco; J. Eisert; S. Montangero
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Open manybody quantum systems play an important role in quantum optics and condensedmatter physics, and capture phenomena like transport, interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate onedimensional open quantum manybody dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is...
Topics: Quantum Physics, Strongly Correlated Electrons, Statistical Mechanics, Condensed Matter
Source: http://arxiv.org/abs/1412.5746
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S. Braun; M. Friesdorf; S. S. Hodgman; M. Schreiber; J. P. Ronzheimer; A. Riera; M. del Rey; I. Bloch; J. Eisert; U. Schneider
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The dynamics of quantum phase transitions poses one of the most challenging problems in modern manybody physics. Here, we study a prototypical example in a clean and wellcontrolled ultracold atom setup by observing the emergence of coherence when crossing the Mott insulator to superfluid quantum phase transition. In the onedimensional BoseHubbard model, we find perfect agreement between experimental observations and numerical simulations for the resulting coherence length. We thereby...
Topics: Quantum Gases, Quantum Physics, Strongly Correlated Electrons, Statistical Mechanics, Condensed...
Source: http://arxiv.org/abs/1403.7199
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M. Cramer; S. Ospelkaus; C. Ospelkaus; K. Bongs; K. Sengstock; J. Eisert
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Mixtures of bosonic and fermionic atoms in optical lattices provide a promising arena to study strongly correlated systems. In experiments realizing such mixtures in the quantum degenerate regime the temperature is a key parameter. In this work, we investigate the intrinsic heating and cooling effects due to an entropypreserving raising of the optical lattice potential. We analyze this process, identify the generic behavior valid for a wide range of parameters, and discuss it quantitatively...
Source: http://arxiv.org/abs/0707.3633v1
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M. M. Wolf; J. Eisert; M. B. Plenio
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We investigate the entangling capability of passive optical elements, both qualitatively and quantitatively. We present a general necessary and sufficient condition for the possibility of creating distillable entanglement in an arbitrary multimode Gaussian state with the help of passive optical elements, thereby establishing a general connection between squeezing and the entanglement that is attainable by nonsqueezing operations. Special attention is devoted to general twomode Gaussian...
Source: http://arxiv.org/abs/quantph/0206171v2
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G. A. Durkin; C. Simon; J. Eisert; D. Bouwmeester
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We analyze the resilience under photon loss of the bipartite entanglement present in multiphoton states produced by parametric downconversion. The quantification of the entanglement is made possible by a symmetry of the states that persists even under polarizationindependent losses. We examine the approach of the states to the set of states with a positive partial transpose as losses increase, and calculate the relative entropy of entanglement. We find that some bipartite distillable...
Source: http://arxiv.org/abs/quantph/0402053v3
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S. C. Benjamin; J. Eisert; T. M. Stace
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We present a scheme for rapidly entangling matter qubits in order to create graph states for oneway quantum computing. The qubits can be simple 3level systems in separate cavities. Coupling involves only local fields and a static (unswitched) linear optics network. Fusion of graph state sections occurs with, in principle, zero probability of damaging the nascent graph state. We avoid the finite thresholds of other schemes by operating on two entangled pairs, so that each generates exactly one...
Source: http://arxiv.org/abs/quantph/0506110v5
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H. B. ColdenstrodtRonge; J. S. Lundeen; A. Feito; B. J. Smith; W. Mauerer; Ch. Silberhorn; J. Eisert; M. B. Plenio; I. A. Walmsley
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Measurement is the only part of a general quantum system that has yet to be characterized experimentally in a complete manner. Detector tomography provides a procedure for doing just this; an arbitrary measurement device can be fully characterized, and thus calibrated, in a systematic way without access to its components or its design. The result is a reconstructed POVM containing the measurement operators associated with each measurement outcome. We consider two detectors, a singlephoton...
Source: http://arxiv.org/abs/0902.4384v1
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C. Krumnow; L. Veis; Ö. Legeza; J. Eisert
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Tensor network states and specifically matrixproduct states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chemistry. A new freedom arising in such nonlocal fermionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this work, we propose a way to overcome this challenge. We...
Topics: Condensed Matter, Quantum Physics, Strongly Correlated Electrons, Chemical Physics, Physics
Source: http://arxiv.org/abs/1504.00042
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M. J. Kastoryano; M. M. Wolf; J. Eisert
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Dissipative engineering constitutes a framework within which quantum information processing protocols are powered by systemenvironment interaction rather than by unitary dynamics alone. This framework embraces noise as a resource, and consequently, offers a number of advantages compared to one based on unitary dynamics alone, e.g., that the protocols are typically independent of the initial state of the system. However, the time independent nature of this scheme makes it difficult to imagine...
Source: http://arxiv.org/abs/1205.0985v2
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L. Zhang; A. Datta; H. B. ColdenstrodtRonge; X. M. Jin; J. Eisert; M. B. Plenio; I. A. Walmsley
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Conventional tomographic techniques are becoming increasingly infeasible for reconstructing the operators of quantum devices of growing sophistication. We describe a novel tomographic procedure using coherent states which begins by reconstructing the diagonals of the operator, and then each successive offdiagonal in a recursive manner. Each recursion is considerably more efficient than reconstructing the operator in its entirety, and each successive recursion involves fewer parameters. We...
Source: http://arxiv.org/abs/1207.3501v3
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H. M. Wiseman; J. Eisert
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Invited contribution to "Quantum Aspects of Life", D. Abbott Ed. (World Scientific, Singapore, 2007).
Source: http://arxiv.org/abs/0705.1232v2
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K. Audenaert; J. Eisert; M. B. Plenio; R. F. Werner
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We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We consider the ground state and thermal states of this system, which are Gaussian states. The entanglement properties of these states can be completely characterized analytically when one uses the logarithmic negativity as a measure of entanglement.
Source: http://arxiv.org/abs/quantph/0205025v2
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J. Eisert; M. M. Wolf
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This article gives an elementary introduction to quantum computing. It is a draft for a book chapter of the "Handbook of NatureInspired and Innovative Computing", Eds. A. Zomaya, G.J. Milburn, J. Dongarra, D. Bader, R. Brent, M. EshaghianWilner, F. Seredynski (Springer, Berlin Heidelberg New York, 2006).
Source: http://arxiv.org/abs/quantph/0401019v3
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M. Arndt; M. Aspelmeyer; H. J. Bernstein; R. Bertlmann; C. Brukner; J. P. Dowling; J. Eisert; A. Ekert; C. A. Fuchs; D. M. Greenberger; M. A. Horne; T. Jennewein; P. G. Kwiat; N. D. Mermin; J. W. Pan; E. M. Rasel; H. Rauch; T. G. Rudolph; C. Salomon; A. V. Sergienko; J. Schmiedmayer; C. Simon; V. Vedral; P. Walther; G. Weihs; P. Zoller; M. Zukowski
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This is a collection of statements gathered on the occasion of the Quantum Physics of Nature meeting in Vienna.
Source: http://arxiv.org/abs/quantph/0505187v4
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J. Gertis; M. Friesdorf; C. A. Riofrio; J. Eisert
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Ultracold atoms in optical lattices provide one of the most promising platforms for analog quantum simulations of complex quantum manybody systems. Largesize systems can now routinely be reached and are already used to probe a large variety of different physical situations, ranging from quantum phase transitions to artificial gauge theories. At the same time, measurement techniques are still limited and full tomography for these systems seems out of reach. Motivated by this observation, we...
Topics: Quantum Physics, Condensed Matter, Quantum Gases
Source: http://arxiv.org/abs/1606.01913
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M. PerarnauLlobet; A. Riera; R. Gallego; H. Wilming; J. Eisert
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Recent years have seen an enormously revived interest in the study of thermodynamic notions in the quantum regime. This applies both to the study of notions of work extraction in thermal machines in the quantum regime, as well as to questions of equilibration and thermalisation of interacting quantum manybody systems as such. In this work we bring together these two lines of research by studying work extraction in a closed system that undergoes a sequence of quenches and equilibration steps...
Topics: Quantum Physics, Statistical Mechanics, Quantum Gases, Condensed Matter
Source: http://arxiv.org/abs/1512.03823
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L. Aolita; C. Gogolin; M. Kliesch; J. Eisert
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A major roadblock for largescale photonic quantum technologies is the lack of practical reliable certification tools. We introduce an experimentally friendly  yet mathematically rigorous  certification test for experimental preparations of arbitrary mmode pure Gaussian states, pure nonGaussian states generated by linearoptical circuits with nboson Fockbasis states as inputs, and states of these two classes subsequently postselected with local measurements on ancillary modes. The...
Topic: Quantum Physics
Source: http://arxiv.org/abs/1407.4817
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K. Kieling; J. L. O'Brien; J. Eisert
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As primitives for entanglement generation, controlled phase gates take a central role in quantum computing. Especially in ideas realizing instances of quantum computation in linear optical gate arrays a closer look can be rewarding. In such architectures, all effective nonlinearities are induced by measurements: Hence the probability of success is a crucial parameter of such quantum gates. In this note, we discuss this question for controlled phase gates that implement an arbitrary phase with...
Source: http://arxiv.org/abs/0909.2057v4
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J. Eisert; S. Scheel; M. B. Plenio
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We show that no distillation protocol for Gaussian quantum states exists that relies on (i) arbitrary local unitary operations that preserve the Gaussian character of the state and (ii) homodyne detection together with classical communication and postprocessing by means of local Gaussian unitary operations on two symmetric identically prepared copies. This is in contrast to the finitedimensional case, where entanglement can be distilled in an iterative protocol using two copies at a time. The...
Source: http://arxiv.org/abs/quantph/0204052v4
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J. Eisert; M. Wilkens
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In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented in [J. Eisert, M. Wilkens, and M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)]. After introducing a general framework, we study quantum games with a classical analogue in order to flesh out the peculiarities of game theoretical settings in the quantum domain. Special emphasis is given to a detailed investigation of different sets of...
Source: http://arxiv.org/abs/quantph/0004076v1
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M. B. Plenio; J. Hartley; J. Eisert
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We study the entanglement dynamics of a system consisting of a large number of coupled harmonic oscillators in various configurations and for different types of nearest neighbour interactions. For a onedimensional chain we provide compact analytical solutions and approximations to the dynamical evolution of the entanglement between spatially separated oscillators. Key properties such as the speed of entanglement propagation, the maximum amount of transferred entanglement and the efficiency for...
Source: http://arxiv.org/abs/quantph/0402004v2
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R. Orus; J. I. Latorre; J. Eisert; M. Cramer
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We establish that the leading critical scaling of the singlecopy entanglement is exactly one half of the entropy of entanglement of a block in critical infinite spin chains in a general setting, using methods of conformal field theory. Conformal symmetry imposes that the singlecopy entanglement for critical manybody systems scales as E_1(\rho_L)=(c/6) \log L (c/6) (\pi^2/\log L) + O(1/L), where L is the number of constituents in a block of an infinite chain and c corresponds to the central...
Source: http://arxiv.org/abs/quantph/0509023v3
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D. Gross; J. Eisert
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We establish a framework which allows one to construct novel schemes for measurementbased quantum computation. The technique further develops tools from manybody physics  based on finitely correlated or projected entangled pair states  to go beyond the clusterstate based oneway computer. We identify resource states that are radically different from the cluster state, in that they exhibit nonvanishing correlation functions, can partly be prepared using gates with nonmaximal entangling...
Source: http://arxiv.org/abs/quantph/0609149v3
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J. S. Lundeen; A. Feito; H. ColdenstrodtRonge; K. L. Pregnell; Ch. Silberhorn; T. C. Ralph; J. Eisert; M. B. Plenio; I. A. Walmsley
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Measurement connects the world of quantum phenomena to the world of classical events. It plays both a passive role, observing quantum systems, and an active one, preparing quantum states and controlling them. Surprisingly  in the light of the central status of measurement in quantum mechanics  there is no general recipe for designing a detector that measures a given observable. Compounding this, the characterization of existing detectors is typically based on partial calibrations or elaborate...
Source: http://arxiv.org/abs/0807.2444v1
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N. de Beaudrap; M. Ohliger; T. J. Osborne; J. Eisert
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We identify a large class of quantum manybody systems that can be solved exactly: natural frustrationfree spin1/2 nearestneighbor Hamiltonians on arbitrary lattices. We show that the entire ground state manifold of such models can be found exactly by a tensor network of isometries acting on a space locally isomorphic to the symmetric subspace. Thus, for this wide class of models realspace renormalization can be made exact. Our findings also imply that every such frustrationfree spin model...
Source: http://arxiv.org/abs/1005.3781v1
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M. Cramer; J. Eisert; M. B. Plenio; J. Dreissig
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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 system of arbitrary spatial dimension, for general finiteranged harmonic interactions, regions of arbitrary shape and states of nonzero temperature. For nearestneighbor...
Source: http://arxiv.org/abs/quantph/0505092v3
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D. Gross; J. Eisert
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We present a simple way to quantize the wellknown Margulis expander map. The result is a quantum expander which acts on discrete Wigner functions in the same way the classical Margulis expander acts on probability distributions. The quantum version shares all essential properties of the classical counterpart, e.g., it has the same degree and spectrum. Unlike previous constructions of quantum expanders, our method does not rely on nonAbelian harmonic analysis. Analogues for continuous variable...
Source: http://arxiv.org/abs/0710.0651v2