Quantum Markov Chain Mixing and Dissipative Engineering
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Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups
1 Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups. Ivan Bardet∗, Marius Junge†, Nichcolas Laracuente†, Cambyse Rouzé‡, Daniel Stilck França§ ∗Institut National de Recherche en Informatique et en Automatique, Paris, France †Department of Mathematics University of Illinois at Urbana-Champaign Champaign, IL, USA ‡Faculty of Mathematics, Technische Universität München, Munich, Germany §Centre for the Mathematics of Quantum Theory, University of Copenhagen, Copenhagen, Denmark Abstract—Capacities of quantum channels and various information processing tasks is of central decoherence times both quantify the extent to which importance to the development of quantum technologies. quantum information can withstand degradation by interactions with its environment. However, calculating The dynamics of open systems are modeled capacities directly is known to be intractable in general. by completely positive trace preserving maps. In the Much recent work has focused on upper bounding certain Markovian approximation, continuous time evolutions capacities in terms of more tractable quantities such as are then modeled by quantum Markov semigroups specific norms from operator theory. In the meantime, there Tt t 0 of such maps. Given a concrete quantum Markov has also been substantial recent progress on estimating ( ) ≥ decoherence times with techniques from analysis and semigroup, it is then important to identify short time geometry, even though many hard questions remain open. versus long time behaviour of the evolution. For example, In this article, we introduce a class of continuous-time it is important to know how long entanglement can be quantum channels that we called transferred channels, preserved. This remains a challenging problem. Even which are built through representation theory from a for classical systems, precise decoherence time estimates classical Markov kernel defined on a compact group. -
Arxiv:0809.1487V2 [Cond-Mat.Stat-Mech] 25 Oct 2008 the first Two Authors Acknowledge Equal Contributions to This Review
Advances in Physics Vol. 00, No. 00, Month-Month 200x, 1–111 Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques P. B. Blakiea∗, A. S. Bradleya;b, M. J. Davisb, R. J. Ballagha, and C. W. Gardinera aJack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin, New Zealand; bThe University of Queensland, School of Physical Sciences, ARC Centre of Excellence for Quantum-Atom Optics, Qld 4072, Australia (Received 00 Month 200x; In final form 00 Month 200x) We review phase space techniques based on the Wigner representation that provide an approximate description of dilute ultra-cold Bose gases. In this approach the quantum field evolution can be represented using equations of motion of a similar form to the Gross-Pitaevskii equation but with stochastic modifications that include quantum effects in a controlled degree of approximation. These techniques provide a practical quantitative description of both equilibrium and dynamical properties of Bose gas systems. We develop versions of the formalism appropriate at zero temperature, where quantum fluctuations can be important, and at finite temperature where thermal fluctuations dominate. The numerical techniques necessary for implementing the formalism are discussed in detail, together with methods for extracting observables of interest. Numerous applications to a wide range of phenomena are presented. Keywords: Ultra-cold Bose gas, quantum and finite temperature dynamics. ∗Corresponding author. Email: [email protected] arXiv:0809.1487v2 [cond-mat.stat-mech] 25 Oct 2008 The first two authors acknowledge equal contributions to this review. 2 Contents 1. Introduction 4 2. Background formalism 7 2.1. -
Exact Calculation of Entanglement in a 19-Site 2D Spin System
Exact Calculation of Entanglement in a 19-site 2D Spin System Qing Xu,1 Sabre Kais∗,2 Maxim Naumov,3 and Ahmed Sameh3 1Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA 2Department of Chemistry and Birck Nanotechnology center, Purdue University, West Lafayette, Indiana 47907, USA 3Department of Computer Science, Purdue University, West Lafayette, Indiana 47907, USA (Dated: October 22, 2018) Using the Trace Minimization Algorithm, we carried out an exact calculation of entanglement in 1 a 19-site two-dimensional transverse Ising model. This model consists of a set of localized spin- 2 particles in a two dimensional triangular lattice coupled through exchange interaction J and subject to an external magnetic field of strength h. We demonstrate, for such a class of two-dimensional magnetic systems, that entanglement can be controlled and tuned by varying the parameter λ = h/J in the Hamiltonian and by introducing impurities into the systems. Examining the derivative of the concurrence as a function of λ shows that the system exhibits a quantum phase transition at about λc = 3.01, a transition induced by quantum fluctuations at the absolute zero of temperature. PACS numbers: 03.67.Mn I. INTRODUCTION Entanglement, which is a quantum mechanical property that has no classical analog, has been viewed as a resource of quantum information and computation [1–8]. Intensive researches of entanglement measurement, entanglement monotone, criteria for distinguishing separable from entangled pure states and all the extensions from bipartite to multipartite systems have been carried out both qualitatively and quantitatively [1]. At the interface between quantum information and statistical mechanics, there has been particular analysis of entanglement in quantum critical models [9–12]. -
Relativistic Fluid Dynamics
Living Reviews in Relativity (2021) 24:3 https://doi.org/10.1007/s41114-021-00031-6 REVIEW ARTICLE Relativistic fluid dynamics: physics for many different scales Nils Andersson1 · Gregory L. Comer2 Received: 1 June 2020 / Accepted: 22 January 2021 © The Author(s) 2021 Abstract The relativistic fluid is a highly successful model used to describe the dynam- ics of many-particle systems moving at high velocities and/or in strong gravity. It takes as input physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process—e.g., drawing on astrophysi- cal observations—an understanding of relativistic features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model sys- tems as “small” as colliding heavy ions in laboratory experiments, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theo- retical physics underpinnings of the relativistic (multi-) fluid model. We focus on the variational principle approach championed by Brandon Carter and collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book deriva- tion of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications This article is a revised version of https://doi.org/10.12942/lrr-2007-1. -
Cluster Mechanisms in the Fully Frustrated Ising Model
Cluster mechanisms in the fully frustrated Ising model Werner Kerler and Peter Rehberg Fachbereich Physik, Universit¨at Marburg, D-35032 Marburg, Germany Abstract The cluster algorithm in the fully frustrated Ising model on the square lattice is essentially different from the ones used in other systems. Thus its better understanding is particularly important for finding new lines of development. Therefore we investigate it in detail. In our simulations of high statistics more appropriate choices of the probability for freezing bonds and of that for flipping spins are seen to lead to better results. In an analysis of the arXiv:cond-mat/9401063v1 27 Jan 1994 topological properties we derive a set of rules for possible cluster configurations and give a classification by pairs of winding numbers. 1 I. Introduction Cluster algorithms, introduced by Swendsen and Wang [1] for ferromagnetic Potts models, allow a considerable reduction of critical slowing down in Monte Carlo simulations. Beginning with O(n)-models [2] and φ4-theory [3] various embeddings of the Ising dynamics have also been successful. In the ferromagnetic Ising model in two dimensions forming blocks of spins [4] has been shown to lead to an improvement. The treatments of Z2 gauge theories in three dimensions [5], which use plaquette variables, profit from the duality to the respective spin models. The combination of cluster dynamics with making the strength parameter dynamical recently has led to a very efficient method [6] to treat first order phase transitions in the Potts model. The algorithm for the fully frustrated Ising model on the square lattice in two dimensions found by Kandel, Ben-Av, and Domany [7, 8] is of rather different type. -
On the Relationship Between a Quantum Markov Semigroup and Its Representation Via Linear Stochastic Schrodinger¨ Equations
Indian J. Pure Appl. Math., 46(4): 399-414, August 2015 °c Indian National Science Academy DOI: 10.1007/s13226-015-0142-7 ON THE RELATIONSHIP BETWEEN A QUANTUM MARKOV SEMIGROUP AND ITS REPRESENTATION VIA LINEAR STOCHASTIC SCHRODINGER¨ EQUATIONS Franco Fagnola¤ and Carlos Mora¤¤ ¤Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy ¤¤CI2MA and Departamento de Ingenier´ıaMatem´atica, Universidad de Concepci´on, Barrio Universitario, Avenida Esteban Iturra s/n, 4089100, Casilla 160-C Concepci´on,Chile e-mails: [email protected], [email protected] Dedicated to Prof. Kalyan B. Sinha on occasion of his 70th birthday. (Received 7 May 2014; after final revision 24 September 2014; accepted 29 September 2014) A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schr¨odingerequation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, stronger than irreducibility of the quantum Markov semigroup, nevertheless, they are equivalent for some important classes of semigroups. Key words : Open quantum systems; quantum Markov semigroups; stochastic Schr¨odinger equations; irreducibility; support of quantum states; control. 1. Introduction ¤ A quantum Markov semigroup (QMS) T is a weakly -continuous semigroup (Tt)t¸0 of com- pletely positive, identity preserving, normal maps on a von Neumann algebra. -
BALANCE BETWEEN QUANTUM MARKOV SEMIGROUPS Contents
BALANCE BETWEEN QUANTUM MARKOV SEMIGROUPS ROCCO DUVENHAGE AND MACHIEL SNYMAN Abstract. The concept of balance between two state preserving quantum Markov semigroups is introduced and studied as an ex- tension of conditions appearing in the theory of quantum detailed balance. This is partly motivated by the theory of joinings. Basic properties of balance are derived and the connection to correspon- dences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical me- chanics, are brieflyexplored. Contents 1. Introduction 1 2. The definition of balance 3 3. Couplings and u.c.p. maps 7 4. A characterization of balance 12 5. Composition of couplings and transitivity of balance 17 6. Balance, detailed balance and non-equilibrium 24 7. An example 28 7.1. The couplings 29 7.2. The dynamics 31 7.3. Balance 34 7.4. A reversing operation 35 8. Further work 36 Acknowledgments 37 References 37 1. Introduction Motivated by quantum detailed balance, we define and study the notion of balance between pairs of quantum Markov semigroups on von Neumann algebras, where each semigroup preserves a faithful normal state. Ideas related to quantum detailed balance continue to play an important role in studying certain aspects of non-equilibrium statistical Date: 2017-7-18. Key words and phrases. balance; quantum Markov semigroups; completely pos- itive maps; quantum detailed balance; joinings; correspondences; non-equilibrium. 1 2 ROCCO DUVENHAGE AND MACHIEL SNYMAN mechanics, in particular non-equilibrium steady states. See for example [2], [3] and [5]. A theory of balance as introduced here is therefore potentially applicable to non-equilibrium statistical mechanics. -
Quantum Simulations of Out-Of-Equilibrium Phenomena
Technische Universitat¨ Munchen¨ Max-Planck–Institut f¨ur Quantenoptik Quantum Simulations of Out-of-Equilibrium Phenomena Birger Christian Helge Horstmann Vollst¨andiger Abdruck der von der Fakult¨at f¨ur Physik der Technischen Universit¨at M¨unchen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzende : Univ.-Prof. Dr. K. Krischer Pr¨ufer der Dissertation : 1. Hon.-Prof. I. Cirac, Ph.D. 2. Univ.-Prof. J. von Delft, Ph.D. 3. Ludwig-Maximilians-Universit¨at M¨unchen Die Dissertation wurde am 07.03.2011 bei der Technischen Universit¨at M¨unchen eingereicht und durch die Fakult¨at f¨ur Physik am 30.03.2011 angenommen. Abstract The quantum world contains many counterintuitive phenomena and trig- gers the development of new technologies. Unfortunately, in most cases classical numerical simulations of quantum systems are intractable. A quantum simulator addresses this problem by running the simulation on a quantum system. In this thesis we discuss three proposals for quantum simulators of the following out-of-equilibrium phenomena: Anderson lo- calization in disordered systems, dissipative quantum phase transitions, and Hawking radiation emitted by black holes. We propose to realize Anderson localization of ultracold atoms in an optical lattice with a disorder potential generated via a secondary species frozen in a superposition of random configurations. We investigate two preparation schemes: the independent preparation of both species and the transfer of a part of the population into a different internal state, for bosonic gases with infinite and weak intra-species interactions. In all setups we observe dynamical localization, i.e., the wavefunctions of the steady states feature exponential tails and reduced coherence. -
Particle Creation and Memory Effects in a Trapped Ion Quantum Simulator
Albert-Ludwigs-Universität Freiburg Particle creation and memory effects in a trapped ion quantum simulator Matthias Wittemer Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg 2019 Dekan: Prof. Dr. Wolfgang Soergel Betreuer und Erstgutachter: Prof. Dr. Tobias Schätz Zweitgutachter: apl. Prof. Dr. Bernd von Issendorff Tag der mündlichen Prüfung: 16. Dezember 2019 Prüfer: Prof. Dr. Marc Schumann apl. Prof. Dr. Thomas Filk Prof. Dr. Tobias Schätz Zusammenfassung Die fundamentalen Gesetze der Quantenmechanik sind von unseren Alltagserfahrun- gen weit entfernt. Sie sind jedoch hervorragend geeignet um Beobachtungen in mikro- skopisch kleinen physikalischen Systemen zu erklären und vorherzusagen und werden daher bereits für verschiedene Anwendungen genutzt. Darüber hinaus wird, in Kom- bination mit relativistischen Überlegungen, grundlegenden Phänomenen der Quan- tenmechanik eine zentrale Rolle in der Entwicklung unseres Universums zugespro- chen. In der vorliegenden Arbeit verwenden wir einen Quantensimulator basierend auf gefangenen Ionen, um zwei fundamentale quantenmechanische Effekte, die seit vielen Jahrzehnten umfangreich studiert werden, zu untersuchen. In unserer ersten Untersuchung implementieren wir nicht-adiabatische Änderungen des Fallenpotenti- als der Ionen, um Fluktuationen des Quantenvakuums zu verstärken. Theoretischen Arbeiten zufolge lässt sich dieser Mechanismus als Analogie zur Teilchenerzeugung während der inflationären Phase des jungen Universums interpretieren. Solche Teil- chen gelten als Keime für die Bildung der makroskopischen Strukturen unseres heu- tigen Universums. In unserer analogen Quantensimulation wird die Erzeugung dieser Teilchen durch die Anregung eines gequetschten Zustands in der Bewegung der Io- nen nachgewiesen. Ein bemerkenswertes Merkmal dieses Mechanismus ist die damit einhergehende Erzeugung einer Quantenverschränkung. Im Falle der kosmologischen Teilchen erstreckt sich diese Verschränkung über große, kosmische Entfernungen. -
Quantum Extensions of Dynamical Systems and of Markov Semigroups Ivan Bardet
Quantum extensions of dynamical systems and of Markov semigroups Ivan Bardet To cite this version: Ivan Bardet. Quantum extensions of dynamical systems and of Markov semigroups. 2015. hal- 01199552 HAL Id: hal-01199552 https://hal.archives-ouvertes.fr/hal-01199552 Preprint submitted on 16 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Quantum extensions of dynamical systems and of Markov semigroups∗ Ivan Bardet† Abstract We investigate some particular completely positive maps which admit a stable commu- tative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the exis- tence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in [1], reducing the problem to the extension of dynamical system. We then apply our method to the same problem in continuous time, proving the existence of a quantum extension for Lévy processes. In the second part of this article, we ∞ focus on the case where the commutative algebra is isomorphic to A = l (1, ..., N) with N either finite or infinite. -
Entropic Fluctuations of Quantum Dynamical Semigroups Vojkan Jaksic, Claude-Alain Pillet, Matthias Westrich
Entropic fluctuations of quantum dynamical semigroups Vojkan Jaksic, Claude-Alain Pillet, Matthias Westrich To cite this version: Vojkan Jaksic, Claude-Alain Pillet, Matthias Westrich. Entropic fluctuations of quantum dynamical semigroups. Journal of Statistical Physics, Springer Verlag, 2013, 154, pp.153-187. 10.1007/s10955- 013-0826-5. hal-00823948v2 HAL Id: hal-00823948 https://hal.archives-ouvertes.fr/hal-00823948v2 Submitted on 28 Aug 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Entropic fluctuations of quantum dynamical semigroups V. JAKŠIC´ a, C.-A. PILLETa,b, M. WESTRICHa aDepartment of Mathematics and Statistics McGill University 805 Sherbrooke Street West Montreal, QC, H3A 2K6, Canada b Aix-Marseille Université, CNRS, CPT, UMR 7332, Case 907, 13288 Marseille, France Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France FRUMAM tL Abstract. We study a class of finite dimensional quantum dynamical semigroups e t≥0 whose generators are sums of Lindbladians satisfying the detailed balance condition.{ Such} semigroups arise inL the weak coupling (van Hove) limit of Hamiltonian dynamical systems describing open quantum systems out of equilibrium. We prove a general entropic fluctuation theorem for this class of semigroups by relating the cumulant generating function of entropy transport to the spectrum of a family of deformations of the generator . -
2 Dec 2017 Relationships Between the Decoherence-Free Algebra And
Relationships between the decoherence-free algebra and the set of fixed points F. Fagnola1, E. Sasso2, V. Umanit`a2 Abstract We show that, for a Quantum Markov Semigroup (QMS) with a faith- ful normal invariant state, the atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we characterize the set of reversible states and explicitly describe the relationship between the decoherence-free subalgebra and the fixed point subalgebra for QMSs with the above equivalent properties. 1 Introduction Starting from the fundamental papers of Gorini-Kossakowski-Sudharshan [19] and Lindblad [23] the structure of uniformly continuous quantum Markov semigroups (QMS) =( t)t≥0, or, in the physical terminology, quantum dynamical semigroups, and theirT generators,T has been the object of significant attention. The increasing interest in mathematical modelling of decoherence, coherent quan- tum computing and approach to equilibrium in open quantum systems continues to motivate investigation on special features of QMS. Special attention is paid to sub- algebras or subspaces where irreversibility and dissipation disappear (see [2, 3, 7, 13, 14, 22, 29] and the references therein). States leaving in such subspaces are promising candidates for storing and manipulating quantum information. arXiv:1712.00637v1 [quant-ph] 2 Dec 2017 The decoherence-free subalgebra, where completely positive maps t of the semi- group act as automorphisms, and the set of fixed points, which is a subalgebraT when there exists a faithful invariant state, also allow us to gain insight into the structure of a QMS, its invariant states and environment induced decoherence. Indeed, its struc- ture as a von Neumann algebra, has important consequences on the structure and the action of the whole QMS.