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Open Quantum Systems and Quantum Information Dynamics UniversitatÄ Ulm Institut fÄurTheoretische Physik Open Quantum Systems and Quantum Information Dynamics Angel¶ Rivas Vargas Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der FakultÄatfÄur Naturwissenschaften der UniversitÄatUlm January 2011 Amtierender Dekan Prof. Dr. Axel Gro¼ Erstgutachter Prof. Dr. Martin B. Plenio Zweitgutachter Prof. Dr. Joachim Ankerhold Tag der Promotion 16. MÄarz2011 Dedication A mis abuelos, F¶elix,Gertrudis, Carlos y Luisa. Et si habuero prophetiam, et noverim mysteria omnia, et omnem scientiam, [:::] caritatem autem non habuero, nihil sum. 1 Cor 13, 2. Abstract Just like other theories of physics, quantum theory was developed to explain some experimental facts that were incomprehensible within the previously existing theo- ries (e.g. the black-body radiation or the photoelectric e®ect). After the Planck's introduction of the concept of quanta at the beginning of the XX century [1], many physical situations have been explained by means of quantum mechanics, ranging from the behavior of elementary particles to the operational mechanism of stars. Nowadays, about 100 years from its foundation, a lot of e®ort is being made in try- ing to exploit the genuine features of quantum systems as a resource to solve a wide variety of problems, such as the secure transmission of information, its processing in a more e±cient way or to make extremely accurate measurements of physical quantities. To carry out this task we require an unprecedented control in the fabrication and manipulation of devices operating in the quantum regime. However, quantum systems are much more fragile in the presence of noise than their classical coun- terparts, and relevant quantum states allowing novel forms to store, transmit and process information vanish very easily. Actually, this is the ultimate reason why we cannot see systems exhibiting quantum behavior in our daily life. As a result, the study of the properties of quantum systems in the presence of noise is a fundamental issue in order to develop any form of reliable quantum technology. Noise in physical systems can be explained theoretically using the concept of \open system". Noise arises due to a lack of insulation, in such a way that it is the result of an e®ective interaction between our system and its environment. In this thesis, we present new results concerning the dynamics of open quantum systems. We propose to de¯ne rigourously the concept of quantum Markov processes and introduce quantitative, novel, ways to detect deviations from strict Markovianity. We explore the validity of the usual techniques employed to describe simple open quantum systems, and their extrapolation to more complicated scenarios involving interacting and driven systems. In addition we present examples of noise assisted processes, and transformations to make these usually tricky problems more e±ciently tractable. In the ¯rst part of this thesis we extensively review the general theory of open quantum systems. We introduce new approaches to describe, in a hopefully more transparent way, some aspects which are usually misleading in a broad range of the literature. In addition, we revise new concepts and methods developed in recent years that complement the general theory mainly developed in the 1970s. In the second part, we focus our attention on the so-called Markovian processes which have been described in the previous part. We study the validity of typical equations used in simple systems such as harmonic oscillators, and propose propose and check di®erent strategies to describe Markovian dynamics in more complicated interacting systems. Finally we propose a counterintuitive situation where some level of noise can be actually useful for information tasks. In the third and last part, we approach the problem of the non-Markovian de- scription of open quantum systems. To quantify how non-Markovian an open quan- tum system is we introduce a computable measure and apply it to some relevant situations. We also propose a way of probing the complexity of an environment by visualizing properties of a pair of qubits embedded in it. In addition, we develop a technique based on orthogonal polynomial theory to transform a wide range of environments into simple chains of particles which can be e±ciently simulated with numerical techniques. Contents Acknowledgments 13 1 Introduction 15 I Time Evolution Theory of Open Quantum Systems 19 2 Mathematical tools 21 2.1 Banach spaces, norms and linear operators . 21 2.2 Exponential of an operator . 22 2.3 Semigroups of operators . 25 2.3.1 Contraction semigroups . 27 2.4 Evolution families . 30 3 Quantum time evolutions 35 3.1 Time evolution in closed quantum systems . 35 3.2 Time evolution in open quantum systems. 37 3.2.1 Dynamical maps . 39 3.2.2 Universal dynamical maps . 42 3.2.3 Universal dynamical maps as contractions . 44 3.2.4 The inverse of a universal dynamical map . 46 3.2.5 Temporal continuity. Markovian evolutions . 47 3.3 Quantum Markov process: mathematical structure . 50 3.3.1 Classical Markovian processes . 50 3.3.2 Quantum Markov evolution as a di®erential equation . 51 3.3.3 Kossakowski conditions . 55 3.3.4 Steady states of homogeneous Markov processes . 58 4 Microscopic derivations 65 4.1 Markovian case . 65 4.1.1 Nakajima-Zwanzig equation . 66 7 4.1.2 Weak coupling limit . 69 4.1.3 Singular coupling limit . 86 4.1.4 Extensions of the weak coupling limit . 89 4.2 Non-Markovian case . 93 4.2.1 Integro-di®erential models . 94 4.2.2 Time-convolutionless forms . 95 4.2.3 Dynamical coarse graining method . 97 II Interacting Systems under Markovian Dynamics 101 5 Composite systems of harmonic oscillators 103 5.1 A closer look into the damped harmonic oscillator . 103 5.1.1 Gaussian states . 103 5.1.2 Damped harmonic oscillator . 106 5.2 Markovian master equations for interacting systems . 116 5.2.1 Two coupled damped harmonic oscillators . 117 5.2.2 Driven damped harmonic oscillator . 125 5.2.3 Some conclusions . 132 6 Stochastic resonance phenomena in spin chains 135 6.1 Introduction . 135 6.2 Steady state entanglement in qubit chains subject to longitudinal decoherence (pure decay) . 136 6.3 XXZ Heisenberg interaction . 139 6.4 Steady state entanglement under transverse decoherence (pure de- phasing) . 140 6.5 SR phenomena under both longitudinal and transverse decoherence . 142 6.6 Conclusion . 144 III Identifying Elements of non-Markovianity 147 7 Measures of non-Markovianity 149 7.1 Introduction . 149 7.2 Previous attempts . 149 7.2.1 Proposal of Wolf et. al. 150 7.2.2 Proposal of Breuer et. al. 150 7.2.3 Geometric measures . 150 7.2.4 Microscopic approach . 150 7.3 Witnessing non-Markovianity . 151 7.4 Measuring non-Markovianity . 154 7.5 Conclusions . 157 8 Probing a composite spin-boson environment 159 8.1 Introduction . 159 8.2 System . 161 8.2.1 Master equation . 162 8.2.2 Observable quantities . 164 8.3 Results: detecting the presence of coupling between the TLFs . 164 ^ 8.3.1 Power spectrum of hMxi . 165 8.3.2 Probe entanglement . 167 8.3.3 Discussion of the results . 169 8.4 Decoherence of entangled states . 171 8.5 E®ect of spin-boson environment on entangling gate operations . 173 8.6 Some conclusions . 174 9 Mapping reservoirs to semi-in¯nite discrete chains 177 9.1 Introduction . 177 9.2 Orthogonal polynomials . 179 9.2.1 Basic concepts . 179 9.2.2 Recurrence relations . 180 9.2.3 Boundness properties of the recurrence coe±cients . 182 9.3 System-reservoir structures . 185 9.4 Applications . 189 9.4.1 Continuous spin-boson spectral densities . 189 9.4.2 Discrete spectral densities. The logarithmically discretized spin-boson model . 191 9.4.3 Linearly-discretised baths . 196 9.5 Conclusions . 198 Prospects 201 Bibliography 203 Journal Publications Parts of this thesis are based on material published in the following papers: ² Introduction to the time evolution theory of open quantum systems A. Rivas and S. F. Huelga In preparation (Part I) ² Markovian Master Equations: A Critical Study A. Rivas, A. D. K. Plato, S. F. Huelga and M. B. Plenio New J. Phys. 12 113032 (2010) (Chapter 5) ² Exact mapping between system-reservoir quantum models and semi- in¯nite discrete chains using orthogonal polynomials A. W. Chin, A. Rivas, S. F. Huelga and M. B. Plenio J. Math. Phys. 51, 092109 (2010) (Chapter 9). ² Entanglement and non-Markovianity of quantum evolutions A. Rivas, S. F. Huelga and M. B. Plenio Phys. Rev. Lett. 105, 050403 (2010) (Chapter 7). ² Probing a composite spin-boson environment N. P. Oxtoby, A. Rivas, S. F. Huelga and R. Fazio New J. Phys. 11, 063028 (2009) (Chapter 8). ² Stochastic resonance phenomena in spin chains A. Rivas, N. P. Oxtoby and S. F. Huelga Eur. Phys. J. B 69, 51 (2009) (Chapter 6). Other papers not covered in this thesis to which the author contributed: ² Precision quantum metrology and nonclassicality in linear and non- linear detection schemes A. Rivas and A. Luis Phys. Rev. Lett. 105, 010403 (2010). 10 ² Nonclassicality of states and measurements by breaking classical bounds on statistics A. Rivas and A. Luis Phys. Rev. A 79, 042105 (2009). ² Practical schemes for the measurement of angular-momentum co- variance matrices in quantum optics A. Rivas and A. Luis Phys. Rev. A 78, 043814 (2008). ² Intrinsic metrological resolution as a distance measure and nonclas- sical light A. Rivas and A. Luis Phys. Rev. A 77, 063813 (2008). ² Characterization of angular momentum fluctuations via principal components A. Rivas and A. Luis Phys. Rev. A 77, 022105 (2008). 11 Acknowledgments First of all, I wish to express my most heartfelt thanks to my supervisors Susana Huelga and Martin Plenio.
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