Quantum Information Theory with Gaussian Systems Von der Fakult¨at f¨ur Physik und Geowissenschaften der Technischen Universit¨at Carolo-Wilhelmina zu Braunschweig zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Ole Kr¨uger aus Braunschweig 1. Referent Prof. Dr. Reinhard F. Werner 2. Referent Prof. Dr. Martin B. Plenio eingereicht am 5.Januar 2006 m¨undliche Pr¨ufung (Disputation) am 6. April 2006 Druck 2006 Vorver¨offentlichungen der Dissertation Teilergebnisse aus dieser Arbeit wurden mit Genehmigung der Fakult¨at f¨ur Physik und Geowissenschaften, vertreten durch den Mentor der Arbeit, in folgenden Bei- tr¨agen vorab ver¨offentlicht: Publikationen N. Cerf, O. Kr¨uger, P. Navez, R. F. Werner und M. M. Wolf, Non-Gaussian Cloning of Quantum Coherent States is Optimal, Phys. Rev. Lett. 95, 070501 (2005). O. Kr¨uger und R. F. Werner, Gaussian Quantum Cellular Automata , in Quantum Information with continuous variables of atoms and light, herausgegeben von N. Cerf, G. Leuchs und E. S. Polzik (Imperial College Press, London/UK, im Druck). Tagungsbeitr¨age J. I. Cirac, G. Giedke, O. Kr¨uger, R. F. Werner und M. M. Wolf, Entanglement of Formation for Gaussian States with 1 1 modes, Third Conference of × esf-qit Advances in quantum information processing: from theory to experiment (Poster, Erice/Italien, 15.–22.3.2003). O. Kr¨uger, R. F. Werner und M. M. Wolf, Cloning Gaussian States , dpg-Fr¨uhjahrstagung 2004 (Vortrag, M¨unchen, 22. – 26. 3. 2004). O. Kr¨uger und R. F. Werner, Gaussian Quantum Cellular Automata , cvqip’ Workshop (Poster, Veilbronn, 2. – 5. 4. 2004). O. Kr¨uger und R. F. Werner, Gaussian Quantum Cellular Automata , ein International Symposium on Entanglement, Information & Noise (Poster, Krzy˙zowa/Polen, 14.– 20.6.2004). O. Kr¨uger und R. F. Werner, Gaussian Quantum Cellular Automata , dpg-Fr¨uhjahrstagung 2005 (Vortrag, Berlin, 4. – 9. 3. 2005). O. Kr¨uger und R. F. Werner, Gaussian Quantum Cellular Automata , iqing (Vortrag, Paris/Frankreich, 23.– 25.7. 2005). Contents Summary 1 1 Introduction 5 2 Basics of Gaussian systems 9 2.1 Phasespace................................ 9 2.1.1 Noncommutative Fourier transf. and characteristic functions 13 2.1.2 Symplectictransformations . 16 2.2 Gaussianstates.............................. 18 2.2.1 Coherent,thermal andsqueezedstates . 19 2.2.2 Spectral decomposition and exponential form . .. 21 2.2.3 Entangledstates ......................... 23 2.2.4 Singularstates .......................... 24 2.3 Gaussianchannels ............................ 25 Cloning 3 Optimal cloners for coherent states 31 3.1 Setup ................................... 33 3.2 Fidelities ................................. 33 3.3 Covariance ................................ 36 3.3.1 Technicalities ........................... 38 3.3.2 Characterization . .. .. .. .. .. .. .. .. .. .. 42 TransformationΩ ........................ 44 3.4 Optimization ............................... 44 3.4.1 Jointfidelity ........................... 45 3.4.2 Single-copyfidelity . 47 Numericaloptimization . 50 BestGaussian1-to-2cloners . 53 Best symmetric Gaussian 1-to-n cloners ............ 54 3.4.3 Classicalcloning ......................... 55 3.4.4 Bosonicoutput .......................... 58 3.5 Opticalimplementation . 61 3.6 Teleportationcriteria. 63 v Contents Quantum Cellular Automata 4 Gaussian quantum cellular automata 69 4.1 Quantumcellularautomata . 71 4.2 Reversible Gaussian qca ......................... 76 4.2.1 Phasespaceandbasics. 76 4.2.2 Transitionrule .......................... 78 4.2.3 Fouriertransform. .. .. .. .. .. .. .. .. .. .. 81 4.2.4 Examplesystem ......................... 82 Convergence ........................... 85 4.3 Irreversible Gaussian qca ........................ 91 Private Quantum Channels 5 Gaussian private quantum channels 101 5.1 Setup ................................... 104 5.2 Securityestimation.. .. .. .. .. .. .. .. .. .. .. .. 106 Singlemode............................ 112 5.3 Resultandoutlook ............................ 113 Bibliography 117 vi List of Figures 2.1 Depicting Gaussianstatesin phasespace. ... 21 3.1 Schematic diagram of achievable worst-case single-copy fidelities . 35 3.2 Numericalsingle-copyfidelities . .. 49 3.3 Optical scheme of a displacement-covariant cloner . ....... 62 3.4 Teleportationscheme.. .. .. .. .. .. .. .. .. .. .. 64 4.1 Depicting the time step of a qca .................... 75 4.2 Depicting the eigenvalues of Γ(ˆ k).................... 83 4.3 Plot of α(k)................................ 84 5.1 Illustrating the continuous encryption of single-mode coherent states 103 5.2 Depicting the discretization TΣ of the cutoff integral in T[] ...... 109 vii List of Figures viii List of Theorems 2.1 Theorem. 12 4.1 Definition . 73 2.2 Lemma ....... 13 4.2 Lemma ....... 74 2.3 Theorem. 13 4.3 Corollary . 75 2.4 Theorem. 17 4.4 Proposition . 80 2.5 Theorem. 17 4.5 Lemma ....... 82 2.6 Theorem. 25 4.6 Lemma ....... 84 4.7 Proposition . 87 3.1 Lemma ....... 37 4.8 Theorem. 87 3.2 Lemma ....... 39 4.9 Theorem. 89 3.3 Corollary . 42 4.10Lemma ....... 94 3.4 Proposition . 43 4.11Lemma ....... 95 3.5 Proposition . 47 4.12Lemma ....... 97 3.6 Proposition . 48 3.7 Lemma ....... 56 5.1 Proposition . 113 3.8 Lemma ....... 57 5.2 Corollary . 113 3.9 Proposition . 58 3.10Lemma ....... 59 3.11 Proposition . 61 3.12 Corollary . 64 3.13 Corollary . 65 ix List of Theorems x Summary This thesis applies ideas and concepts from quantum information theory to systems of continuous-variables such as the quantum harmonic oscillator. In particular, it is concerned with Gaussian states and Gaussian systems, which transform Gaussian states into Gaussian states. While continuous-variable systems in general require an infinite-dimensional Hilbert space, Gaussian states can be described by a finite set of parameters. This reduces the complexity of many problems, which would otherwise be hardly tractable. Moreover, Gaussian states and systems play an important role in today’s experiments with continuous-variable systems, e.g. in quantum optics. Examples of Gaussian states are coherent, thermal and squeezed states of a light field mode. The methods utilized in this thesis are based on an abstract characterization of Gaussian states, the results thus do not depend on the particular physical carriers of information. The focus of this thesis is on three topics: the cloning of coherent states, Gaussian quantum cellular automata and Gaussian private channels. Correspondingly, the main part of the thesis is divided into three chapters each of which presents the results for one topic: 3 Cloning An unknown quantum state can in general not be duplicated perfectly. This impossibility is a direct consequence of the linear structure of quantum mechan- ics and enables quantum key distribution. The approximate copying or cloning of quantum states is possible, though, and raises questions about optimal cloning. Bounds on the fidelity of cloned states provide restrictions and benchmarks for other tasks of quantum information: In quantum key distribution, bounds on cloning fi- delities allow to estimate the maximum information an eavesdropper can get from intercepting quantum states in relation to noise detected by the receiver. Beyond that, any communication task which aims at the complete transmission of quantum states has to beat the respective cloning limits, because otherwise large amounts of information either remain at the sender or are dissipated into the environment. Cloning was investigated both for finite-dimensional and for continuous-variable systems. However, results for the latter were restricted to covariant Gaussian opera- tions. This chapter presents a general optimization of cloning operations for coherent input states with respect to fidelity. The optimal cloners are shown to be covariant with respect to translations of the input states in phase space. In contrast to the finite-dimensional case, optimization of the joint output state and of weighted combi- nations of individual clones yields different cloners: while the former is Gaussian, the latter is not. The optimal fidelities are calculated analytically for the joint case and numerically for the individual judging of two clones. For classical cloning, the opti- 1 Summary mum is reached by a measurement and preparation of coherent states. The bound on classical cloning is turned into a criterion for the successful transmission of a coherent state by quantum teleportation. 4 Quantum Cellular Automata Quantum cellular automata (qcas) are a model for universal quantum computation in translationally invariant lattice systems with localized dynamics. They provide an alternative concept for experimental realiza- tion of quantum computing as they do not require individual addressing of their constituent systems but rather rely on global parameters for the dynamics. Quan- tum cellular automata seem to be particularly fitted for implementation in optical lattices as well as for the simulation of lattice systems from statistical mechanics. For this purpose the qca should be able to reproduce the ground state of a different dynamics, preferably by driving an initial state into a suitable stationary state in the limit of large time. This chapter investigates abstract Gaussian qcas with respect to irreversibility. As a basis, it provides methods to deal with translationally invariant systems on infinite lattices with localization conditions. A simple example of a reversible Gaus- sian qca (a nonsqueezing dynamics