Quantum Information Theory with Gaussian Systems

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1 Introduction electromagnetic field ofbright2 laser beams. A field mode is described as a quantum harmonic oscillator with field operators Q, P corresponding to the quadrature components of the complex amplitude. Since Q and P have continuous eigenvalue spectrum, the mode is a continuous-variable system, which cannot be represented on a finite-dimensional Hilbert space. 3 Gaussian continuous-variable states are characterized by a Gaussian Wigner quasiprobability function. They naturally arise as the ground and thermal states of quadratic bosonic Hamiltonians, in particular for the standard harmonic oscillator, H = 1 2 (Q2 + P 2 ). (Throughout this thesis, we set = 1. Similarly, we do not distinguish different modes by their frequency but always assume m, ω = 1. Units of physical quantities are chosen accordingly.) Hence Gaussian states are relevant wherever quantum systems are described by such Hamiltonians. Examples of Gaussian states in quantum optics include coherent states (pure states with minimal uncertainty,displaced vacuum), thermal states (coherent states with additional classical Gaussian noise) and squeezed states (with reduced variance for Q or P). In particular, the output states of lasers are approximated by coherent states. Gaussian states are also mathematically appealing, because they can be described by a finite number of parameters for each mode. The underlying phase space related to the canonical commutation relation, [Q, P] = i , provides a rich mathematical structure. This makes Gaussian states much easier to handle than general continuous-variable states, which require tools for infinitedimensional Hilbert spaces: Restricting questions to Gaussian states allows to investigate problems which would otherwise be hardly tractable. Moreover, Gaussian states are extremal among all states with the same first and second moments with respect to certain functionals: It is a standard result of statistical mechanics that Gaussian states maximize the von Neumann entropy S(ρ) = − tr[ρ log ρ] for fixed energy. Only recently, Wolf et al. [3] have proved that a similar result holds for a more general class of functionals, which comprises important examples from quantum information theory (entanglement measures, key distillation rates, channel capacities). One can thus assume an unknown quantum state to be Gaussian in order to obtain reliable bounds on such quantities. For these reasons, Gaussian states are of particular relevance for the study of continuous-variable systems. While quantum information theory for finite-dimensional systems is quite far developed, continuous-variable systems have not yet attracted equal attention. In this 2 This emphasizes the contrast to very weak laser pulses with approximately 0.1 photons per pulse, which are used to emulate single-photon sources. 3 Consider e.g. position and momentum operators Q and P, which obey the canonical commutation relation [Q, P] = i . If Q and P could be described by finite-dimensional matrices, the trace of the commutator would vanish, tr[Q P −P Q] = tr[Q P]−tr[P Q] = tr[Q P]−tr[Q P] = 0. This contradicts tr[ ] = dim H. 6
thesis, three well-established concepts of quantum information theory are transfered to the continuous-variable Gaussian world: the cloning of coherent states (chapter 3), Gaussian quantum cellular automata (chapter 4) and Gaussian private quantum channels (chapter 5). In addition, chapter 2 provides the common ground for all chapters with an overview of the basic tools of phase space as well as Gaussian states and systems. The main chapters can be read independently of each other and provide a selfcontained introduction to the respective topics. 7
Page 1 and 2: Quantum Information Theory with Gau
Page 3: Vorveröffentlichungen der Disserta
Page 6 and 7: Contents Quantum Cellular Automata
Page 8 and 9: List of Figures viii
Page 10 and 11: List of Theorems x
Page 12 and 13: Summary mum is reached by a measure
Page 14 and 15: Summary 4
Page 18 and 19: 1 Introduction 8
Page 20 and 21: 2 Basics of Gaussian systems Since
Page 22 and 23: 2 Basics of Gaussian systems Theore
Page 24 and 25: 2 Basics of Gaussian systems For a
Page 26 and 27: 2 Basics of Gaussian systems for co
Page 28 and 29: 2 Basics of Gaussian systems 2.2 Ga
Page 30 and 31: 2 Basics of Gaussian systems As pur
Page 32 and 33: 2 Basics of Gaussian systems unitar
Page 34 and 35: 2 Basics of Gaussian systems In pha
Page 36 and 37: 2 Basics of Gaussian systems Remark
Page 38 and 39: 2 Basics of Gaussian systems 28
Page 41 and 42: 3 Optimal cloners for coherent stat
Page 43 and 44: 3.1 Setup 3.1 Setup A deterministic
Page 45 and 46: f2(T) 1 0 (a) 1 f1(T) f2(T) 1 0 (
Page 47 and 48: the expectation value of an arbitra
Page 49 and 50: 3.3 Covariance clone). A more gener
Page 51 and 52: 3.3 Covariance In the case of joint
Page 53 and 54: 3.3 Covariance where the second ide
Page 55 and 56: 3.4 Optimization search to covarian
Page 57 and 58: 3.4 Optimization Hence for the case
Page 59 and 60: 1 2 3 1 2 0 f2 1 2 (a) 2 3 1 f1 0.0
Page 61 and 62: 3.4 Optimization tained without app
Page 63 and 64: = π R 2 0 dt ǫ + R2 = π log , ǫ
Page 65 and 66: 3.4 Optimization where the bound is
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3.4 Optimization wheren is the matr
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3.4 Optimization Lemma 3.10: The ou
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3.5 Optical implementation The wei
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3.6 Teleportation criteria can be t
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3.6 Teleportation criteria carry th
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Quantum Cellular Automata
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4 Gaussian quantum cellular automat
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5 Gaussian private quantum channels
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p Figure 5.1: Illustrating the encr
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characteristic function χrand(ξ)
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= − 1 2 2Nf i,j=1 where M ′ =
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p δ ξk ξ x 5.2 Security estimati
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5.2 Security estimation where in th
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5.3 Result and outlook 5.3 Result a
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Bibliography
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Bibliography [7] M. Reed and B. Sim
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Bibliography [37] R. F. Werner,Opti
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Bibliography [69] N. Takei, H. Yone
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Bibliography 124
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Quantum Information Theory with Gaussian Systems

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