Quantum Information Theory with Gaussian Systems

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Contents Quantum Cellular Automata 4 Gaussian quantum cellular automata 69 4.1 Quantum cellular automata . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Reversible Gaussian qca . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.1 Phase space and basics . . . . . . . . . . . . . . . . . . . . . . 76 4.2.2 Transition rule . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.3 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.4 Example system . . . . . . . . . . . . . . . . . . . . . . . . . 82 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Irreversible Gaussian qca . . . . . . . . . . . . . . . . . . . . . . . . 91 Private Quantum Channels 5 Gaussian private quantum channels 101 5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Security estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Single mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3 Result and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Bibliography 117 vi
List of Figures 2.1 Depicting Gaussian states in phase space . . . . . . . . . . . . . . . . 21 3.1 Schematic diagram of achievable worst-case single-copy fidelities . . 35 3.2 Numerical single-copy fidelities . . . . . . . . . . . . . . . . . . . . . 49 3.3 Optical scheme of a displacement-covariant cloner . . . . . . . . . . . 62 3.4 Teleportation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1 Depicting the time step of a qca . . . . . . . . . . . . . . . . . . . . 75 4.2 Depicting the eigenvalues of ˆ Γ(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 T Σ of the cutoff integral in T [ ] . . . . . . 109 vii
Page 1 and 2: Quantum Information Theory with Gau
Page 3: Vorveröffentlichungen der Disserta
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 16 and 17: 1 Introduction electromagnetic fiel
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
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3.4 Optimization Hence for the case
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1 2 3 1 2 0 f2 1 2 (a) 2 3 1 f1 0.0
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3.4 Optimization tained without app
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= π R 2 0 dt ǫ + R2 = π log , ǫ
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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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