Quantum Information Theory with Gaussian Systems

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4 Gaussian quantum cellular automata In the case of one mode per site, the requirements of (4.11) simplify readily: The conditions for u = ±2, meaning σ(Γ − ξ, Γ + η) = σ(Γ + ξ, Γ − η) = 0 for all ξ, η ∈Ê2 , imply that Γ − and Γ + project onto the same, one-dimensional subspace ofÊ2 . Hence both are multiples of a common matrix Γ ± with rank one. For any real 2×2 matrix M we get M + · M = (det M) and thus immediately have Γ + = 0 as − · Γ = Γ+ − + · Γ + well as Γ + 0 ·Γ0 = from the condition for u = 0. But for one mode, this is equivalent to Γ0 being a symplectic matrix. If we choose the one-dimensional subspace of Γ− and Γ + as the direction of the position variable, 6 we get Γ + = Γ − = f Γ ± with Γ ± = (Γ0)2,1 (Γ 0)2,2 0 0 , (4.12) where f is a common, arbitrary, real-valued coupling parameter and (Γ 0)i,j denotes the respective matrix entries of Γ 0. The shape of Γ ± is a consequence of the conditions for u = ±1. We summarize these results in Proposition 4.4: The quasi-free quantum channel T W(ξ) = W(Γ ξ), where Γ is translationally invariant by (4.8) and symplectic by the conditions in (4.11), results in a reversible qca on an infinite linear chain of harmonic oscillators with nearest-neighbor interaction. For the case of one mode per site, the on-site transformation Γ 0 is symplectic and determines the interaction Γ ± , except for the coupling constant f, according to (4.12). Remark: The fact that in this case the coupling is identical in both directions implies that theleft-andright-shiftermentioned as examples in the introduction cannot be realized with one mode per site. Instead, they require a spareswap system and an alternating partitioning scheme in order to avoid collision problems. For details, see [70]. Proof: These definitions result in a qca in the sense of Definition 4.1. The local observable algebra Ax is spanned by the Weyl operators on single lattice sites, wx(ξx) with ξx ∈Ê2 . The global Weyl operators W(ξ) with ξ ∈ Ξ span the quasi-local algebra A(). Since Γ is a symplectic transformation and translationally invariant, T as defined above is a translationally invariant automorphism of A(). The requirement of locality and finite propagation speed is met by the nearest-neighbor coupling inherent in Γ. The local rule is the restriction of T to the algebra of singlesite observables. A single time step of the system is implemented by applying T to the observable in question. For Weyl operators, this is by the definition in (4.7) the same as applying 6 This choice can be interpreted either as a specification of the interaction Γ± or as a choice of the symplectic basis in the phase spaceÊ2 of a single site. 80
4.2 Reversible Gaussian qca Γ to the phase space argument ξ. Further iteration of the dynamics for t time steps is equivalent to an overall transformation Γt+1 = Γ Γt. Due to the translational invariance, this is a convolution-style operation, 4.2.3 Fourier transform (Γt+1)x,z = (Γt+1)x−z = +1 y=−1 Γ (x−z)−y · (Γt)y . (4.13) Since the system obeys translational invariance, it can be diagonalized together with the momentum operator generating the translations. Hence we can simplify expressions like the iteration relation (4.13) by turning to the Fourier transform of the phase space, i.e. we decompose the phase space elements ξ into plane-wave modes as the eigenstates of the momentum operator and consider the resulting weight functions ˆ ξ with values ˆ ξ(k) ∈Ê2 : ξx = 1 2π π dk ˆ ξ(k)e +ikx −π and ˆ ξ(k) = x∈ξx e −ikx . (4.14) Due to the discrete structure, k is unique only up to multiples of 2π, hence the Fourier transform is determined by k ∈ [−π, π]. All other translationally invariant quantities are treated similarly. This casts the iteration relation (4.13) into an ordinary multiplication of matrices, ˆΓt(k) = ˆ Γ(k) t, where ˆ Γ(k) = Γ0 + 2f cos(k)Γ ± (4.15) is the Fourier transform of Γx according to (4.14). The Fourier transform also simplifies the state condition (4.6) for γ. To properly define the transformed γ(k), we restrict γ(x) to be absolutely summable, i.e. x∈γ(x) < ∞. This condition excludes problematic correlation functions, e.g. those with singular portions but retains the important cases of product and clustering initial states. From a mathematical point of view, it requires γ(x) to decrease faster than 1/|x| and makes γ(k) continuous. With this, the state condition (4.6) on the correlation function reads in terms of Fourier transforms 1 2π π dk µ T (k) · γ(k) + iσs · µ(k) ≥ 0. (4.16) −π This is equivalent to the condition on 2×2 matrices that γ(k) + iσs ≥ 0 for all k ∈ [−π, π]: if this condition holds for all k, then the l.h.s. of (4.16) is indeed positive semi-definite; if, however, γ(k0) + iσs is not positive semi-definite for some k0, then the l.h.s. of (4.16) can be made negative by choosing an appropriate µ(k), e.g. the sharply peaked Fourier transform of a flat Gaussian which is centered around k0 and has been restricted to finite support. Moreover, if γ(k0) + iσs is strictly positive for 81
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Quantum Information Theory with Gau
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Vorveröffentlichungen der Disserta
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Contents Quantum Cellular Automata
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List of Figures viii
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List of Theorems x
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Summary mum is reached by a measure
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Summary 4
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1 Introduction electromagnetic fiel
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1 Introduction 8
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2 Basics of Gaussian systems Since
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2 Basics of Gaussian systems Theore
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2 Basics of Gaussian systems For a
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2 Basics of Gaussian systems for co
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2 Basics of Gaussian systems 2.2 Ga
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2 Basics of Gaussian systems As pur
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2 Basics of Gaussian systems unitar
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2 Basics of Gaussian systems In pha
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2 Basics of Gaussian systems Remark
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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
Page 67 and 68: 3.4 Optimization wheren is the matr
Page 69 and 70: 3.4 Optimization Lemma 3.10: The ou
Page 71 and 72: 3.5 Optical implementation The wei
Page 73 and 74: 3.6 Teleportation criteria can be t
Page 75 and 76: 3.6 Teleportation criteria carry th
Page 77: Quantum Cellular Automata
Page 80 and 81: 4 Gaussian quantum cellular automat
Page 111 and 112: 5 Gaussian private quantum channels
Page 113 and 114: p Figure 5.1: Illustrating the encr
Page 115 and 116: characteristic function χrand(ξ)
Page 117 and 118: = − 1 2 2Nf i,j=1 where M ′ =
Page 119 and 120: p δ ξk ξ x 5.2 Security estimati
Page 121 and 122: 5.2 Security estimation where in th
Page 123 and 124: 5.3 Result and outlook 5.3 Result a
Page 125: Bibliography
Page 128 and 129: Bibliography [7] M. Reed and B. Sim
Page 130 and 131: Bibliography [37] R. F. Werner,Opti
Page 132 and 133: Bibliography [69] N. Takei, H. Yone
Page 134 and 135: Bibliography 124
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Quantum Information Theory with Gaussian Systems

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