Quantum Information Theory with Gaussian Systems

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3 Optimal cloners for coherent states The power iteration described above does not work well in the vicinity of the trivial cloners with f1 = 1 or f2 = 1. Instead, we use a different family of non- Gaussian, highly squeezed states φc and directly evaluate 〈φc| F |φc〉, varying the squeezing parameter c. These states are described by φc(x1, x2) = c φ(c x1, c x2) in a representation on L 2 (Ê2 , dx1 dx2), such that φc2 = φ2. In momentum space L 2 (Ê2 , dp1 dp2), they are represented by the Fourier transformed function ˆφc(p1, p2) = ˆ φ(p1/c, p2/c)/c. According to Eq.(3.29b), the single-copy fidelities for a cloner determined by these states in the limit c → ∞ are f1(c) = 〈φc| e −(Q2 1 +Q2 2 )/2 |φc〉 = dx1dx2 |φ(x1, x2)| 2 e −(x2 1 +x2 2 )/(2c2 ) → 1 − 1 2c 2 f2(c) = 〈 ˆ φc| e −(P2 1 +P2 2 )/2 | ˆ φc〉 = = 2π c 2 dx1dx2 |φ(x1, x2)| 2 (x 2 1 + x2 2 ), dp1dp2 | ˆ φ(p1, p2)| 2 e −(p2 1 +p2 2 )c2 /2 dp1dp2 | ˆ 2 c2 φ(p1, p2)| 2π e−(p2 1 +p2 2 )c2 /2 → 2π c 2 |ˆ φ(0, 0)| 2 . (3.30a) (3.30b) This case describes the cloner in the vicinity of f1 = 1. Differentiating both quantities with respect to c2 yields the slope s = df2/df1 = f2/(f1 − 1). In order to show that s approaches −∞, we choose the family of functions generated by φ(x1, x2) = 1/(ǫ + x2 1 + x22 ). Introducing polar coordinates, we approximately evaluate the relevant quantities in (3.30) as 52 dx1dx2 |φ(x1, x2)| 2 (x 2 1 + x2 2 R ) ≈ 2π dr r 3 0 1 (ǫ + r 2 ) 2 ǫ+R = π 2 dt (t − ǫ) ǫ 1 t2 ǫ + R2 ǫ = π log + π − 1 ǫ ǫ + R2 2π ˆ φ(0, 0) = dx1dx2 φ(x1, x2) R ≈ 2π dr 0 r ǫ + r 2 ,
= π R 2 0 dt ǫ + R2 = π log , ǫ + t ǫ 3.4 Optimization where the approximations become exact for R → ∞. Using these expressions to compute the slope yields s → −∞ for R → ∞ and arbitrary ǫ, c. By the argument in Section 3.2, this implies that the optimal cloners in the vicinity of f1 = 1 do not become singular. Since the problem is symmetric with respect to interchange of f1 and f2, the result can be shown to also hold for f2 = 1 by exchanging the squeezing parameter c for 1/c. The solid curve in Fig. 3.2 was complemented with fidelity pairs (f1, f2) from the above expressions sampled at R = 1000, ǫ = e l , c = 100 l −4 for 0.1 ≤ l ≤ 0.9 and 0.6 ≤ l ≤ 1.4 with increments of 0.1. Best Gaussian 1-to-2 cloners The best Gaussian cloners for a given weighted single-copy fidelity λf1 + (1 − λ)f2 maximize the expectation value of F in (3.29b) with respect to Gaussian states ρT. Since F is invariant under simultaneous rotation of the Qi and Pi, an averaging argument similar 7 to that in Section 3.3 implies that the maximizing states ρT are also rotation invariant and thus are described by a rotation invariant Gaussian function φc(x1, x2) ∝ exp −c (x 2 1 + x2 2 ) for ρT = |φc〉〈φc| in the L 2 (Ê2 , dx1 dx2) representation. Depending on the squeezing parameter c, these cloners yield fidelities (f1, f2) = 2/(2 + c −1 ), 2/(2 + c) . The squeezing copt which yields an optimal weighted fidelity λf1 + (1 − λ)f2 can be calculated analytically from the weight λ, copt = 2 − 4 λ + 3 λ(1 − λ) 5 λ − 1 for 1 4 5 < λ < 5 . The resulting fidelities are plotted as the dotted curve in Fig. 3.2. At the intersection with the dash-dotted diagonal lies the best Gaussian symmetric cloner with λ = 1 2 , fidelities f1 = f2 = 2 3 and squeezing c = 1. This is the cloner already known from [53,54,55] (see also its optical implementation in Section 3.5, where the state φc is explicitly used as the idler mode of an opa). In the regimes of λ ≥ 1 4 5 and λ ≤ 5 , the above expression yields values copt = ∞ and copt = 0, respectively. The corresponding cloners are no longer described by a density matrix ρT = |φc〉〈φc|, but by singular,infinitely squeezedstates [44]. This implies that for strongly asymmetric single-copy fidelities with λ ≤ 1 1 5 or (1 −λ) ≤ 5 , the singular cloners mapping the input state exactly into one of the output systems are optimal. Geometrically, this result corresponds to a finite slope of the dotted curve in Fig. 3.2 at the end points. The discussion in Section 3.2 connects this slope in this to the weight λ0 up to which the singular cloners are optimal. From λ0 = 1 5 case, the slope computes to s = −1 4 . 7 However, since the symmetry group in this case is compact, the averaging does not have to resort to an invariant mean but can use the Haar measure of the group. 53
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Quantum Information Theory with Gau
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Vorveröffentlichungen der Disserta
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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 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
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: 3.4 Optimization tained without app
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
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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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