Quantum Information Theory with Gaussian Systems

More documents

Recommendations

Info

2 Basics of Gaussian systems for complex vectors ξ ∈2f together with the operator L = 2f k=1 ξk Rk. Then A is positive-semidefinite due to 〈ξ|A|ξ〉 = tr[ρ L ∗ L] ≥ 0. Since γ is real, this is equivalent to γ − iσ ≥ 0. Moreover, as σ is antisymmetric, the inequality implies γ ≥ 0. Due to the symplectic scalar product ξ T · σ · R in the definition (2.5) of the Weyl operators, these relations are written in terms of modified field operators R ′ = σ · R. In the standard basis of (2.3), this transformation is local to each mode, ′ Q j = σ0 · P ′ j Qj Since we are usually not concerned with specific physical realizations but rather with qualitative results for all continuous-variable systems, we mostly drop the distinction between R ′ k and Rk. Note, however, the effect of displacing a state ρ with Weyl operators, ρ ′ = W η ρ W ∗ η, on the characteristic function: χ ′ ρ(ξ) = tr[W η ρ W ∗ η Wξ] = tr[ρ W ∗ η W η Wξ] = χρ(ξ)e −iξT ·σ·η . (2.19) In field operators R ′ k , the state is displaced by the vector −σ · η, which corresponds to a translation by −η in Rk; cf. also Eq. (2.8). 2.1.2 Symplectic transformations While an orthogonal transformation leaves the scalar product over a (real) vector space unchanged, a real symplectic or canonical transformation S preserves the symplectic scalar product of a phase space, Pj . σ(S ξ, S η) = σ(ξ, η) for all ξ, η ∈ Ξ. By this definition, a symplectic transformation for f degrees of freedom is a real 2f × 2f matrix such that S T · σ · S = σ. The group of these transformations is the real symplectic group, denoted as Sp(2f,Ê). Moreover, with S ∈ Sp(2f,Ê) also S T , S −1 , −S ∈ Sp(2f,Ê), where the inverse of S is given by S −1 = σ S T σ −1 . Symplectic transformations have determinant detS = +1. In addition, the symplectic matrix σ itself is a symplectic transformation, as can be seen from one of its standard forms (2.3) or (2.4). The inverse is σ −1 = σ T = −σ. For the special case of a single mode, the symplectic group consist of all real 2×2 matrices with determinant one, i.e. Sp(2,Ê) = SL(2,Ê). Extensive discussions of the symplectic group, including the topics of this section, can be found e.g. in [11,12,13]. By (2.2), symplectic transformations of the vector of field operators, R ′ = S R, do not change the canonical commutation relations; they do not alter the physics of a continuous-variable system but merely present a change of the symplectic basis. Since σ is itself a symplectic transformation, this argument justifies neglecting the distinction between R ′ k and Rk in the computation of the moments above. Under a symplectic transformation S, displacement vector and covariance matrix are modified according to d ↦→ S·d and γ ↦→ S T ·γ·S. Weyl operators are mapped to Weyl operators 16
2.1 Phase space by a linear transformation of the argument, Wξ ↦→ WS ξ. By Theorem 2.1, the two families of Weyl operators are connected by a unitary transformation US such that WS ξ = U ∗ S Wξ U S . The operators US form the so-called metaplectic representation of the symplectic group Sp(2f,Ê). 6 Every symplectic transformation S can be decomposed in several ways of which we only consider the Euler decomposition into diagonal squeezing transformations and symplectic orthogonal transformations: Theorem 2.4: Every symplectic transformation S ∈ Sp(2f,Ê) can be decomposed as (written in standard ordering of R) ⎛ f rj S = K · ⎝ e 0 0 e−rj ⎞ ⎠ · K ′ , (2.20) j=1 where K, K ′ ∈ Sp(2f,Ê) ∩ SO(2f) are symplectic and orthogonal and rj ∈Êare called squeezing parameters. Remark: This implies that Sp(2f,Ê) is not compact. In fact, Sp(2f,Ê) ∩ SO(2f) is the maximal compact subgroup of Sp(2f,Ê). Similar to real-valued normal matrices, which can be diagonalized by orthogonal transformations, symmetric positive matrices can be diagonalized by symplectic transformations. This corresponds to a decomposition into normal modes, i.e. into modes which decouple from each other: Theorem 2.5 (Williamson): Any symmetric positive 2f × 2f matrix A can be diagonalized by a symplectic transformation S ∈ Sp(2f,Ê) such that SAS T = f j=1 aj 2 , where aj > 0. The symplectic eigenvalues aj of A can be obtained as the (usual) eigenvalues of iσA, which has spectrum spec(iσA) = {±aj}. 6 Due to an ambiguity in a complex phase, the operators US form a faithful representation of the metaplectic group Mp(2f,Ê), which is a two-fold covering of the Sp(2f,Ê). 17
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 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 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
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 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
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
show all

Quantum Information Theory with Gaussian Systems

Create successful ePaper yourself

Delete template?

Save as template?