Quantum Information Theory with Gaussian Systems

More documents

Recommendations

Info

3 Optimal cloners for coherent states and S (i,j) acts on modes i and j. Recall that by (2.15) the expectation value ofÈin ρ is obtained from the Wigner function, which in turn by (2.14) is a classical Fourier transform of χρ: Hence tr[ρÈ] = π f Wρ(0), Wρ(ξ) = (2π) −2f tr ρ ′ (i) ⊗È(j) = (2π) −2f = (2π) −2f where χ (i,j) ρ ′ (0, η) = trρ ′ W (i,j) (0, η) = tr dη e iξT ·σ·η χρ(η). dη χ (i,j) ρ ′ (0, η) dη χ (i,j) √ √ ρ η/ 2, −η/ 2 , ρ W (i,j) η+ξ √2 , η−ξ √ 2 = χ (i,j) ρ η+ξ √2 , η−ξ √ 2 (3.38) Since ρ is the output state of a cloner determined by a state ρT, its characteristic function can be decomposed into χρ(ξ) = t(ξ)χin( i ξi) = χT(Ω ξ)χin( i ξi) according to (3.16) and (3.17). Continuing (3.38), this yields tr ρ ′ (i) ⊗È(j) = (2π) −2f dη t (i,j) η/ √ 2, −η/ √ 2 √ √ χin η/ 2 − η/ 2 = (2π) −2f dη χ (i,j) √ √ T η/ 2, −η/ 2 . (3.39) Note that χin(0) = 1 and furthermore, Ω −1 from (3.22) has been applied to the argument of t together with a suitable substitution for η. Traveling back along the lines of (3.39), (3.38) and (3.37) for ρ and ρT, we get tr ρ(i,j) = tr ρ ′ (i) ⊗È(j) = tr ρ ′ T (i) ⊗È(j) = tr ρT(i,j) . We now prove that all optimized symmetric cloners which were discussed in this chapter are described by a bosonic state ρT. Hence their output states are bosonic, too. Starting with the optimal joint fidelity cloner from Section 3.4.1, note that the fidelity operator Fjoint commutes with all flip operators(i,j) : The flip acts by interchanging the phase space arguments of modes i and j, see (3.36), and the Gaussian characteristic function exp(−ξ T · Γ · ξ/4) describing Fjoint is invariant with respect to interchange of modes since its covariance matrix Γ from (3.26) is invariant. Hence the eigenvectors of Fjoint are eigenvectors to all flip operators(i,j) . But since the eigenstate to the maximal eigenvalue is pure and unique (cf. Section 3.4.1), it must be an eigenvector with eigenvalue +1 for all flips and thus lies in the bosonic sector. 60 .
3.5 Optical implementation The weighted, symmetric single-copy fidelity is represented by an operator F = i Fi, where Fi = exp(−P 2 i /2 − i=j Q2j /2) from (3.28). This operator is invariant under permutations of the modes and thus commutes with all flip operators(i,j) . Just as for the joint fidelity, its eigenvectors are eigenvectors to the flip operators. For the optimal symmetric 1-to-2 cloner, the eigenvector to the maximal eigenvalue is unique by the arguments given in Section 3.4.2 (see discussion of the numerical optimization). Hence it is an eigenvector with eigenvalue +1 for all flip operators and thus bosonic. Symmetric Gaussian 1-to-n cloners are described by a state ρT with characteristic function χT(Ω ξ) = exp −ξ T · (a n ⊗ 2 + bn ⊗ 2) · ξ/4 by (3.31). Applying the transformation Ω−1 from (3.22) to the covariance matrix shows that the state commutes with all permutations of modes. By the above arguments, ρT as well as the output of the cloner is thus bosonic for all a and b. In particular, this is true for the best symmetric Gaussian 1-to-n cloner with a = 1, b = (n−1−a)/n (cf. Section 3.4.2) and the best classical cloner with a = 1, b = 1 (cf. Section 3.4.3). These results are summarized in Proposition 3.11: The optimal joint fidelity cloner, the optimal 1-to-2 cloner, the best symmetric Gaussian 1-to-n cloners and the best classical cloners are described by a bosonic state ρT in (3.18) and thus yield bosonic output states by Lemma 3.10. 3.5 Optical implementation An implementation of the optimal 1-to-2 cloners for single-copy and joint fidelity was briefly described by Cerf and Navez in [a]. A more detailed discussion is provided in e.g. [47,48]. For reference and completeness, we sketch their ideas in this section. Note that optical implementations of the best symmetric Gaussian cloners have been described in [49] as well as in [50], where also the best asymmetric Gaussian 1-to-2 cloner is discussed. The implementation is based on an optical parametric amplifier (opa). In the setup depicted in Fig. 3.3 (taken from [a], see also [47]), it effectively acts as a linear amplifier [52] of intensity gain 2 for the signal in ain, mixing in one part of the state ψ as the idler in b1. This results in a signal output described by the annihilation operator a ′ in = √ 2ain + b∗ 1 (not indicated in the picture). The idler output, given by b ′ 1 = √ 2b1 + a∗ in , is discarded. The signal is then mixed with the other part of ψ in b2 at the beam splitter bs. Its output constitutes the two clones in modes a1 and a2. The state ψ characterizes the cloner and is equivalent to ρT in Eq.(3.17) and (3.23). In the simplest setting, ψ is the vacuum state. This corresponds to the best symmetric Gaussian cloner [49,50]. For the general case, the input–output relations of the system yield as annihilation operators for the output modes a 1 = a in + (b ∗ 1 + b 2 )/√ 2, a 2 = a in + (b ∗ 1 − b 2)/ √ 2. 61
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 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
Page 67 and 68: 3.4 Optimization wheren is the matr
Page 69: 3.4 Optimization Lemma 3.10: The ou
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
show all

Quantum Information Theory with Gaussian Systems

Create successful ePaper yourself

Delete template?

Save as template?