Quantum Information Theory with Gaussian Systems

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4 Gaussian quantum cellular automata Note that every system which is a qca by Definition 4.1 falls into category I. In addition, II has to hold with N − N ⊆ M in order to assure causality. 9 In order to establish which conditions should be imposed to guarantee the desired properties, consider connections between the cases: Lemma 4.10: The above notions of locality constitute a partial hierarchy in the sense that (i) III implies I with N = K, (ii) IV implies III with K = D, (iii) IV implies II with M = D − D, (iv) IV implies I with N = D. (v) However, II does not imply I and vice versa. Remark: It remains open if any of the lower cases imply higher ones, e.g. if I and II together already require IV. Proof: (i) By the embedding described in Section 4.1, A ∈ A(Λ) is also in A(Λ + K). Since Ki ∈ A(Λ + K), obviously T(A) = i K∗ i AKi ∈ A(Λ + K) and hence T A(Λ) ⊂ A(Λ + K). (ii) As every channel, the dynamics of IV can be described by Kraus operators, see Section 2.3. For A ∈ A(Λ), we have by definition T1(A) ∈ A ′ (Λ + D) and T(A) = A(Λ + D) since the ancilla systems do not introduce correlations. Hence Ki ∈ A(Λ + D). (iii) For A1 ∈ A(Λ1), A2 ∈ A(Λ2) and (Λ1 + D) ∩ (Λ2 + D) = ∅, the observables T1(A1) and T1(A2) are localized on different regions A ′ (Λ1+D) and A ′ (Λ2+D) without overlap. Hence their product can be written as a tensor product with respect to the sites by implicit embedding. Since T1 is an automorphism, this yields: ⊗s T(A1 A2) = trE ⊗ ρ0 T1(A1 ⊗ )T1(A2 ⊗ ) ⊗(Λ1+D) = trE ⊗ ρ0 T1(A1 ⊗ ) ⊗(Λ2+D) trE ⊗ ρ0 T1(A2 ⊗ ) = T(A1)T(A2). 9 Note that (Λ1 + D − D) ∩ Λ2 = ∅ ⇐⇒ (Λ1 + D) ∩ (Λ2 + D) = ∅. Causality is the notion that operations on sufficiently far separated areas should be independent of each other. In our terms, this requires for observables A1 ∈ A(Λ1) and A2 ∈ A(Λ2), where T(Ai) ∈ A(Λi + N) and (Λ1 + N) ∩ (Λ2 + N) = ∅ that 94 T(A1 ⊗ |Λ2+N ) = T(A1) ⊗ |Λ2+N, T( |Λ1+N ⊗ A2) = |Λ1+N ⊗ T(A2). (For details and a brief discussion, see e.g. [89].) Note that under this conditions in II we get T(A1 ⊗ A2) = T(A1) T(A2) = T(A1) ⊗ T(A2) by implicit embedding. (ii) (i) III IV I (iv) II (iii)
4.3 Irreversible Gaussian qca Note that the tensor products above are with respect to the decomposition of main system and ancilla at each site. (iv) This follows from (ii) and (i). (v) Consider a dynamics which completely depolarizes an initial state ρin to a Ns translationally invariant product state ρ0 . For A1 ∈ A(Λ1), A2 ∈ A(Λ2) and (Λ1 + M) ∩ Λ2 = ∅ we get tr ρin T(A1 ⊗ A2) = tr N N Λ1 Λ2 ρ0 A1 tr ρ0 A2 = tr ρin T(A1) tr ρin T(A2) . Hence T(A1 ⊗ A2) = T(A1)T(A2) and II holds true independently of I and M (as long as 0 ∈ M). For the converse, the Gaussian dynamics from (4.32) serves as a counterexample by virtue of (4.35) if g(x) is not restricted to a finite M. For Gaussian irreversible qcas, the cases I and II are easily expressed in terms of Γ and g: Lemma 4.11: A Gaussian irreversible qca, described by dynamics Γ and noise form g, complies with type I or II, respectively, if I. Γ(x) = 0 for x /∈ N, II. g(x) = 0 for x /∈ M and ∀∆ ∈ (N − N) \ M: x∈N Γ + x Γ∆+x = 0 . (4.36) Remark: If N − N ⊆ M, case II does not impose a condition on Γ. Otherwise, (4.36) corresponds to part of the condition (4.10) for Γ to be symplectic. Consider in particular the important case M = {0}; as for reversible qcas, this allows to reconstruct the global rule from the local rule by the arguments from the proof of Lemma 4.2. For a nearest-neighbor interaction, in detail Γ has to obey the conditions (4.11b–e) but not (4.11a). Hence for systems with one mode per site a deviation from symplectic transformations is possible by choosing e.g. Γ + = . Proof: Compliance with case I was already considered above and corresponds to finite support for Γ, i.e. Γ(x) = 0 for x /∈ N. For case II to apply, the dissipation form C from (4.34) has to vanish due to (4.35) if the supports suppξ and suppη are separated by M, (suppξ + M) ∩ suppη = ∅ =⇒ C(ξ, η) = g(ξ, η) + iσ(ξ, η) − iσ(Γ T ξ, Γ η) = 0 . Since we assume 0 ∈ M, this implies suppξ ∩ suppη = ∅ and σ(ξ, η) ≡ 0. As Γ and g are real-valued, the condition can be split into real and imaginary parts 0 Γ 0 g(ξ, η) = 0 and σ(Γ T ξ, Γ η) = 0 , (4.37) 95
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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
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3 Optimal cloners for coherent stat
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3.1 Setup 3.1 Setup A deterministic
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f2(T) 1 0 (a) 1 f1(T) f2(T) 1 0 (
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the expectation value of an arbitra
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3.3 Covariance clone). A more gener
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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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