Quantum Information Theory with Gaussian Systems

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2 Basics of Gaussian systems In phase space, transposition of Hermitian density operators is the same as complex conjugation, which in turn can be identified with inversion of sign for the momenta [15], i.e. P ↦→ −P while Q ↦→ Q under Θ. This corresponds to areversal of timeor rather a reversal of time evolution. As a density operator ρ of a bipartite Gaussian state is positive if its covariance matrix obeys the state condition (2.22), γ + iσA ⊕ σB ≥ 0 (where σA, σB are the symplectic forms for systems A and B), the partial transpose ρ T A is positive if the covariance matrix obeys γ + i(−σA) ⊕ σB ≥ 0 , where the sign on σA reflects the change of sign for momenta in the ccr (2.2). While the ppt criterion is necessary for separability, it is sufficient only forsmall systems:2 ⊗2 and2 ⊗3 in finite dimensions [17], Gaussian states with 1×n modes for continuous-variable systems [18]. In particular, the criterion fails if both parties A and B of a Gaussian states have more than one mode (an explicit example is presented in [18]). Since the entanglement of entangled states with positive partial transpose cannot be freely converted into other forms, the entanglement isbound and the sates are calledppt-bound entangled. 2.2.4 Singular states In a general sense, a quantum state ω is a normalized positive linear functional 11 on the algebra of observables [19], i.e. here on ccr(Ξ, σ) for f degrees of freedom: ω: ccr(Ξ, σ) →, where ω(X ∗ X) ≥ 0 for all X ∈ ccr(Ξ, σ) and ω( ) = 1 . Note that = W0 ∈ ccr(Ξ, σ). A state is normal if it can be described by a density operator, i.e. a positive trace class operator ρ on the representation Hilbert space H ⊗f : ω(X) = tr[ρ X] . Otherwise, the state ω is singular and can be decomposed into a normal part ωn given by a density operator and a purely singular contribution ωs, which has expectation value zero for all compact operators12 : ω = ωn + ωs (cf. Section 3.3.1). Singular states have a characteristic function by χ(ξ) = ω Wξ and can thus be Gaussian if χ(ξ) is a Gaussian (2.21). If the ccr algebra is represented on the Hilbert space H⊗f , then the normalized positive linear functionals ω on ccr(Ξ, σ) form the space B∗H⊗f . Similarly, the ⊗f linear space generated by the density operators is denoted by B∗ H , whose closure in the weak topology is B∗H⊗f . 11 Normalized positive linear functionals are automatically bounded and continuous. 12 Compact operators on a Hilbert space are those which can be approximated in norm by finite rank operators, i.e. operators represented as a finite sum of terms |φ〉〈ψ|, cf. e.g. [7, Vol.I]. 24
2.3 Gaussian channels 2.3 Gaussian channels Quantum channels describe transformations between quantum states which correspond to physical operations. For example, applying a unitary transformation U to a state ρ as Uρ U ∗ is a channel and corresponds to a change of basis or a symmetry transformation. Formally, a quantum channel T∗ in the Schrödinger picture is a trace-preserving, completely positive linear map on the trace class operators. For Hilbert spaces H and K of input and output systems, respectively, T∗: B∗(H) → B∗(K), tr T∗(ρ) = tr[ρ] . T∗ has to be positive, i.e. map positive trace class operators to positive trace class operators, and it has to preserve the trace to assure normalization. However, positivity alone is not enough. In addition, applying T∗ to part of a quantum state has to yield an admissible quantum state for the whole system. This is assured by complete positivity: A map T∗ is completely positive if (T∗ ⊗ id)(ρ ′ ) is positive for every positive trace class operator ρ ′ on a composite Hilbert space H ⊗ H ′ and id is the identity on H ′ . Rather than transforming states (Schrödinger picture), a corresponding transformation can be applied to observables (Heisenberg picture), such that both yield the same expectation values. Instead of preserving the trace, this transformation is unital, i.e. it preserves . The Heisenberg picture variant T of a channel is thus determined by tr ρ T(A) = tr T∗(ρ)A , where T : B(K) → B(H), T( ) = . (2.37) For simplicity, we will also refer to input and output spaces by the respective ccr algebras, e.g. for a channel in the Heisenberg picture T : ccr(Ξout, σout) → ccr(Ξin, σin). Gaussian channels have been considered e.g. in [20,21,22,23,24,53]. A channel is Gaussian if it maps Gaussian states to Gaussian states in the Schrödinger picture. In the Heisenberg picture, such channels are quasi-free, i.e. they map Weyl operators to multiples of Weyl operators. A general Gaussian channel for f degrees of freedom acts by T(Wξ) = WΓ·ξ e −g(ξ,ξ)/4+iξT ·d , (2.38) where Γ is a real 2f ×2f matrix, g is a real, symmetric bilinear transformation and d is a real vector of length 2f. The transformations Γ and g cannot be chosen arbitrary, but are subject to a restriction in order for T to be completely positive. In [20], this condition is stated and proven. For ease of reference, we repeat the theorem in our notation: Theorem 2.6: A unital map T : ccr(Ξout, σout) → ccr(Ξin, σin) of the form (2.38) is completely positive if and only if g + iσout − iΓ T σin Γ ≥ 0 . (2.39) 25
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 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
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4 Gaussian quantum cellular automat
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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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