Quantum Information Theory with Gaussian Systems

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3 Optimal cloners for coherent states all pure input states with equal fidelity. Remarkably, these cloners simultaneously maximize both the joint and the single-copy fidelity [39]. For continuous-variable systems, a universal cloner with finite fidelities for all pure input states cannot exist. As explained in Section 3.4.1 below, for every cloner there are pure squeezed states which yield a fidelity of zero. To facilitate handling of the mathematical structures, the set of input states is further restricted to Gaussian states, which are also important from a practical point of view (cf. the discussion in the Introduction). The set of pure, nonsqueezed Gaussian states is the set of coherent states, which we take as our input states. Similar to the finite-dimensional case, the cloning of coherent states was studied in depth, see [48,49,50,53,54,55] and references mentioned below. However, the cloners considered were restricted to Gaussian operations and were also assumed to be covariant with respect to phase space translations of the input state. It remained unclear if this set of cloners includes the optimal one. In particular, the results include a proof [55] that under this presumptions the best symmetric Gaussian 1-to-2 cloner is limited to a single-copy fidelity of 2 3 as well as its optical implementation [48,49,50]. While mostly only deterministic cloners are studied, [51] investigates probabilistic finite-dimensional and continuous-variable cloning. In the following we optimize the worst-case joint fidelities and weighted single-copy fidelities for deterministic 1-to-n cloning of coherent input states. These quantities do not depend on a priori information about the probability distribution of the input states (as long as all coherent states can occur). We show that the optimal fidelities can indeed be reached by cloners which are covariant with respect to phase space translation (Sec. 3.3). These cloners are necessarily quasi-free, i.e. they map Weyl operators onto multiples of Weyl operators in the Heisenberg picture (Sec. 3.3.2). Contrary to the finite-dimensional case, the optimization of single-copy and joint fidelity for coherent input states requires different cloners. While the joint fidelity is analytically maximized by a Gaussian cloner, the single-copy fidelity can be enhanced by non-Gaussian operations (Sec. 3.4.1, 3.4.2). For the case of a symmetric cloner which takes one copy of the input state into two clones, the maximal fidelity is approximately 0.6826, compared to 2 3 for the best Gaussian cloner. We also analytically derive the best single-copy fidelities reached by Gaussian cloners for the 1-to-2 cloning with arbitrary weights and symmetric 1-to-n cloning. In addition, we show that classical cloning is limited to a fidelity of 1 2 (Sec. 3.4.3). This can be reached by a Gaussian scheme, namely by a heterodyne measurement on the input state and repreparation of coherent states according to the measurement result. Furthermore, the fidelity cannot be enhanced by the use of supplemental ppt-bound entangled states. The results on cloning fidelities give rise to success criteria for continuous-variable teleportation. One of these criteria proves and extends an important conjecture in the literature (cf. Sec. 3.6). The main results and arguments presented in this chapter have been published in [a]. 32
3.1 Setup 3.1 Setup A deterministic 1-to-n cloner abstractly is a completely positive, trace-preserving map which in the Schrödinger picture transforms a single input state into an output state of n subsystems, the clones. In the Heisenberg picture, these channels map observables on the output systems onto observables on the input system. Our task is to characterize these cloning maps and to optimize them with respect to suitable fidelities. To more formally describe the class of relevant cloning maps, we start by setting up the involved phase spaces. If Ξin =Ê2 denotes the phase space of the one-mode input system equipped with the nondegenerate symplectic form σin, then the output is described in terms of the phase space Ξ = n j=1 Ξin with symplectic form n σ(ξ, η) = σ j=1ξj, n j=1ηj = n σin(ξj, ηj). Where appropriate, we identify a vector in Ξ with the n-tuple of its components in Ξin, i.e. Ξ ∋ ξ ≡ n j=1 ξj ≡ (ξ1, . . . , ξn). Recalling the discussion of Section 2, a channel between continuous-variable systems is a map between (states on) the respective ccr algebras. The cloning map T in the Heisenberg picture maps the output ccr algebra onto the input ccr algebra, i.e. T : ccr(Ξ, σ) → ccr(Ξin, σin). In the Schrödinger picture, the cloner T maps input states onto output states, j=1 T∗: S ccr(Ξin, σin) → S ccr(Ξ, σ) , where S ccr(Ξ, σ) denotes the state space of the ccr algebra. For general states, including singular states, this is the space of positive linear functionals on the representation Hilbert space H, i.e. S ccr(Ξ, σ) = B ∗ (H). If only normal states are involved, it can be restricted to the space of trace class operators on H, i.e. S ccr(Ξ, σ) = B∗(H). Due to the Stone-von Neumann Theorem 2.1, the representation Hilbert space is essentially unique: for ccr(Ξin, σin) and ccr(Ξ, σ) we have Hin = L 2 (Ê2 , dx) and H = H ⊗n in ≃ L2 (Ê2n , dx), respectively, where dx is understood to have appropriate dimension. 3.2 Fidelities The fidelity quantifies how well two quantum states coincide [40, 41]. For general states described by density matrices ρ1 and ρ2, it is defined as f(ρ1, ρ2) = ρ1/2 tr 1 ρ2 ρ 1/2 1/2 1 2 . 33
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: 3 Optimal cloners for coherent stat
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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