Quantum Information Theory with Gaussian Systems

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4 Gaussian quantum cellular automata Since our notion of a quantum cellular automaton is based on an infinite lattice, any attempt to define a qca has to deal with the infinite number of quantum systems at the lattice sites. As discussed in [70], several previous definitions found in the literature suffer from conceptual shortcomings which prevent a successful application to infinite lattice systems. In particular, the notion of localization as implemented by states on the infinite lattice is problematic. For example, the basic operation of applying the same unitary transformation to each cell separately would require the multiplication of an infinite number of phase factors, which does not allow for a well-defined unitary operator describing the global state change. In order to circumvent these problems, we work in the Heisenberg picture and define the dynamics of observables. This approach was motivated by methods used in statistical mechanics of quantum spin systems, where infinite arrays of simple quantum systems play a prominent role [70]. In contrast to a notion of localized states, localized observables are clearly defined: they require a measurement of a finite collection of cells only. If the lattice sites are labeled by s-tuples of integers, where s is the lattice dimension, we denote by Ax the algebra of observables which are localized on the single lattice site x ∈s . This algebra could be an algebra of d × d matrices for a spin system or a ccr algebra for a continuous-variable system. The set of all observables which are localized on a finite region Λ ⊂s of the lattice constitutes the algebra A(Λ) = x∈Λ Ax associated with this region. For two regions Λ1 ⊂ Λ2, we take A(Λ1) as a subalgebra of A(Λ2) by tensoring with unit operators as necessary, i.e. on Λ2 \ Λ1. This allows us to properly define the product of two operators A1A2 from different local algebras A(Λ1) and A(Λ2), respectively, as the corresponding element from A(Λ1 ∪ Λ2). Since this procedure does not affect the norm, all local algebras are normed and their completion is the quasi-local algebra [84], denoted by A(s ). This inclusion of algebras is especially instructive in connection with the neighborhood. If N ⊂s is defined as the finite neighborhood of the cell x = 0, we can install it as the uniform neighborhood scheme and obtain the neighborhood of any cell x as the set x + N ≡ {x + n | n ∈ N }. Accordingly, the neighborhood of a finite region Λ ⊂s of the lattice is the set Λ + N ≡ {x + n | x ∈ Λ, n ∈ N }. The observables on any finite region Λ are contained in the algebra on the region enlarged by its neighborhood, A(Λ) ⊂ A(Λ + N), if and only if Λ ⊂ Λ + N. This is only true if the neighborhood scheme explicitly contains the origin. While this need not necessarily be the case, we can formally enlarge the neighborhood without actually considering the additional elements in the interaction. Hence we can always assume 0 ∈ N. By the same argument, we can w.o.l.g. assume the neighborhood N to be simply connected. Note that by the above definition thepointwise difference of two sets is in general not empty, e.g. N − N = {x − y | x, y ∈ N }. The dynamics of the system is implemented as linear transformations on the observable algebras. In particular, one time step in the global evolution of the qca is a transformation T on the observable algebra A(s ) of the infinite system. To describe a proper time evolution, T has to be completely positive. Since we only consider deterministic dynamics, it has also to be unital, T( ) = , i.e. it has to be a quantum channel. In addition, uniformity of the whole system requires that T 72
4.1 Quantum cellular automata is translationally invariant. It has thus to commute with all lattice translations τx, where x ∈s and τx is the isomorphism from Ay to Ay+x. Hence we have to require that T(τxA) = τx T(A). If T is to arise from a local interaction coupling a cell to its neighborhood, it has to obey a suitable locality condition: For any observable A localized on a finite region Λ, the observable T(A) obtained after one time step has to be localized in Λ + N: T A(Λ) ⊂ A(Λ + N). (4.1) While T implements the global rule, i.e. one time step of the whole system, the local rule as the time evolution of a single cell x is obtained as the restriction Tx of T to this cell. Due to the translational invariance, it suffices to consider the origin; hence given T, the local rule is determined as T0: A0 → A(N). A qca is called reversible if the global rule T has an inverse which also is a quantum channel. This is equivalent to T being an automorphism of the quasi-local algebra. The above considerations give rise to the following definition of a qca: Definition 4.1: A (deterministic) quantum cellular automaton (qca) on the lattices with finite neighborhood scheme N ⊂s , where 0 ∈ N, is a quantum channel T : A(s ) → A(s ) on the quasi-local algebra which is translationally invariant and satisfies the locality condition T A(Λ) ⊂ A(Λ + N) for every finite region Λ ⊂s . A qca is called reversible if T is an automorphism of A(s ). While T constitutes the global rule, the local rule is its restriction to a single cell, T0: A0 → A(N). This definition essentially complies with the respective definition from [70]. However, we do not restrict it to reversible qcas. Moreover, a qca can be proven to be reversible if T is only a homomorphism. 1 For an extended discussion, including qcas on finite lattices, see [70]. The elements of this definition correspond to the characteristics of a ca given at the beginning of this section as follows: ⊲ lattice of discrete cells: an infinite lattice labeled by x ∈s with local observable algebras Ax ⊲ discrete, synchronous global time evolution: a quantum channel T : A(s ) → A(s ) on the quasi-local algebra A(s ) ⊲ uniformity: translational invariance of T ⊲ locality and finite propagation speed: for every finite set Λ ⊂s and the algebra of observables A(Λ) localized on this region, T A(Λ) ⊂ A(Λ + N) with the finite neighborhood scheme N ⊲ local transition rule: the restriction of T to a single site, T0: A0 → A(N) ⊲ reversibility: T is an automorphism. 1 This is a corollary of the structure theorem for reversible qcas [70], which states that the inverse in this case is again a qca. 73
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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
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 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
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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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