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53 General Setting: Quantum Processing of a Measure Space
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mechanics However, it is precisely at this stage that the quantum processing of X differs from signal processing in at least three points: (i) not all square-integrable functions are eligible as quantum states, (ii) a quantum state is de ned up to a nonzero factor, (iii) among the functions f (x), those that are eligible as quantum states and that are of unit norm, X | f (x)|2 (dx) = 1, give rise to a probabilistic interpretation: the correspondence X | f (x)|2 (dx) is a probability measure, which is interpreted in terms of localization in the measurable set and which allows one to determine mean values of quantum observables, which are (essentially) self-adjoint operators de ned in a domain that is included in the set of quantum states The rst point lies at the heart of the quantization problem (to which we devote the second part of the book): what is the more or less canonical procedure allowing us to select quantum states among simple signals In other words, how should we select the true (projective) Hilbert space of quantum states, denoted by K, that is, a closed subspace of L2 (X , ), or equivalently the corresponding orthogonal projector I K This problem can be solved if one nds a map from X to the Hilbert space K, x |x K, de ning a family of states {|x }x X obeying the following two conditions: normalization x|x = 1 , resolution of the unity in K |x x| (dx) = I K ,
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where (dx) is another measure on X, usually absolutely continuous with respect to (dx): this means that there exists a positive measurable function h(x) such that (dx) = h(x) (dx) The explicit construction of such a set of vectors as well as its physical relevance are clearly crucial It is remarkable that signal and quantum formalisms meet again on this level, since the family of states is called, in a wide sense, a wavelet family [56] or a coherent state family [11] according to the practitioner s eld of interest Two methods for constructing such families are generally in use The rst one rests upon group representation theory: a speci c state or probe, say, |x 0 , is transported along the orbit {|g x 0 = x , g G} by the action of a group G for which X is a homogeneous space Irreducibility (Schur lemma) and unitarity conditions, combined with square integrability of the representation in some restricted sense, automatically lead to properties (54) and (55) Various examples of such group-theoretical constructions are given in [10, 11] The second method has a wave-packet
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5 Coherent States: a General Construction
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avor in the sense that the state |x is obtained from the superposition of elements in a xed family of states {| } that is total in H: |x =
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Here, the complex-valued x-dependent measure has its support contained in the support of the spectral resolution E (d ) of a certain self-adjoint operator A, and the | s are precisely eigenstates of A: A| = | The choice of the operator A is ruled by the existence of the experimental device that allows us to measure all possible and exclusive issues Sp(A) of the physical quantity precisely encoded by A The eigenstates can be understood in a distributional sense so as to put into the game of the construction portions belonging to the possible continuous part of the spectrum of A Examples of such wave-packet constructions are given in [57 59], and here we will follow a similar procedure For pedagogical purposes, we now suppose that A is a self-adjoint operator in a Hermitian space (with nite dimension, say, N + 1) or a separable Hilbert space (with in nite dimension N = ), say, H, of quantum states or of something else, it does not matter Let us assume that the spectrum of A has only a discrete component, say, {a n , 0 u n u N } Normalized eigenstates of A are denoted by |e n and they form an orthonormal basis of H Next, suppose that the basis {|e n }0unuN is in one-to-one correspondence with an orthonormal set { n (x)}0unuN of elements of L2 (X , ) The generic H could be the Hilbert space K, subspace of L2 (X , ), but we keep our freedom in the choice of realization of H Furthermore, and this a decisive step in the wave-packet construction, we assume, in the case N = , that 0 < N (x) =
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