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arXiv:quant-ph/0004026 · Journal of Mathematical Physics

Group theoretical quantum tomography

G. Cassinelli, G. M. D’Ariano, E. De Vito, A. Levrero

Published Dec 1, 2000• 29 citations
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Abstract

The paper is devoted to the mathematical foundation of quantum tomography using the theory of square-integrable representations of unimodular Lie groups.
DOI
10.1063/1.1323497

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. The paper is devoted to the mathematical foundation of the quantum tomography using the theory of square-integrable representations of unimodular Lie groups. 1. introduction In Quantum Mechanics a physical system is associated with a Hilbert space H: the states are described by positive trace-class trace-one operators T on H, the physical quantities by self-adjoint operators A on H and the physical content of the theory is given by the expectation values Tr (AT ). The state T is completely determined by Tr (Q n T ) for Q n running on a suitable set fQ n g of observables and, for arbitrary operator A, Tr (AT ) can be computed in terms of Tr (Q n T ). In order to implement this scheme one has to estimate Tr (Q n T ) experimentally, facing the problems arising from statistical errors and instrumental noise. Moreover, the number of experimental data is clearly nite, while A and T are operators on an innite dimensional Hilbert space and the set fQ n g is innite. The problem of determ...
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.