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Journal of Mathematical Physics

Sharp reconstruction of unsharp quantum observables

Roberto Beneduci, Giuseppe Nisticò

Published Nov 22, 2003• 21 citations
Publisher

Abstract

A well defined procedure exists which allows us to “reconstruct” a sharp, i.e., standard, quantum observable A starting from a given commutative unsharp observable F. In this work we prove that the outcomes of measurements of F can be consistently interpreted as the result of a stochastic diffusion of outcomes of its sharp reconstruction A. Furthermore, for every sharp observable B, such that F is unsharp realization of B, we explicitly construct a real mapping g such that A=g(B).

DOI
10.1063/1.1623615

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A well defined procedure exists which allows us to “reconstruct” a sharp, i.e., standard, quantum observable A starting from a given commutative unsharp observable F. In this work we prove that the outcomes of measurements of F can be consistently interpreted as the result of a stochastic diffusion of outcomes of its sharp reconstruction A. Furthermore, for every sharp observable B, such that F is unsharp realization of B, we explicitly construct a real mapping g such that A=g(B).
Given a commutative POV measure [Formula: see text] on the Borel σ-algebra of the reals it is possible to construct a PV measure [Formula: see text] (the sharp reconstruction of F) such that the outcomes of the measurement of F can be interpreted as the random diffusion of the outcomes of the measurement of E. On the other hand Neumark's theorem ensures the existence of an extended Hilbert space [Formula: see text] and of a PV measure [Formula: see text] such that [Formula: see text] for every [Formula: see text]. We consider the projection [Formula: see text] of the self-adjoint operator [Formula: see text] corresponding to E+ and the self-adjoint operator A corresponding to the sharp reconstruction E and show the existence of a function G: [0,1] → [0,1] such that [Formula: see text]. An example for which the last relation can be inverted is shown.
A commutative positive operator valued (POV) measure F with real spectrum is characterized by the existence of a projection valued measure E (the sharp reconstruction of F) with real spectrum such that F can be interpreted as a randomization of E. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark’s extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of F and the sharp reconstruction of F. The relevance of this result to the theory of nonideal quantum measurement and to the definition of unsharpness is analyzed.
The so called spectral decomposition method of quantization involving spectral decompositions of the identity operator has been proposed. The quantization yields the standard (sharp) quantum observables in the form of Hermitean operators acting in the Hilbert space of square-integrable functions. Here, the quantizable classical observables are limited as in the case of geometric quantization. It has been shown that this quantization leads to the same results as those yielded by geometric quantization in the position representation as well as in the momentum representation. The method of quantization also has been formulated in the representation concerning any pair of canonical observables. Then, the quantization has been generalized by replacing the spectral measure appearing in the spectral decompositions of the identity operator with positive-operator-valued measure constructed by using suitable smearing function (e.g. confidence distribution) so that the quantization yields unsharp quantum observables. It has been shown that the unsharp quantum observables are yielded from the quantization of the so called unsharp quantizable classical observables. Some properties of unsharp classical observables have been investigated. The explicit expressions of unsharp quantum observables have been obtained.
The so-called spectral decomposition quantization (SDQ) involving spectral decompositions of the identity operator has been proposed. The quantization yields the standard (sharp) quantum observables in the form of Hermitian operators acting in the Hilbert space of square-integrable functions. Here, the quantizable classical observables are limited as in the case of geometric quantization (GQ). It has been shown that this quantization leads to the same results as those yielded by geometric quantization in the position representation as well as in the momentum representation. The method of quantization also has been formulated in the representation concerning any pair of canonical observables. Then, the quantization has been generalized by replacing the spectral measure appearing in the spectral decompositions of the identity operator with positive-operator-valued measure constructed by using suitable smearing function (e.g. confidence distribution) so that the quantization yields unsharp quantum observables. It has been shown that the unsharp quantum observables are yielded from the quantization of the so-called unsharp quantizable classical observables. Some properties of unsharp classical observables have been investigated. The explicit expressions of unsharp quantum observables have been obtained.