An unsharp quantum observable can be considered a realization of a sharp observable if and only if it is commutative. In this paper we describe an explicit procedure for reconstructing such a sharp observable and for establishing the probabilistic correlations between the sharp reconstruction and the given unsharp observable.
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.
refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be
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.