Gianni Cassinelli

Unitary measurements of discrete quantities in quantum mechanics

The pure measurements of discrete physical quantities are characterized within quantum theory of measurement and their unitary representations are given. Probabilistic aspects of measurements related to the so‐called strong correlation conditions and a probabilistic characterization of the first kind measurements are examined. The problem of the objectification of the measurement result is analyzed in terms of a classical behavior of the measuring apparatus. As a by‐product a generalization of the Wigner–Araki–Yanase theorem is given.

Positive operator valued measures covariant with respect to an irreducible representation

Given an irreducible representation of a group G, we show that all the covariant positive operator valued measures based on G/Z, where Z is a central subgroup, are described by trace class, trace one positive operators.

Square-Integrability of Induced Representations of Semidirect Products

We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form where x0 is in the dual group of A, G0 is the stability group of x0 and m is an irreducible unitary representation of G0∩H. We give a new selfcontained proof of the following result: the induced representation is square-integrable if and only if the orbit G[x0] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of .

Group theoretical quantum tomography

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

Covariant localizations in the torus and the phase observables

We describe all the localization observables of a quantum particle in a one-dimensional box in terms of sequences of unit vectors in a Hilbert space. An alternative representation in terms of positive semidefinite complex matrices is furnished and the commutative localizations are singled out. As a consequence, we also get a vector sequence characterization of the covariant phase observables.

A Theorem of Ludwig Revisited

Using a recent result of Busch and Gudder, we reconsider a theorem of Ludwig which allows one to identify a class of effect automorphisms as the symmetry transformations in quantum mechanics.

On the quantum theory of sequential measurements

The quantum theory of sequential measurements is worked out and is employed to provide an operational analysis of basic measurement theoretical notions such as coexistence, correlations, repeatability, and ideality. The problem of the operational definition of continuous observables is briefly revisited, with a special emphasis on the localization observable. Finally, a brief overview is given of possible applications of the theory to various fields and problems in quantum physics.

Wavelet transforms and discrete frames associated to semidirect products

We consider a semidirect product G=Rn×′H and its unitary representations U of the form IndG0G(p0m) where Ind is the unitary induction, p0 is in the dual group of Rn, G0 is the stability group of p0, and m is a unitary representation of G0∩H. We give sufficient conditions such that U defines a wavelet transform and a discrete frame.

ON ACCARDI'S NOTION OF COMPLEMENTARY OBSERVABLES

We discuss some examples of complementary observables, in the sense of Accardi. We show that the trace property is not sufficient to determine such a pair of observables.

Positive operator valued measures covariant with respect to an Abelian group

Given a unitary representation U of an Abelian group G and a subgroup H, we characterize the positive operator valued measures based on the quotient group G/H and covariant with respect to U.