A pedagogical presentation of a $C^\star$-algebraic approach to quantum tomography

TL;DR

This paper introduces a pedagogical approach to quantum tomography using the $C^ riangleright$-algebraic framework, connecting states and observables through group functions and illustrating with simple examples.

Contribution

It presents a novel, accessible presentation of quantum tomography within the $C^ riangleright$-algebraic approach, emphasizing the role of group functions and criteria for identifying quantum tomograms.

Findings

Tomographic distributions for finite and compact groups are formulated.

A criterion for recognizing quantum state tomograms is established.

The $C^ riangleright$-algebraic approach provides a unified view of states and observables.

Abstract

It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrodinger-Dirac picture of quantum mechanics on Hilbert spaces. In this picture states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture,which has evolved in the $C^\star-$algebraic approach to quantum mechanics, starts with the algebra of observables and introduce states as a derived concept. The equivalence between these two pictures amounts essentially, to the Gelfand-Naimark-Segal construction. In this construction, the abstract $% C^\star-$algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation one defines may be reducible or irreducible, but in either case it allows to identify an unitary group associated with the $C^\star-$algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group, it follows that also quantum tomograms are strictly related with appropriate functions (positive-type)on the group. In this paper we present, by means of very simple examples, the tomographic description emerging from the set of ideas connected with the $C^\star-$algebra picture of quantum mechanics. In particular, the tomographic probability distributions are introduced for finite and compact groups and an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state is formulated.