title: Directly measuring mean and variance of infinite-spectrum observables such as the photon orbital angular momentum.
abstract: The standard method for experimentally determining the probability distribution of an observable in quantum mechanics is the measurement of the observable spectrum. However, for infinite-dimensional degrees of freedom, this approach would require ideally infinite or, more realistically, a very large number of measurements. Here we consider an alternative method which can yield the mean and variance of an observable of an infinite-dimensional system by measuring only a two-dimensional pointer weakly coupled with the system. In our demonstrative implementation, we determine both the mean and the variance of the orbital angular momentum of a light beam without acquiring the entire spectrum, but measuring the Stokes parameters of the optical polarization (acting as pointer), after the beam has suffered a suitable spin-orbit weak interaction. This example can provide a paradigm for a new class of useful weak quantum measurements.
authors.affiliation: Dipartimento di Fisica, Università di Napoli Federico II, Compl. University di Monte S. Angelo, Napoli 80126, Italy.
authors.forename: Bruno
authors.initials: B
authors.lastname: Piccirillo
authors.affiliation: Dipartimento di Fisica, Università di Napoli Federico II, Compl. University di Monte S. Angelo, Napoli 80126, Italy.|Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology, Griffith University, Brisbane, Queensland 4111, Australia.
authors.forename: Sergei
authors.identifier: http://orcid.org/0000-0002-5318-3790
authors.initials: S
authors.lastname: Slussarenko
authors.affiliation: Dipartimento di Fisica, Università di Napoli Federico II, Compl. University di Monte S. Angelo, Napoli 80126, Italy.|CNR-SPIN, Compl. University di Monte S. Angelo, Napoli 80126, Italy.
authors.forename: Lorenzo
authors.identifier: http://orcid.org/0000-0002-1154-8966
authors.initials: L
authors.lastname: Marrucci
authors.affiliation: Dipartimento di Fisica, Università di Napoli Federico II, Compl. University di Monte S. Angelo, Napoli 80126, Italy.
authors.forename: Enrico
authors.identifier: http://orcid.org/0000-0002-9130-4151
authors.initials: E
authors.lastname: Santamato