  In quantum theory, observables with a continuous spectrum are known to be
fundamentally different from those with a discrete and finite spectrum. While
some fundamental tests and applications of quantum mechanics originally
formulated for discrete variables have been translated to continuous ones, this
is not the case in general. For instance, despite their importance, no
experimental demonstration of nonlocality exists in the continuous variables
regime. Attempts to bridge this gap and put continuous variables on a closer
footing to discrete ones used dichotomization. However, this approach considers
only discrete properties of the continuum, and its infinitesimal properties are
not fully exploited. Here we show that it is possible to manipulate, detect and
classify continuous variable states using observables with a continuous
spectrum revealing properties and symmetries which are analogous to finite
discrete systems. Our approach leads to an operational way to define and adapt,
to arbitrary continuous quantum systems, quantum protocols and algorithms
typical to discrete systems.
