  Binary quantum information can be fault tolerantly encoded in states defined
in infinite dimensional Hilbert spaces. Such states define a computational
basis, and permit a perfect equivalence between continuous and discrete
universal operations. The drawback of this encoding is that the corresponding
logical states are unphysical, meaning infinitely localized in phase space. We
use the modular variables formalism to show that, in a number of protocols
relevant for quantum information and for the realization of fundamental tests
of quantum mechanics, it is possible to loosen the requirements on the logical
subspace without jeopardizing their usefulness or their successful
implementation. Such protocols involve measurements of appropriately chosen
modular observables that permit the readout of the encoded discrete quantum
information from the corresponding logical states. Finally, we demonstrate the
experimental feasibility of our approach by applying it to the transverse
degrees of freedom of single photons.
