  In 1924 David Hilbert conceived a paradoxical tale involving a hotel with an
infinite number of rooms to illustrate some aspects of the mathematical notion
of "infinity". In continuous-variable quantum mechanics we routinely make use
of infinite state spaces: here we show that such a theoretical apparatus can
accommodate an analog of Hilbert's hotel paradox. We devise a protocol that,
mimicking what happens to the guests of the hotel, maps the amplitudes of an
infinite eigenbasis to twice their original quantum number in a coherent and
deterministic manner, producing infinitely many unoccupied levels in the
process. We demonstrate the feasibility of the protocol by experimentally
realising it on the orbital angular momentum of a paraxial field. This new
non-Gaussian operation may be exploited for example for enhancing the
sensitivity of N00N states, for increasing the capacity of a channel or for
multiplexing multiple channels into a single one.
