Quantum Hilbert hotel

Potoček, Václav and Miatto, Filippo M. and Mirhosseini, Mohammad and Magaña-Loaiza, Omar S. and Liapis, Andreas C. and Oi, Daniel K. L. and Boyd, Robert W. and Jeffers, John (2015) Quantum Hilbert hotel. Physical Review Letters, 115 (16). 160505. ISSN 1079-7114 (https://doi.org/10.1103/PhysRevLett.115.160505)

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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 realizing 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 NOON states, for increasing the capacity of a channel, or for multiplexing multiple channels into a single one.