Quantum Hilbert Hotel


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.

Coherent OAM multiplication.—Top row: Near field of input coherent superpositions. Bottom row: Tripled output states. The number of petals is 6|ℓ|, as expected from a coherent operation.

Although not in the form of a real hotel made of brick and cement, the Czech physical Václav Potoček, Quantum Theory department researcher at the University of Glasgow, now recreated a Hilbert Hotel in quantum version, using a beam of light.

In Hilbert experience, the mathematician explains that in a depleted hotel, but with an infinite number of rooms, new rooms can always be created, and can always be accommodated more guests because the hotel manager could simply "change" all the guests present for a new room and put more guests in rooms that are vacant.

Hilbert even proposes two rules for changing guests.

With one of the rules, it creates a new room and all guests move to the room with the number above the room they are in, freeing the room number 1 for additional guests.

With the other rule, guests move to the room that has the number that is twice the number of the room they are in, creating an infinite number of new rooms and freeing the odd rooms.

In their study, published in the journal Physical Review Letters, the team of Václav Potoček has now proposed two ways to model this paradox - a theoretical and experimental.

Both use the infinite number of quantum states of a quantum system to represent the infinite number of rooms in a hotel.

The theoretical proposal Potoček uses the infinite number of energy levels of a particle in a quantum system, known as potential well, while the experimental demonstration using infinite number of orbital angular momentum states of the light.