The Quantum Hilbert Hotel

TL;DR

This paper demonstrates a quantum analog of Hilbert's hotel paradox using infinite-dimensional state spaces, introducing a protocol that coherently maps infinite eigenstates and experimentally realizes it with orbital angular momentum, with potential applications in quantum sensing and communication.

Contribution

It introduces a novel quantum protocol that maps infinite eigenstates coherently, experimentally demonstrating the quantum Hilbert hotel concept in orbital angular momentum states.

Findings

Successfully implemented the protocol experimentally.

Produced infinitely many unoccupied levels in a quantum system.

Potential applications in quantum sensing and multiplexing.

Abstract

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.