Infinity – the mathematical notion of no limits. Our physical world has many limits. Yet mathematically infinity is reality.

To demonstrate this counterintuitive notion of infinity, David Hilbert conceived of a thought experiment – the Hilbert Hotel Paradox. In his experiment, the Hilbert Hotel contains an infinite number of rooms, all of which are occupied. One night, an infinite number of guests arrive looking for rooms. To accommodate the new guests, each current guest is moved to a room that is twice their current room number, leaving all the odd-numbered rooms for the new guests. The tale illustrates the particular nature of infinity where it can be summed to itself without leading to any change.

A group of researchers from the universities of Strathclyde, Glasgow, Rochester, Ottawa and the Institute for Quantum Computing have proved, for the first time, that this paradoxical tale is experimentally feasible by using quantum mechanics to mimic what happens in the Hilbert Hotel.

While visiting the University of Strathclyde in Scotland, IQC postdoctoral fellow Filippo Miatto heard colleagues discussing how to realize a physical version of the Hilbert Hotel by associating the rooms with quantum states. By taking a given system that has an infinite amount of quantum states, they knew that they would need some way to vacate every second level. Miatto had the idea of how to use the Orbital Angular Momentum (OAM) states of light to implement the Hilbert Hotel protocol. OAM is associated with the rotation of an object around a fixed axis, and in the case of a light beam to the rotation of the optical phase around the direction of propagation.

In their theoretical analysis, the team found a suitable transformation in an infinite square well – similar to a box within which a particle cannot escape. Although the particle cannot escape the well, it can occupy an infinite number of energy levels and it can be in any superposition of a specified set of states while the well is being transformed.

Without the ability to actually build an infinite potential well, the team nevertheless realized the protocol using photons and OAM by interpreting it as a potential well wrapped around itself. To do this, researchers needed to cut the light beam using astigmatic lenses, copy it with phase modulators and use an OAM sorter in reverse to stitch it back together (this last step was another novelty in the experiment).

Each time the protocol was run, the researchers could coherently multiply any OAM superposition by three – similar to the concierge multiplying current guests’ room numbers to fit more people in rooms that were initially full. The multiplication left open rooms, or gaps, and that is what occurred in the spectrum of OAM. Following multiplication, the researchers had an infinite set of free states that weren’t occupied and could carry more information.

“Philosophically, after our findings, one could argue that the real world can accommodate for the mathematical notion of infinity in the sense of Hilbert’s paradox,” said Miatto. “Or at least, the formalism of quantum mechanics allows for that.”

By showing that they could physically realize the Hilbert Hotel Paradox, the researchers also found that they could perform deterministic non-linear interactions on the OAM by pre-sorting and then applying distinct Hilbert Hotel operation to the distinct OAM components. This opens up new possibilities for quantum information and computation operations.

The paper, *The Quantum Hilbert Hotel* was published in PRL on October 15, 2015.