Physicists have created a quantum version of a famous thought experiment: the so-called Hilbert Hotel, which is always full and yet a vacancy can always be found for additional unexpected guests.

Back in 1924, mathematician David Hilbert devised a paradoxical tale to play with notions of infinity in mathematics. He imagined a hotel with an infinite number of rooms. Even when it was completely full, the manager could always accommodate more guests through a simple trick. He would have each guest switch to the next room up. So if they were in room N, they would move to N+1. *Voila*! There is now a vacancy for the arriving guest.

From there, things get a bit more complicated in Hilbert’s classic paradox. What if you get 40 new guests all at once? Or a bus holding an infinite number of new guests? Or an infinite number of buses holding an infinite number of guests? I’ll let Jeff Dekofsky explain how it works in this charming video from TED-Ed:

Does your brain hurt yet? Infinity isn’t a concept that comes naturally to most of us, even in a thought experiment. And now there is a new twist. In a recent paper in *Physical Review Letters*, a group of physicists described how they recreated a quantum version of all that room-switching in the Hilbert Hotel with laser beams.

A quantum system has an infinite number of states, represented by a wave function. The amplitude of any one of those states determines the probability of that state being realized once the wave function collapses. In the latest experiment, the infinite states in the quantum system represent the infinite number of rooms in the Hilbert Hotel. The amplitudes of those states represent the room numbers. The physicists then created “vacancies” by changing one quantum state into another using a laser beam to alter the state.

The physicists even extended the experiment to triple and quadruple infinity, which produced pretty “petals” of light. Check out the photo atop this post. Each “petal” in the top row is a quantum state with an infinite number of values corresponding to an infinite number of hotel rooms. When the physicists multiplied by three, for example, they got triple the number of petals in the bottom row.

Weird, right? You’re doubling or tripling an infinite number of things and getting infinitely many more of them, which seems like it shouldn’t be possible outside the abstract mathematical realm.

“As far as there being an infinite amount of ‘something’ it can make physical sense if the things we can measure are still finite,” co-author Filippo Miatto (University of Waterloo and University of Ottawa) explained to Phys.org. “For example, a coherent state of a laser mode is made with an infinite set of number states, but as the number of photons in each of the number states increases, the amplitudes decrease, so at the end of the day when you sum everything up, the total energy is finite. The same can hold for all of the other quantum properties, so no, it is not surprising to the trained eye.”

It’s an interesting experiment that could eventually prove useful in information processing. Mostly, though, it demonstrates that even at the quantum level, there’s always room at the Hilbert Hotel.

**References:**

Gamow, George. *One Two Three... Infinity: Facts and Speculations of Science*. New York: Viking Press, 1947.

Hilbert, David. (1925) “Über das Unendliche,” *Mathematische Annalen* 95: 161-190.

Kragh, Helge. (2014) “The True (?) Story of Hilbert’s Infinite Hotel.” [arXiv]

Potocek, V. et al. (2015) “Quantum Hilbert Hotel,” *Physical Review Letters* 115: 160505.

[Via *Physical Review Letters*]

*Image: V. Potocek et al./PRL*

## DISCUSSION

From what I recall of my second year quantum courses in university 35 years ago, Hilbert space, as a way of understanding the infinite number of states for most quantum systems, has long been a feature in quantum mechanics.

Anyway, I kinda wish this article went into more detail on the specific scientific applications of this rather than spending so much time on unpacking infinity for the general public.

I mean I get the reasons and the need to do that. Infinity can be very counterintuitive even if our understanding of aspects of it are as logically airtight as any other area of mathematics. I can see that people would have to have that explained to them to wrap their heads around the rest but, to me, it’s a little elementary.

I wanted to know more about how this might apply to things like quantum discord, quantum chaos, the measurement problem and other areas of quantum theory. I mean never mind the technical stuff that makes new widgets for people, instead I want to know how does this new trick illuminate the rest of quantum theory?