# You shouldn’t try to pigeonhole quantum physics

## Subatomic particles violate a basic principle underlying the concept of numbers and counting

Just when you thought quantum physics couldn’t get any weirder, it violates the pigeonhole principle.

No, it’s not about a pigeon in a hole that is simultaneously alive and dead. The pigeonhole principle is a basic tenet of mathematics. It illustrates what the very idea of numbers is all about. And it’s easy to state: If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole.

How can anybody, even a quantum physicist, argue with that? All you have to do is be able to count. Guess again, say quantum physicist Yakir Aharonov and collaborators in a new paper about the pigeonhole principle. “It seems … to be an abstract and immutable truth, beyond any doubt,” they write. “Yet … for quantum particles the principle does not hold.”

The basic idea was worked out in rigorous form in 1834 by Peter Gustav Lejeune Dirichlet, a gifted 19th century German mathematician of Belgian descent. He called it *Schubfachprinzip, *which means something like the drawer principle. Nowadays mathematicians whodon’t like pigeons call it the Dirichlet box principle. In that form the principle is stated as whenever more than n objects are distributed in n boxes, then there will be at least one box containing two objects.

Dirichlet showed that this seemingly simple principle could be used to prove more complex mathematical propositions. It has become widely used and appreciated among mathematicians for its power and scope.

“It … encapsulates abstract mathematical notions that go to the core of what numbers and counting are,” Aharonov and colleagues point out. “So it underlies, implicitly or explicitly, virtually the whole of mathematics.”

But in the quantum world, it’s wrong.

To be fair, the quantum spoilsports note, it’s not always wrong. But in some circumstances the quantum math shows that upon further review, Dirichlet’s decision must be reversed.

You could demonstrate this point at home yourself if you are properly equipped with the right lasers and mirrors on a perfectly stable quantum optics laboratory table. Or you can just do the math. Start with three particles and two boxes. Prepare the particles so that each one is in both of the two boxes at the same time — your basic simple quantum mechanical superposition of locations. You can now write the quantum equation for the state of all three particles. You can use it to calculate the probabilities for what will happen when you actually look for the particles.

After all, particles can be in two places at once only when you’re not looking. When you observe a particle to see where it is, it acquires a specific position in one box or the other.

Aharonov and colleagues consider situations in which you choose to independently measure the locations of any two of the particles. Depending on how you conduct your measurements, in some situations, you will find that particles 1 and 2 are both in the same box. But in other circumstances, your measurements will find that they aren’t. And applying those conditions gives the same result no matter which two particles you choose to measure. So you can have a situation where you have more particles than boxes, but no more than one particle in either box.

“In other words,” the physicists write, “we have three particles in two boxes, yet no two particles can be found in the same box — our quantum pigeonhole principle.”

Aharonov and friends say that this new quantum conundrum can be of value in providing deeper insights into entanglement, the mysterious connection that links properties of each of a pair of particles even when they are separated by great distances. And it has implications for other quantum processes.

“The quantum pigeonhole effect has major implications for the understanding of the very nature of quantum interactions,” the physicists write.

But perhaps an even deeper implication of the new work is as a signal of further surprises yet to come. As quantum mechanics nears its 90th birthday, physicists are still uncovering seemingly absurd aspects of the reality it describes. Which, like it or not, is the reality we all live in.

So be on the alert for new quantum assaults on conventional logic. You won’t have to wait long. Just go read the Aharonov paper. At one place the authors point out that you can prepare the particles and conduct measurements in such a way that you get another confusing result: Particle 2 is in the same box as particle 1, particle 3 is in the same box as particle 1, but particles 2 and 3 are not in the same box.

Why? Too bad Walter Cronkite is not still alive. He gave the answer every night at the end of each newscast: That’s the way it is.

*Follow me on Twitter: **@tom_siegfried*