September 28, 2022

# Playing games with quantum interference

One of the most striking features of quantum mechanics is the superposition principle. This principle can be most easily illustrated via the double-slit experiment, which involves a particle that is sent through a plate pierced with two slits. According to our common everyday intuitions, one might expect the particle to always pass either through one slit, or through the other. However, quantum mechanics implies that the particle can in a certain sense pass through both slits at the same time, that is, it can be in a superposition of two locations at the same time. This possibility underlies the phenomenon of quantum interference. Physicists also analyzed multi-slit experiments.

Researchers at the University of Vienna and IQOQI-Vienna (Austrian Academy of Sciences) have made a significant step towards understanding the superposition problem by reformulating interference experiments in terms of information-theoretic games. Their analysis provides an intuitive way of thinking about interference phenomena and its limitations, thereby paving the way towards solving the aforementioned puzzle.

Namely, they have reformulated interference phenomena and multi-slit experiments in terms of parity games.

The game involves two players, Alice and Bob, who are separated by a wall pierced with four pairs of tubes. Each pair of tubes can either be straight or twisted, and the number of twisted pairs is unknown both to Alice and to Bob. Furthermore, Alice has at disposal a certain number of marbles that she can flick through the tubes towards Bob; the players can use these marbles to learn something about the structure of the tubes. The goal of the game is for the players to cooperate and to find out whether the total number of twisted pairs is even or odd, by using the least possible number of marbles.

Now, suppose that Alice throws one marble through one of the tubes, for example through the second one. Bob can then easily infer whether the first pair of tubes is straight or twisted by simply checking whether the marble has fallen through the second tube or through the first one. Analogously, if Alice has at disposal four marbles, she can flick each of them through the right tube of each pair (as it is the case in the figure). Bob can then straightforwardly infer the number of twisted pairs, and thus whether this number is even or odd, thereby winning the game. However, if the number of tubes’ pairs exceeds the number of marbles that Alice has at disposal, then the game cannot be won, as there will always be at least one tubes’ pair, about which Bob can gather no information whatsoever. Therefore, in order to win the game, the players need to use as many marbles as there are pairs of tubes.

On the other hand, quantum mechanics, and more specifically, the superposition principle, enables the players to win the game illustrated in the figure by using only two “quantum marbles!” One way of understanding where this enhancement is coming from is to remember, as it was stated earlier, that a quantum particle can “pass through two locations at the same time.” Two quantum marbles can thus “simultaneously pass through four locations,” thereby mimicking the behavior of four ordinary (classical) marbles. “In this game, marbles behave analogously to tokens that can be inserted through the tubes.

When Alice inserts an ordinary classical marble, it is as if she inserted 1 penny. On the other hand, as quantum theory allows marbles to pass through 2 tubes at the same time, each quantum marble is worth 2 pennies. The value of the tokens is additive: for example, in order to win the game, Alice can either insert 4 classical marbles or 2 quantum marbles, as the total token value is in both cases equal to 4 pennies.

On the other hand, recall that a quantum particle cannot pass through more than two locations at the same time: this is reflected in the fact that Alice and Bob cannot win the game by using less than two quantum marbles. Hence, in order to win the game, the number of quantum marbles sent by Alice needs to be equal to at least half of the total number of tubes’ pairs.

In their work, the researchers have analyzed more general formulations of this game and have studied the players’ performance depending on the number of particles and on whether the particles are classical, quantum, or of more general and hypothetical kinds.

All in all, parity games provide an alternative description of quantum interference within a more general and intuitive framework, which will hopefully shed light on novel features of quantum superposition, similarly to how the study of quantum entanglement has been deepened through the formulation of the so-called nonlocal games. (SciTechDaily)

Their work has been published in the journal Quantum.