Quantum tic tac toe
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In physics, the game of Quantum tic tac toe is a quantum generalization of tic-tac-toe in which the players' moves are quantum superposition of plays in the classical game. The game was invented by Allan Goff of Novatia Labs.
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[edit] Quote
Quantum tic tac toe offers a way of introducing quantum physics without advanced mathematics, provides a conceptual foundation for understanding the meaning of quantum mechanics, and is fun to play. —Allan Goff, 2006
[edit] The motivation
The rules of Quantum Tic-Tac-Toe are not particularly hard, but they take gaming in a new direction, and the unfamiliarity initially makes learning them somewhat more challenging than for conventional games. It may help, therefore, to consider why this game was invented.
The motivation to invent Quantum Tic-Tac-Toe was to explore what it means to be in two places at once. In classical physics, a single object cannot be in two places at once. In quantum physics, however, the mathematics used to describe quantum systems seems to imply that when we are not looking, quantum particles can be in multiple places at once. (The textbook example of this is the double-slit experiment.) How the universe can be like this is a bit mysterious. There is a disconnect between the mathematics and our mental images of reality, a disconnect that is absent in classical physics. This is why quantum mechanics supports multiple "interpretations." An interpretation is a formal effort to explain what a theory means, to articulate a model. By definition, interpretations are not testable. Testability implies a theory or at least a hypothesis. The success of quantum mechanics is not in dispute, it has been validated over an incredibly wide range of phenomenon and to astonishingly high levels of precision. The problem of multiple mutually incompatible interpretations is therefore a bit of a professional embarrassment.
Because of this conundrum, we sought to devise abstract quantum systems, formal systems whose axiomatic foundation included only a few of the axioms of quantum mechanics. Quantum Tic-Tac-Toe is our most thoroughly studied abstract quantum system, and while it has offered insights that have spawned new research, it also turned out to be a fun and engaging game, a game that provided good pedagogy in the classroom. The rules of Quantum Tic-Tac-Toe attempt to capture several phenomena of quantum systems. These phenomena are superposition, entanglement and collapse. Superposition is the ability of quantum objects to be in two places at once. Entanglement is the phenomenon where distant parts of a quantum system display correlations that cannot be explained by either timelike causality or common cause. Collapse is the phenomenon where the quantum states of a system are reduced to classical states. Collapses occur when a measurement happens, but the mathematics of the current formulation of quantum mechanics is silent on the measurement process. Many of the interpretations of quantum mechanics derive from different efforts to deal with the measurement problem.
[edit] The rules
Quantum Tic-Tac-Toe captures the three quantum phenomena discussed above by adding a single rule to classical tic-tac-toe; the rule of superposition. On each move, two squares must be marked. The pair of marks are called "spooky marks" and consist of either a pair of X's or a pair of O's subscripted with the number of the move. X always moves first, so the X's get subscripted with the odd integers and the O's get subscripted with the even integers. The phenomenon of entanglement is captured by allowing more than one spooky mark in a square. The phenomenon of collapse is captured by specifying that a cyclic entanglement (we'll define that in a moment) causes a measurement. A measurement requires a new type of move where one of the players decides which of the two possible collapses will happen. In a collapse, one spooky mark of every pair connected to the cyclic entanglement evaporates, the other spooky mark converts to a single classical mark. A classical mark fills a square, so future spooky marks may not be played in any square containing a classical mark. The first player to get a 3-row of classical marks wins. We'll discuss later what happens when both players get 3-rows, as can happen since marks become classical at least two at-a-time.
If that all seems a bit complicated, it is, and it isn't. Most of the verbiage is simply defining what it means to be a quantum game. It helps to have a tutor who can demonstrate move by move what is going on.
[edit] Commentary
In our experience, once someone gains experience with the game, the quantum rules come be seen as rather obvious. Perhaps we forget how hard it was as very young children to learn classical tic-tac-toe. We had to learn about boards and pieces, and making moves and taking turns, as well concepts of strategy and tactics. We were actually learning the foundations of games in general. In Quantum Tic-Tac-Toe, a similar phenomenon is occurring, we are learning the foundations of quantum games. Once the foundations of quantum games are learned, the rules to Quantum Tic-Tac-Toe should appear much simpler.
[edit] References
Allan Goff 2006. Quantum tic tac toe: A teaching metaphor for superposition in quantum mechanics. American Journal of Physics, volume 74, issue 11 [1].
