Allan Goff’s Entanglements

          Quantum tic-tac-toe adds just one rule to the venerable child’s game of classical tic-tac-toe, a rule of superposition. On every move, two marks must be placed in separate squares. These two marks are subscripted with the number of the move so X gets the odd number moves {(X₁, X₁), (X₃, X₃),…} and O gets the even number moves {(O₂, O₂), (O₄, O₄)…}. These pairs of marks are called “spooky” marks in allusion to Einstein’s comment about how the nonlocality of matter in quantum systems implies “spooky action at a distance.” The quantum tic-tac-toe board is drawn with the dividing lines doubled (to reinforce the idea that play requires a pair of spooky marks) and with the squares numbered from 1 to 9. Quantum moves are indicated with hyphens, 1–3, 6–9, … Classical boards are smaller with single lines, unnumbered squares, and utilize only unsubscripted marks.

            Superposition in quantum systems seems to imply that an object can be in two places at once, but only when we are not looking at it! Whenever we do look, that is, whenever a measurement is performed, a particle always ends up in only one place. The act of measurement yields classical values, not quantum ones. To predict the pattern of observations, the formalism of quantum mechanics implies the particle was in two places at once before we looked. 

            Quantum tic-tac-toe provides superposition with an immediate and obvious interpretation. Figure 1 shows the first move of a game where X places his spooky marks in squares 1 and 2. A superposition in quantum tic-tac-toe means that we are really playing two games of classical tic-tac-toe at once. In the first classical game X has moved to square 1, in the second classical game X has moved to square 2. The two classical games are in simultaneous play; they are not independent. Together they are called the classical ensemble and are isomorphic with the state of the game on the quantum board. 

Figure 1.

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            Quantum tic-tac-toe was developed in part to provide both a metaphor and an interactive activity for students to grapple with the weirdness of the quantum world. It requires neither mathematical training nor experimental apparatus of any kind. Although it can be played on paper or on a whiteboard, the most effective medium is a Web-based refereed board.

            Quantum tic-tac-toe is a variation on classical tic-tac-toe that formally adds only one rule, a rule of superposition. The paper consists of three main sections: rules, play, and metaphors. The rules section introduces superposition and the concepts that directly derive from it. Section III elaborates on common aspects and situations that result during the course of play. Section IV explores the features of the quantum world that quantum tic-tac-toe mimics and constitutes the heart of the paper.

            The intent of this paper is to provide an introduction to quantum tic-tac-toe in sufficient depth for classroom use. The teacher can mix and match lectures, play, and challenges as appropriate for the students’ level and coursework. Our expectation is that many students will advance quickly in their understanding of quantum tic-tac-toe and begin contributing observations to the class sometimes ahead of their teacher.

Quantum tic-tac-toe was developed as a metaphor for the counterintuitive nature of superposition exhibited by quantum systems. It 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. A single superposition rule is added to the child’s game of classical tic-tac-toe. Each move consists of a pair of marks subscripted by the number of the move (“spooky” marks) that must be placed in different squares. When a measurement occurs, one spooky mark becomes real and the other disappears. Quantum tic-tac-toe illustrates a number of quantum principles including states, superposition, collapse, nonlocality, entanglement, the correspondence principle, interference, and decoherence. The game can be played on paper or on a white board. A Web-based version provides a refereed playing board to facilitate the mechanics of play, making it ideal for classrooms with a computer projector.