QT3 - The Rules (Superposition)
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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