Quantum Tic-Tac-Toe

This month's challenge originated in some thoughts about causality. Let us first review those thoughts.

When two things are observed always to happen together, the question arises of whether (and how) the two are connected. If they are connected, it can be in one of two ways. Either one of the things caused the other to happen, or some other thing caused both things to happen. The name for the two situations are, respectively, time-like causality and common-cause causality.

Early in the twentieth century, the new discipline of quantum mechanics suggested a third type of causality, which Einstein disparaged as "spooky action at a distance." This kind of causality resembles common-cause, in that two things appear to be linked; but there doesn't seem to be a common cause. It resembles timelike causality, but the separation is actually spacelike (that is, the things are separated by a distance greater than light could span in the available time). This new kind of causality is demanded by quantum mechanics, and indeed has been observed (Aspect, in 1982, and Gisin, in 1997). But it has never been explained.

So now, let us return to the Challenge. We can model all three kinds of causality, using classical and quantum tic-tac-toe, if we make two assumptions: we assume that both players are playing to win, and will not pass up an opportunity if it is offered; however, both players are only moderately competent – in particular, neither can “look ahead” more than two moves.

As an illustration of what we are looking for, consider the game to the right. What is O's next move? Since she can look ahead two moves, she knows that she must place her mark in square 1 – or X will do so and win. We can, in the context of the situation, claim that O's move in square 1 is a consequence of X's last move.

Here is another classical game. What are X's best choices for his next move? How will O respond, in each case? What kind of causality does this illustrate?

Here is a third classical game. What are O's best choices for her next move? How will X respond in each case? What kind of causality does this illustrate?

The third type of causality cannot be illustrated in classical tic-tac-toe. Can you devise a quantum tic-tac-toe game that illustrates spooky action at a distance? Hint: You may assume that “lightspeed” is one square per move.

Last month's challenge was to figure out, from a present classical reality, which classical reality was discarded in the collapse.

For the two-move situation, this is easy. Both moves have to be in the same two squares to cause a collapse, and so the reality that was discarded is the one where the marks are exchanged. That is, if the reality you see has an X in square 1 and an O in square 2, then the discarded reality had an O in square 1 and an X in square 2.

The three-move situation is not as easy. To get a collapse, we must have three moves in three squares. There are three ways to put a quantum move in three squares. But we cannot have a collapse on move 2, so O only has two choices on move 2. So there are 3×2×3 = 18 possible quantum situations. But there are only three ways to arrange 2 Xs and an O in three squares, so the likelihood that we can uniquely identify the discarded reality, is small.

In fact, the three-move case is small enough to examine exhaustively, and it turns out that there are ten three-move quantum games that yield each of the three patterns of three marks in three squares. So the discarded reality simply cannot be determined by looking at the remaining reality.

The four-move case is even worse: there are six ways to arrange two Xs and two Os in four squares. There are also six ways to put a quantum move in four squares, so there are 6×5×4×6 = 720 four-move games that collapse completely. Again, there are too many choices to uniquely determine the discarded reality, after the collapse.