Quantum Tic-Tac-Toe

Actually ... there is not a challenge this month. Instead we are going to take an extended look at the implications of the January challenge.

January's challenge was to devise examples of several kinds of causality. The simpler kinds are time-like causality (a physics-jargon name for ordinary cause-and-effect) and common-cause causality (a special case of time-like, where we are interested more in the associated "effect" events than we are in the event that caused them all).

We illustrated time-like causality using a simple classical tic-tac-toe game. In the game shown to the left, X has moved twice and O once, so it is O's turn. What should she do? Clearly, she has to block X in square 1, or she loses. One could argue — we do argue — that X's second move causes O's second move to be in square 1. Granted, it is not strictly deterministic; we have allowed "causality" to include not just the rules, but also tactics.

This next game also illustrated timelike causality. X's only reasonable moves are in square 8 or square 9; they are the only moves that offer him a chance of winning. O's response will be to mark the remaining unoccupied square in the bottom row: if X marks square 8, she will mark square 9, and vice versa, because otherwise he wins. Even though there are two possibilities here, X's move is therefore still the cause of O's response; and the same response will follow the same move, however many times this game is repeated. We can borrow a term from physics, and say that X's move and O's move are correlated.

This example illustrates common-cause causality: O's next move and X's move after that, are both determined by the situation you see on the board (again allowing tactics, as well as the rules, to determine what may happen). O may place her mark in square 8 or square 9; X will mark square 9 or square 8, and win. X has two ways to win; O can block either one of them and he will take the other. Again we can say that the two moves are correlated, because the same move will always be followed by the same response.

Now, what about the third type of causality, which was first revealed by quantum mechanics? This type of causality appears unlimited by distance, with no temporal separation between cause and effect (as long as the observer is at rest with respect to the two events). So, what we want is a cause, and an immediate effect, that are as widely separated as possible. On a Quantum Tic-Tac-Toe board, that means that cause and effect must be in opposite corners.

So, let us construct such a game. Here is the move list, and a picture of the board after move 5.

Move X O 1 4 - 2 2 - 3 3 4 - 9 6 - 1 5 1 - 8    7 8 - 7    9		6 8 9 - 6 1 - 2 9 - 6 1 - 4 9 - 8 1 - 2 9 - 8 1 - 4 1 - 2 9 - 6 1 - 2 9 - 8 1 - 4 9 - 6 1 - 4 9 - 8

This game looks unbalanced, because two of the moves are not shown. Moves 6 and 8 are cause and effect — or effect and cause — and there are two choices for each move. There is also the choice of which corner to move in first, so we have eight possible games. The choices for moves 6 and 8 are shown to the right of the move list. In all eight cases, move 8 will make a cyclic entanglement that collapses the whole game. Here is the list of possible moves 6 and 8 again, showing both classical games that could result from the collapse.

6 8 Classical Games 9 - 6 1 - 2 9 - 6 1 - 4 9 - 8 1 - 2 9 - 8 1 - 4		6 8 Classical Games 1 - 2 9 - 6 1 - 4 9 - 6 1 - 2 9 - 8 1 - 4 9 - 8

Let us see what we can learn from these games. We are interested in the top left corner and the bottom right corner. The bottom right corner contains an X or an O in all the games, depending on how the cycle is collapsed. The top left corner contains an X or an O, in some games, but always an O in other games. If the move on the lower right of the board is in squares 9 and 8, then square 1 may contain an X or an O. If the move on the lower right is in squares 9 and 6, then square 1 will always an O, regardless of how the cycle is collapsed.

Let us look at it from the other direction. If the move on the upper left of the board is in squares 1 and 2, then square 9 may contain an X or an O. If the move on the upper left is in squares 1 and 4, then square 9 may contain ... an X or an O.

The move in the upper left corner has no effect on what happens in the lower right. The move in the lower right corner, on the other hand, can change what happens in the upper left corner. It is reasonable to infer that the move in the lower right corner, is the cause of an effect in the upper left corner.

Now, notice something very strange about this. It does not matter whether the move in the lower right corner happens before, or after, the move in the upper left corner. The cause could precede the effect, or the effect could precede the cause! Even stranger, the cause and effect happen at the same time, from a quantum point of view: they occur when the cycle collapses. It is from the classical point of view, looking at the classical move list, that it seems they could occur in reverse order.

In physics, similar behavior is characteristic of spacelike causality, also known as nonlocality because the causal connection cannot be local (that is, mediated by something moving at lightspeed or slower). This idea that two related events can be separated by a distance greater than light could traverse (implying that ordinary causes cannot explain the connection), comes directly out of quantum mechanics. It was a controversial idea. Nonlocality implies two impossible things: faster-than-light causality, and backwards-in-time causality. FTL is a straightforward development. If an effect occurs simulataneously with its cause, over a great distance, then of course it is faster than light. Backwards-in-time is not as straightforward, but is still clearly implied by nonlocality. Causality can be time-reversed only from some points of view, however; it depends on how fast and in which direction the observer is moving with respect to the events being observed.

From any point of view, backwards-in-time causality was abhorrent to physicists. This idea caused nonlocality in general to be regarded as a wrong idea, and therefore quantum mechanics to be flawed (incomplete, it was called, since there was no question of arguing that quantum mechanics was wrong; it had already yielded too many correct predictions of experimental results). In 1935, Einstein and two colleages (Podolsky and Rosen) wrote a paper to demonstrate that quantum mechanics could not be complete. The paper started with quantum entanglement, and deduced phenomena that should result (such as "instantaneous" effects over great distances). Einstein hoped to provoke the development of a quantum mechanics that would be complete (and that would, of course, disallow such preposterous things as "spooky action at a distance" and its correlaries).

It took a number of years, but eventually other physicists determined that Einstein and his colleagues were wrong, and why. J.S. Bell determined that common-cause causality could not explain what seemed to be nonlocal causality. N. Gisin determined that timelike causality could not in any way be twisted to explain nonlocal causality. A. Aspect, in 1982, and Gisin, in 1998, also performed experiments that demonstrated nonlocal causality; cause and effect were measured over a spacetime interval much greater than light could traverse. Such experiments could not be performed, of course, until technology allowed precise measurement of the brief times involved over terrestrial distances. In Einstein's day, such an experiment would have to be set up over a distance equalling the span of the entire solar system.

One final oddity of all this: relativity clearly showed that space and time are symmetric. If you calculate what an observer would see, travelling at high speed past some pair of timelike-connected events, he would always see that the cause preceded the effect. But depending on his direction of motion, he might see the cause located to the left or to the right of the effect. Time-ordering is preserved; space ordering is not. But if the events were spacelike-connected ("faster than light" from our point of view), he would always see the events in the same spatial relationship (if he saw the cause to the left, effect to the right, he would always see them that way no matter which direction he was moving). But he might see the cause precede the effect, if he passed in one direction, and the effect precede the cause if he passed in another direction. For spacelike-connected events, spatial ordering is preserved, while temporal ordering is not. Which is exactly what we saw on the Quantum Tic-Tac-Toe board.