Quantum Tic-Tac-Toe

The Many Worlds Interpretation

One of the more audacious attempts to explain what quantum physics means is Everett’s Many Worlds Hypothesis. In this interpretation, every measurement outcome is actually realized, spawning off a new universe for every possible outcome. Every measurement splits reality. You, and everything else, exist simultaneously in each universe, a multi-verse of parallel dimensions.

As in all interpretations of quantum mechanics, the Many Worlds view does not actually predict any new phenomenon. It isn’t a testable idea, and so is not formally actually a hypothesis. There are also significant philosophical and theological grounds for rejecting this idea. If you make every choice you are faced with, then each "you" in every universe can dodge personal responsibility for your actions. Taken seriously, the Many Worlds idea destroys any possible foundations for morality or ethics. If people behaved in this way, in the limit, it would destroy civilization, and with it, science itself.

That said, the idea is not that crazy. Let us approach the reasonableness of this idea via a real world example - Chess. Consider a player up against a superior opponent. The weaker player has just made a poor move, and realizing it just a little too late, begs his opponent to let him take it back. Being good-natured, and truly desiring to play an interesting game, the superior player is inclined to agree, but then he gets a radical idea. "OK" he says, "but with a twist" and pulling out a second board he sets up the identical position letting his opponent change his weak move in only one of them. "Now we play both games together", he says. They are now playing two separate games of chess, just with a common history. Parallel games, parallel universes.

In Quantum Tic-Tac-Toe, the rule of superposition can be interpreted to imply the existence of multiple games of Classical Tic-Tac-Toe. Call this the classical ensemble. Each successive pair of spooky marks that does not entangle with others splits each classical game in two, doubling the number of games in the ensemble. However, unlike the chess example above, different moves cannot be made in the different classical games. At each move the games in the classical ensemble are duplicated, one spooky mark indicating the classical move to be made in one half, the other spooky mark indicating the classical move to be made in the other half. In the web version the ensemble of classical games is shown below the Quantum Tic-Tac-Toe board, in the Palm Pilot version, the classical ensemble is shown on a separate screen accessible via the Real Games button or the QT3 menu.

In Quantum Tic-Tac-Toe, the classical games implied by superposition form an integrated whole. Superpositions expand the ensemble; entanglements may expand or contract it depending on the details; when an entanglement becomes cyclic, the ensemble is pruned, often to only two classical realities; while collapses (measurements) may prune the ensemble to just one classical reality. In this way, the classical ensemble is different from the Many World’s interpretation, where the parallel universes are created upon measurements and their number steadily grows throughout all time.

Applied to physics, you exist across all the realities in the ensemble. There are not multiple copies of you, the nature of you and of all other objects, is defined by the entire set of realities in the classical ensemble. This avoids the philosophical and theological problems of the traditional Many World’s Hypothesis.

To see this a little more clearly, it is useful to introduce the idea of the state tree. In a state tree, each node represents a possible state of the system, and each branch a legal transition from one state to another. In a classical reality, the evolution of the system describes a single path from the root (starting position or condition) to the present. This is shown in the figure below.

While a quantum system also has a state tree, it is exponentially larger. It is possible, however, to map a quantum system onto the state tree for the underlying classical system. It is just that now; the evolutionary path is a many-branched path, as shown in the next figure.

One each quantum move, superposition branches the existing branches. If the spooky marks entangle this move with others, some branches will branch, others won’t, and some may be terminated. On a cyclic entanglement (a collapse) many branches will be pruned. At any moment, the quantum state of the system consists of all the active branches. But at the end of the game, there is only one branch that extends from root to the end, and looked at from below, it specifies a single classical game. That game was influenced in its development however, by classical realities that now no longer "exist". No wonder quantum games work out differently. This is the source of the ascertainity principle. Note also how, while similar, the classical ensemble idea is substantially different from the Many World’s model.

This idea is novel enough that it is worth a little bit of time just playing with Quantum Tic-Tac-Toe to see the effect of various moves and entanglements on the set of classical games in the ensemble. See if you can make the six moves that lead to the maximum possible number of games (27) in the classical ensemble.