Quantum Tic-Tac-Toe

Superposition, Entanglement, & Collapse

Consider a classical system. It is characterized by being in a particular state at a particular time. There are rules it must follow for changing from one state to another. We call them the laws of nature. The state of a classical system is specified by a complete listing of the locations and properties of all the objects of interest.

Classical games provide a good metaphor here. The pieces, cards, dice, etc. are the objects of interest. They all have a location, or a value, or other properties of significance to the game. The game starts in a specified state, and the rules specify how on each move the current state can be evolved into a new state. That is, what are the legal moves. This could involve moving a piece, or changing a property, etc. The sequence of moves is a fair measure of the idea of time, although some games directly incorporate real time as an element of the game.

In the case of Classical Tic-Tac-Toe, the only objects are the marks, X’s and O’s, and their only property is which square they are in. The state of the game is just the position of marks on the board at any time, i.e., after so many moves. The rule for changing the state is simply to place a new mark, different from the previous mark, into any unoccupied square. The board starts empty, X moves first, and the game is over when the first player succeeds in getting 3 of their own marks in a row, or the board is filled without a winner.

Given a classical game, it is possible to construct a quantum game from it by adding three types of rules:
        1. Rules of Superposition
        2. Rules of Entanglement
        3. Rules of Collapse
There may be one or more rules of each type. In Quantum Tic-Tac-Toe, there is only one rule of each type.

	The Rule of Superposition – On every move, two squares are indicated. Such a quantum move consists of a pair “spooky” marks, one in each square. They are identified with the number of the move so that it is clear which spooky marks go together. The move isn’t really in both squares, and it isn’t really in neither. It is potentially in either, but we won’t find out till later which square that move actually ends up in. Think of it as the square root of a move. Another good way to think about is to regard the state on the quantum board as a shorthand for an entire ensemble of classical games of tic-tac-toe in simultaneous play. After the first move of the quantum game, there will be two such classical games. In one of them, X is in one square, in the other, X is in the other square. The first move is in two different positions on the two classical boards. It is in a “super”-position.
	The Rule of Entanglement – Because of the classical rule that a move must be played into an unoccupied square, the pieces on the Tic-Tac-Toe obey a restriction called Fermionic statistics. All this means is that two objects, if they happen to be Fermions, cannot be found in the same state. This restriction applies to the classical real marks, but not to the spooky marks. There is no limit on the number of spooky marks that can be in a single square. However, since only one of them can become real in that square, what happens to a move when it collapses affects other moves. They are “entangled” with each other, like links on a chain.
	The Rule of Collapse – In quantum physics, the quantum state is never actually observed. All nature ever lets us see are classical values. We must deduce what the quantum state was in order to explain the classical results actually found. As you might suspect, this can be fairly difficult, and scientists have had to be very clever to figure out reliable ways of doing this. Mathematics plays a crucial role here, and it is one of the great achievements of science that we have been able to figure out how quantum systems work from experiments that literally provided only clues.

That said, there still remains a fundamental unsolved problem in quantum mechanics. It is called the measurement problem. Scientists don’t actually have a mathematical description of what causes a measurement. We have good rules of thumb, but it remains a situation of, “we know a measurement when we see one.” This is not a terribly satisfactory state of affairs, from a conceptual point of view, but as a practical matter it has still been possible to figure out a great deal about how quantum systems behave, and to build effective technologies off that understanding.

One of the reasons we developed Quantum Tic-Tac-Toe is that we wanted to explore a possible solution to the measurement problem, but in an abstract space away from the current paradigm barriers. This possibility proposes that self-referential quantum states should collapse to classical values. In Quantum Tic-Tac-Toe this idea is captured in cyclic entanglements. In a cyclic entanglement, there is a path from one spooky mark back to itself. Therefore where a move ends up is decided by where that move ends up. It is its own cause. In logic, this is called a circular argument, a type of self-reference.

Part of why this idea is attractive is that self-reference is a kind of nonlinearity. The mathematics of quantum mechanics is strictly linear. Linear equations have many nice properties, one of which is they are generally solvable. Nonlinear equations often are not. Several current approaches to solving the measurement problem add nonlinear terms to the equations of quantum mechanics, and while such ideas are worth exploring, such additions have an ad hoc nature to them scientists have found over the centuries to be unlikely candidates. The nonlinearity of self-reference seems more natural and requires fewer assumptions.

In the formalism of quantum mechanics, the evolution of the state of the system is driven by a linear equation called the wave equation. It is like the rules of a quantum game, but includes the superposition rule. When a measurement occurs, however, this equation is discarded, and a totally different rule is used to predict the outcomes of experiments. This rule is probabilistic and as different from the wave equation as one could imagine. In Quantum Tic-Tac-Toe, making quantum moves is like using the wave equation. It specifies how the quantum state of the system (game) is going to look next. But when a cyclic entanglement occurs, one of the players must make a totally different kind of move, a collapse move, where they choose how the entanglement collapses to classical values. Half the spooky marks vanish, the other half becomes real, and each previously entangled move has collapsed to being in but a single square. The state of each move has been “measured” and found to have a classical value, not a quantum one.

While it remains to be seen if this concept of self-referential entanglement can be made to work in quantum mechanics as the measurement mechanism, it is leading to new research. Quantum Tic-Tac-Toe certainly points out that the radical difference between evolution and collapse of quantum states is a characteristic of quantum systems, even abstract ones such as quantum games.

The Correspondence Principle

When a successful physical theory is replaced with an improved theory, there are still areas where the original theory is good enough. Under these conditions, the new theory must predict the same results as the original theory. When Einstein’s theory of General Relativity supplanted Newton’s theory of gravity, it had to replicate Newton’s results for slow speeds and modest densities of matter. In the limit, it has to correspond with the earlier theory. ndeed, NASA only uses Newton’s theory of gravity for spacecraft trajectories; it being mathematically simpler and accurate enough that other sources of error completely swamp the difference.

Since quantum mechanics is a theory that supercedes classical physics, in the limit of large objects, it too must predict the same results as classical physics. This is called the correspondence principle. Stars, rocks, bumble bees, even bacteria, follow classical physics. It isn’t until we reach the scale of molecules, atoms, and subatomic particles that classical physics fails and must be supplanted by quantum mechanics.

Because Tic-Tac-Toe has only 9 squares, the correspondence principle is not dramatic, but it is still there. Consider a game where O decides on the strategy of immediately collapsing X’s previous move. She places both her spooky marks in the same squares as X did. After a few moves, the board has many classical marks on it, but at most only one pair of spooky marks. It looks very much like a game of Classical Tic-Tac-Toe. That’s the correspondence principle. Even when collapses are rare, any game that goes through all nine moves, ends up with nine collapsed squares, i.e., nine classical marks, and not a spooky mark to be found.

The appearance of the board is not the only place where the correspondence principle applies; it also applies to strategy. In general, the strategy for Quantum Tic-Tac-Toe is significantly different than for Classical Tic-Tac-Toe. However, the smaller the entanglements and the more rapidly they are collapsed, the more the strategy resembles that in Classical Tic-Tac-Toe. So the correspondence principle applies to both states and the evolution from state to state.

The Ascertainity Principle Consider the Quantum Tic-Tac-Toe game shown to the right. On move one, X has staked out the center and a corner, a good strategy based on what works in Classical Tic-Tac-Toe. O does similarly, taking advantage of the fact that X is not yet deterministically in control of the center square, and at the same time preventing X from ever getting a 3-row down the anti-diagonal. Moves three and four proceed for similar tactical reasons. On move five X makes a very threatening play that O must respond to. Her decision to create a cyclic entanglement gives selection of the collapse to X, but any other move would allows X to force a win. X chooses to collapse the entanglement such that he ends up with a single horizontal 2-row, even though this gives O two 2-rows. X can force a win, 1 to ½ from here by playing in squares 4 and 8. However, the classical listing of moves (shown to the right of the game board) makes no tactical sense at all. O could have easily won the classical game on move six simply by playing in square 4, and X should have played there previously on move five to prevent it, but neither X nor O could force these outcomes in the Quantum game. These are not the only poor moves in the classical version. O should have played her first move in the center, and when she didn’t, X should have played his second move in either remaining corner. Of the first six moves in the implied classical game, four of them were tactically dumb. If all an observer has access to is the classical listing, this is a very mysterious result. If the observer is persuaded that X and O are both competent Tic-Tac-Toe players intent on winning, he must conclude that the classical game he witnessed was actually played under quantum rules. Thus it is possible in general, just from the classical listings, to ascertain whether a game of Tic-Tac-Toe was played under quantum or classical rules. Although we usually take it for granted, it is actually a little remarkable that we have been able to deduce that we live in a quantum universe and not a classical one. As it is, this revelation is less than 100 years old. Call this the ascertainity principle.

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.