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.