When [[TiqTaqToe with d4 dice]], it is possible to include an extra rule that allows the player, upon making a move, to assign each part of the wavefunction a binary phase, i.e. "plus" or "minus". This is how: If the d4 is placed on the board like a right-side up triangle ($\triangle$), then the phase is positive, and if the d4 is placed like an upside-down triangle ($\nabla$), then the phase is negative. ![[image 23.jpg]] The player can furthermore choose the phase independently for each part of the wavefunction: ![[image 22.jpg]] Of course, global phases are unobservable, so if you measure a single die it doesn't matter which way it is pointing, and we have therefore not yet introduced any new game mechanic of consequence. This changes when we add the rule[^2] that *opposite phases cancel between players*. For example, if a player entangles with the opponent using opposite phases ![[image 24.jpg]] then after measurement, both squares will be empty with 100% probability: ![[image 25.jpg]] *The amplitudes[^1] have cancelled!* The player can also choose to cancel only part of the wavefunction: ![[image 26.jpg]] Here, measuring the board yields an X or O in the middle square with 50/50 probability, and nothing else, equivalent to the following non-unitary[^3] board. ![[image 29.jpg]] Note, however, that the dice stay on the board until measurement, even when the phases will eventually cancel completely—such squares can not be treated as empty until *after* the board has been fully measured. Entangling with a superposition using opposite phases, meanwhile, cancels only part of the opponents wavefunction, and thus only part of your own: ![[image 27.jpg]] Upon measuring, the board will be equivalent to the following (non-unitary) one: ![[image 28.jpg]] As we see, unlike the base game, the probability that a colour you place on the board will end up being observed *somewhere* on the board can now be less than 100%—and even 0%! Now, it is natural to wonder if this might not lead to a game that simply never ends. However, remember [[TiqTaqToe with d4 dice|the first of the game's two main rules]]: Every move must involve an empty square. This means that there will typically be at least *some* probability that measuring the board yields a new X or O, which progresses the game. There is one exception: If there are an even number of empty squares left on the board, and the first player insists on using classical moves only, then the second player can prolong the game indefinitely: ![[image 30.jpg]] Measuring takes us back to the start, and given that the second player went last, the first player remains first after measurement: ![[image 31.jpg]] However, this (rather annoying) strategy can be thwarted simply by using at least one quantum move, at which point the probability that at least one new piece is observed on the board is guaranteed to be larger than zero. Take the following example. Here, it is O's turn, and O is guaranteed to win *if* it can only get to place its piece in the middle square: ![[image 36.jpg]] If O is limited to classical moves, then the optimal strategy for X is simply to stall by continuously cancelling O's amplitude. ![[image 35.jpg]] On the other hand, a single superposition move is enough for O to win the game. ![[image 37.jpg]] And so I conjecture (though I'm certainly not sure!) that indefinite games are completely avoidable. It is definitely true, however, that including phase cancellations will *prolong* the game! I leave it up to the reader to decide whether this is a bug or a feature. [^1]: The relationship between a quantum amplitude $\alpha$ and the classical probability $P$ that the part of the wavefunction that is multiplied by $\alpha$ will be observed, is $P = \alpha^*\alpha$, where the star denotes complex conjugation. Since we only deal with real-valued amplitudes here, this can be written $P = \alpha^2$. And so we get a positive probability, even when $\alpha$ is negative. As [[Scott Aaronson]] is fond of saying, quantum theory is simply the square root of probability theory! [^2]: This is more of a "quantum-inspired" rule than a strictly quantum mechanical one, as in order to cancel, the phases would really have to relate to the same colour, or at the very least the same X/O value, rather than the opponent's. [^3]: Quantum physics insists on unitarity, which in our case translates roughly to a board configuration being allowed if, by ignoring the phases, the probabilities sum to one for each colour on the board.