## Level 3 – Advanced: TiqTaqToe with Interference It is time for the *red* numbers to enter the stage. As mentioned in [[TiqTaqToe with Entanglement]], one sense in which quantum probabilities differ from classical probabilities is that they can be both positive and negative (and, as it turns out, anywhere "in between"!) This is where the red and blue numbers come in: Simply put, if the die belongs to player $X$, then a blue number represents "$-Xquot;, while a red number represents "$-Oquot;. And vice versa for player $O$: Blue numbers are $-O$ and red numbers are $-X$. In other words, red numbers represent *minus the opponent's symbol*, while blue numbers represent *minus your own symbol*. Blue and red $2$s, then, are as you might expect: half-amplitudes of such states. Quarter amplitudes, meanwhile, are represented by blue and red *$3$s*, since we have run out of $1$s to use. (In later versions of the dice, $3$s will be replaced with blue and red $1$s.) We will meet the blue $2$s in the next level, [[TiqTaqToe with Complementarity]], and negative *quarter* amplitudes will not become relevant until Level 5. *This* level is all about the red $2$s! ### The Cancelling Move The cancelling move is performed exactly like the entangling move, except that now one of the two $2$s that you place as part of the move is red (you choose which). The other $2$ is blank, as usual. ![[IMG_0004.jpeg|400]] ![[IMG_0001.jpeg|400]] The game then adjusts itself, just like it would for an entangling move, ![[IMG_0002.jpeg|400]] but the resulting *state* is a very different one! Now, the square that contains the red $2$ will be guaranteed to end up *empty* in the observation phase. Why? Well, recall that the red $2$ represents *minus half the opponent's piece*: What we have here is then half $X$ minus half $X$. That gives empty! ![[IMG_0003.jpeg|400]] The other square, meanwhile, is either $X$ or $O$ with 50/50 probability. This means that one of the two players will *not* end up with a piece on the board after the cancelling move! Note, however, that the middle square remains filled-in until the observation phase, so that even though it is guaranteed to *end up* empty, no player can place in it until it has been observed, as that would break the *max 2 dice per square* rule (see [[TiqTaqToe Player Reference]]). ### The Half-cancelling Move Again, the half-cancelling move is performed just like the half-*entangling* move, except that now one of the two $2$s is red. ![[IMG_0005 1.jpeg|400]] ![[IMG_0006.jpeg|400]] Again, the board adjusts itself as it would for the half-entangling move. ![[IMG_0007.jpeg|400]] And just like with half-entangling, should you so wish, you could go on to perform a half-cancelling move on the second half of the opponent's superposition as well on a later turn. ![[IMG_0008.jpeg|400]] But we appear to be in an unfortunate situation here: What the board is telling us is that we have "quarter $X$ *minus half* $Xquot;. We have cancelled too much! What we end up with, then, seems to be a square containing "minus a quarter $Xquot;. What does this mean? Well, first off, minus signs are irrelevant when we observe the state (at least in the standard basis), so you don't have to worry about the state becoming going negative. Does this mean that the result is the same as "a quarter $Xquot;, that is, do we get a $25\%$ probability that the attempted cancellation failed and the $X$ piece is still in there? The situation turns out to be quite a bit more fortuitous for the cancelling player: The laws of quantum physics tell us that the cancellation attempt, though it *can* fail, fails with only $3\%$ probability! In essense, this is because quantum probabilities are calculated as the *square roots* of their classical counterparts. The result is that calculations involving interference can give quite different answers to what one might have expected! (To see the full derivation of the $3\%$ probability, check out [[The Physics of TiqTaqToe]].) We will see how to roll for these tiny probabilities in the observation phase below. But first, let us talk a little bit about what is actually happening on the board! ### A New Kind of Superposition Until now, we have thought of the pieces as having a definite value, either $X$ or $O$, but possibly in superpositions over different positions on the board. However, after performing a cancelling move, one square will definitely end up empty, and the other square will definitely contain a piece—and this piece will be an $X$ or an $O$ with 50/50 probability! We see that we now have a single piece in a definite position, but in an equal superposition over the *symbols!* This matches a situation found in nature: Quantum particles can be found in superpositions not only of space, but in fact of *any* observable feature, such as velocity, energy level, or the quanta of magnetic moment known as "spin".[^1] In fact, it is exactly this kind of superposition that sits at the heart of quantum computing, where the fundamental elements, the *qubits*, are in well-defined *positions* on the quantum chip, but can be in arbitrary superpositions over the *symbols* $0$ and $1$! The bottom line is that, unlike the earlier moves, the cancelling move does not *add* a piece to the board: After a cancelling move, the total number of squares that will end up filled after an observation phase is *unchanged*. Instead, the move *alters the state of an existing piece*, effectively moving it to from one square to another, while placing it in an equal superposition of values. The same holds true for the half-cancelling move: No piece is added to the board; instead, the state of an existing piece is altered. The resulting state is a somewhat more complicated one, however, as the half-cancelled piece will end up in a non-equal superposition over both squares and symbols! ### The Observation Phase Let us observe the following board. ![[IMG_0033.jpeg|400]] As before, start by replacing any classical pieces with their corresponding symbols. ![[IMG_0035.jpeg|400]] Next, remove the dice from any fully cancelled squares. ![[IMG_0036.jpeg|400]] Note again how the partner square to the cancelled square, the top square, now houses a superposition over *symbols*. Player 1 points to the top square and asks, "is this square mine?" They roll, and get a $5$: The answer i *no*, and so square goes to Player $2$. ![[IMG_0037.jpeg|400]] Next, Player 2 checks whether the middle square was successfully cancelled; as mentioned above, half-cancelling has a $3\%$ chance of failing. This can be checked for by rolling both the d6 dice: If you get double ones, which happens with probability $1$ in $36 \approx 3\%$, the cancelling failed, and the piece, whose state was spread over the middle, bottom, and rightmost squares, resolves to the opponent's symbol in the middle square—the worst possible outcome for Player 2! ![[IMG_0038.jpeg|400]] But Player 2 rolls a $4$ and a $1$: The cancelling succeeded, and the square is empty. ![[IMG_0039.jpeg|400]] Now, how do we resolve the state's two remaining squares? Well, notice that the sum of the values on the dice in the bottom and rightmost squares is $1+2+2=5$. The total "probability weight" is five, with two of the outcomes having twice as high probability as the third outcome; therefore, two of the outcomes must have probability $2$ in $5$, and the last one must have probability $1$ in $5$.[^2] With a d5 die, this can be resolved with a single roll! "A d5?" I hear you ask. "How do I get hold of that?" Well, luckily for us, d5 dice are simple to implement using normal d6 dice: Simply roll as usual, and reroll any time you hit a $6$! So Player 1 does the following: First, decide on the outcomes and state them out loud, for example: "On a $1$ or a $2$, the rightmost square is yours; on a $3$, the *bottom* square is yours; and on a $4$ or a $5$, the bottom square is mine!" They roll, and get a $5$. Things are starting to look good for player 2! ![[IMG_0040.jpeg|400]] All that remains now is to resolve Player 1's final superposition state. They point to the top-right square and ask, "is this square mine?" They roll a $6$: The answer is yes. ![[IMG_0041.jpeg|400]] There is still no winner, and there are now *four* empty squares remaining. The game continues! Player 1 initated the observation phase by placing the last piece, and so the turn goes to Player 2... #### Double Half-cancelling What if both halves of the superposition had been cancelled? ![[IMG_0042.jpeg|400]] Then we proceed as before: For both of the cancelled squares, we first check whether the cancellation succeeded by rolling two d6. If one of the rolls is double ones and the other one isn't, then the opponent's piece is in that square, and the other three squares are empty. ![[IMG_0043.jpeg|400]] If *both* rolls gave double ones (which only happens with probability $1$ in $1296$!), then the result is a superposition state over both half-cancelled squares. ![[IMG_0044.jpeg|400]] If neither roll yielded double ones, then both cancellations succeeded. ![[IMG_0048.jpeg|400]] Now there are four possible outcomes, two of which have double the probability of the rest. The total "probability weight" is $1+2+1+2=6$, so this can be resolved with a single d6 roll. Player 2 states the outcomes out loud, "on a $1$, the bottom square is yours; on a $2$ or a $3$, the bottom square is mine; on a $4$, the rightmost square is yours, and on a $5$ or a $6$, the rightmost square is mine!" They roll, and get a $5$. Good for them! ![[IMG_0049.jpeg|400]] The crucial thing to remember about half-cancelling moves is that, even though they can spread the opponent's superposition state over three and even four squares, they never add a new piece to the board, so only one square will house a symbol in the end! ### Balance and Strategy In terms of gameplay, the main feature of the cancelling move is that you now get access to a move that has a *certain* outcome (or near-certain, in the case of the half-cancelling move), namely emptying a square. This means that if Player 1 attempts to capture the coveted middle square, instead of settling for a $50\%$ chance of stealing it from them with an entangling move, Player 2 can cancel the square altogether, and hope to capture it themselves after the first observation phase. Effective use of cancelling moves also means that more squares will end up empty after each observation phase, which lowers (or even eliminates) the chance that one player will be declared the winner before the other one has had a chance to recapture the emptied squares. This means that there will usually be several rounds of gameplay and observation, with the "true" board being gradually revealed to both players. This adds additional layers of strategy. For example, should you make sure to place as many (or more) pieces as the opponent; or should you prioritize *not* placing last, to ensure you get to go first after the observation phase; or should you cancel as much as possible, so that it matters less who ends up going first, as the final placement of the pieces is slowly revealed? Once you've played a few rounds, I think you will agree that this added layer of strategy is a welcome one. As one playtester remarked: "Now it feels like a *game!*" ### Levelling up In this level, we saw a new kind of superposition: Now, pieces could be in superpositions over *symbols*, as well as over space. This gives us the opportunity to showcase one of the strangest effects in all of quantum physics—one that sits at the heart of a century of unsettled philosophical debates about what quantum theory *means* for the nature of reality—namely the *basis dependency of observations.* Using the blue numbers, you will create two different *kinds* of superpositions over symbols, and then, the player who initiates the observation phase, will get to decide *how* to observe them. To learn what this means, take a leap and enter **Level 4: [[TiqTaqToe with Complementarity]]!** [^1]: In [[The Physics of TiqTaqToe]], we interpret the pieces as *electrons* existing in a discretized space of nine squares, and say that if the electron is found to be in a spin-up state, then it is an $X$, and if it is observed in a spin-down state, it is an $O$. [^2]: The fact that probabilities adjust themselves the more we learn more about a piece, so that probabilities always add up to $100\%$ per piece, is known as *renormalization*, and it is essential to making quantum theory physically sensible.