## Level 2 – Adept: TiqTaqToe with Entanglement Welcome to the second level of TiqTaqToe! Here, we introduce one new quantum move—one that illustrates the phenomenon of *entanglement*. What is entanglement? In short, it is *interdependent superpositions*, meaning superpositions over two interdependent *pieces*, such that the *state* can not be described without talking about both pieces at once.[^6] For example, consider the situation "If piece A is at position 1, then piece B is at position 2, and if piece A is at position 2, then piece B is at position 1". Note the dependence: If the superposition states were *independent*, then the statement would instead be that "piece A is either at position 1 or 2, and so is piece B", meaning that both pieces could end up in the same square. Instead, we have that the position of piece B depends on the position of piece A, and vice versa.[^1] But first things first: Let's hand out the pieces. ### Handing Out the Pieces Start by handing each player the same dice as in the previous level, [[TiqTaqToe with Superpositions]]: Ten dice each of mainly either white or gold numbering, where the $2$s and $4$s are *not* coloured red or blue, as well as one d6 each of the same main colour. Next, place the remaining d4 dice next to the board, in a "bank" of a sort. These will not be used by the players as part of a move, but will come on the board as the game adjusts itself after each certain moves. ![[IMG_9925 1.jpeg|400]] Finally, place the observation token between the players with the "XO" side up. ### The Entangling Move The classical superposition moves are both still available to the player, but now they also have access to the *entangling* move. Unlike the previous moves, the entanglement move can *not* be the first move, as it is fundamentally a *reactive* move. It works as follows. If the first player performs a classical move, meaning they place a die with the number $4$ up in one of the squares, then the second player can (as before) place two dice of the same colour with the number $2$ up in two different squares. The difference is that now, as long as one of the two dice are placed in an empty square, the player is allowed to "invade" the square of the opponent with the other die. ![[IMG_9927 1.jpeg|400]] If we now look at the board, we see that we have a conundrum: The dice currently tell us that there is an $X$ in the middle with $100\%$ probability, *and* an $O$ in the middle with $50\%$ probability. Well, it can't be both! The game resolves this by automatically adjusting itself: The $4$ in the middle is *split in halves*, meaning into two $2$s, where one $2$ remains in the original square and the other $2$ goes with the second player to the second square. ![[IMG_9928 1.jpeg|400]] I like to think that Player 2's piece "invaded" Player 1's square, and "dragged" half their probability over to the other square. Regardless, the result is that there are now two possible, equally probable outcomes: Either $X$ is in the first square and $O$ is in the second square, *or* $O$ is in the first square and $X$ is in the second square. Note that learning where $X$ is immediately reveals where $O$ is, and vice versa, meaning we only have to roll for one piece to learn the position of both. This is exactly what it means for objects to be entangled! ### The Half-entangling Move The same procedure applies if Player 1 instead started out with a superposition move: As long as one of the two dice you are placing goes in an empty square, you are free to place the other die in a square already occupied by the opponent. ![[IMG_9929.jpeg|400]] Again, the game adjusts itself: Player 1's $2$ is split in halves, this time into $1$s, with one $1$ remaining in place and the other going to the other square. (Use one of the same-coloured dice from the "bank" for this.) ![[IMG_9930.jpeg|400]] Just like before, your piece "invaded" the square of the opponent, and "dragged" half their probability over to another square. Thus, there is now a $1$ in $4$, or $25\%$, probability that the piece will end up in either of these squares in the end. The result is something "half-entangled": You have entangled with one half of a superposition, while the other half of the superposition remains independent of your piece. There are thus three possible outcomes: *If* the opponent's piece is found to be in the unentangled square (which happens with $50\%$ probability), *then* your piece is just in a normal superposition over otherwise empty squares, and there is no entanglement anymore. Otherwise, the opponent's piece is in one of the entangled squares, and the situation is as if you had performed a full entangling move. (If checked as part of the observation phase, the $1$s would become $2$s in this case, to show that the piece is in one of the two squares with 50/50 probability.) ![[IMG_9931.jpeg|400]] Now, there is nothing stopping you from later *also* entangling with the other half the opponent's superposition: Simply follow the same procedure, using the final remaining die of the relevant colour from the bank. ![[IMG_9933.jpeg|400]] Note how the opponent's piece is now maximally spread out over the board: Their yellow dice are all $1$s, meaning the piece has a $25\%$ probability of being in either of the four squares. What's more, the "no self-entanglement" rule stops you from splitting the piece any further (into, say, eights).[^2] ### The Observation Phase Let's look at how one would go about observing the following board. The order in which the pieces are resolved is still somewhat arbitrary, but here's a suggested procedure. ![[IMG_9938.jpeg|400]] First, replace all classical pieces with their respective symbols. (Here, as before, white numbering means $X$ and gold numbering means $O$.) ![[IMG_9939.jpeg|400]] Next, take turns resolving each of your pieces (i.e. colours), by phrasing yes/no-questions like "is this square mine?" As before, answer the question by rolling the d6, interpreting even numbers as "yes" and odd numbers as "no". Note that the order is irrelevant: Only when the observation phase concludes will a possible winner be declared![^3] Let us start with Player 2's grey dice, in the leftmost and lower right squares, respectively—the result of a superposition move. Player 2 asks, "is the leftmost square mine?" They roll a $3$, so the answer is *no*—then the piece must be in the lower-right. ![[IMG_9941.jpeg|400]] Next, let's look at the purple and white dice. These describe two pieces in an entangled superposition between the top and bottom squares—the result of an entangling move. Since it's Player 1's turn to roll, they ask, "is my white $X$ piece in the top square?" They roll a $6$, so the answer is yes! Then, the purple $O$ piece has to be in the bottom square. ![[IMG_9944.jpeg|400]] Finally, we have the yellow $O$ piece, which has been spread over four squares by two half-entangling moves. Player 2 asks, "is my blue $O$ piece in the bottom-left square?" They hope for a yes, since then they will get a three-in-a-row, but they roll a $1$, so the answer is *no*—it is in the top-left square. ![[IMG_9947.jpeg|400]] Notice how the yellow die in the lower-left was changed from a $1$ to a $2$: Since the yellow piece is now known *not* to be in the top-left square, it must have twice the likelihood of being in the bottom-left square, since this is the part of the superposition that was entangled with Player 2's blue piece. The resolution of the blue piece does not affect the probabilities to the right on the board, because that part of the yellow piece's superposition state was independent of the blue piece. (In other words, the piece still has a 50/50 chance of being to the left or to the right on the board, just like it did before each half of the superposition got entangled; resolving the entangled pieces does not change this fact.) It is player 1's turn to roll. They ask, "is the lower-left square mine?" They roll a $2$, so the answer is yes. ![[IMG_9949.jpeg|400]] As you can see, this also means that the yellow piece is *not* in either of the squares to the right, so that Player 2 is left with a normal, unentangled superposition. They ask, "is the top-right square mine", and make the final roll. It is a $5$—a no. ![[IMG_9950.jpeg|400]] The game is undecided, and there are still empty squares left, which means the game continues! Looking at the board, we see that both players have precisely one opportunity at winning, namely if they can capture the top-right square. Well, $X$ placed last,[^7] so it is Player 2's turn. What can they do? If they make a classical move in the top-right square, the opponent will surely entangle with it, giving each player a 50/50 chance of winning. If, instead, they make a superposition move on both of the remaining squares, the game moves straight to another observation phase, and Player 2 gets a $50\%$ chance of winning, while Player 1 gets nothing. At first glance this may seem better, but what happens if Player 2 *doesn't* get the square? Well, then the game is undecided with an empty square left, and so the game continues, with the turn going to Player 1. They will then perform a classical move in the coveted square, at which point the game will enter the observation phase again, and Plater 1 will win. And so this *also* gives each player a 50/50 chance at winning! The only thing that we can be sure of here is that a *draw* is impossible. Player 2 opts for a classical move, and Player 1 responds as expected. ![[IMG_9951.jpeg|400]] The game has come down to a single roll. Player 2 takes a deep breath, and asks: "Is the top-right square mine?" They roll, and— ![[IMG_9952.jpeg|400]] A three. The answer is no, which means the red $O$ piece must be in the left square, which in turn means that the yellow $X$ piece must be in the top-right square. Player 1 wins! ![[IMG_9953.jpeg|400]] ### Balance and Strategy Just like the classical game, [[TiqTaqToe with Superpositions]] had a definite first-player advantage—but thanks to the fact that Player 2 can now alter the states of the first player's pieces, the situation is suddenly much less clear. True, Player 1 *can* place one more piece on the board than Player 2, which could translate to a higher total win probability—but in doing so they are also placing the last piece on the board, giving the second player the opportunity to place first after the observation phase *if* the game is not yet decided by then. As we already saw in the previous level, this can be a very fortuitous position to be in! In fact, it seems to be quite easy for two skilled players to end up with a board that has almost an exact 50/50 win probability between them, for example if they end up with a highly symmetric board. ![[IMG_9954.jpeg|400]] The trick, then, is to find ways to edge the probabilities in your own favour, at the expense of your opponent's. It remains unknown whether there are optimal strategies for achieving this, but anecdotal evidence suggests that something like a 70/30 advantage *should* achievable for skilled players. As for developing good strategies, it is usually easier to spot *bad* moves than good ones. In that sense, good play might simply come down to avoiding bad moves, with the player that avoids the most bad moves coming out in the lead. As an example, consider the following board. Do you see why placing a die in the rightmost square would be a bad move for either player? If you were next, where would you place instead to maximize future chances for a three-in-a-row? (See footnote for answer.[^4]) ![[IMG_9956.jpeg|400]] As another example, a superposition move, which was a decent opening move in [[TiqTaqToe with Superpositions]], is now suddenly a poor way opener. Do you see why? (See footnote for answer.[^5]) ![[IMG_9957.jpeg|400]] ### Levelling Up This version of the game was the first to be developed, and by far the one that has been playtested the most. It shares the features of many great games of the past: Easy to learn, yet hard to master! Indeed, even young students (who were self-proclaimed "bad at math") have found that after only a single round of play, they have already mastered the moves. And yet, mastering *strategy* remains a challenge even for seasoned players! Additionally, the game provides a good illustration of the world of quantum physics, and how even simple situations can quickly become hard to reason about. (In fact, we have found it easier to teach rules of the game *first*, without ever referencing quantum physics, before revealing to the player that they were just tricked into learning quantum theory!) And yet, the game is still *really* only using classical conditional probabilities. The *real* weirdness of the quantum world comes from the fact that, unlike classical probabilities, quantum probabilities[^8] can not only be positive, but also negative, as well as in between positive and negative! (The latter are known as "imaginary" or "complex" numbers, though these won't be relevant for this game.) This means, among other things, that quantum probabilities can *cancel out*, much like how two colliding waves *interfere* with each other to create pockets of strengthened and cancelled-out regions, depending on whether wave-tops meet other wave-tops, or they meet wave-bottoms instead. It is precisely this effect the next move will exploit, which we introduce in **Level 3: [[TiqTaqToe with Interference]]!** [^1]: If you are already familiar with probability theory, you may recognize this as nothing but conditional probabilities. Indeed, entanglement might be said to be the quantum generalization of conditional probabilities! Still, this version of the game involves nothing but classical conditional probability theory rephrased in a quantum language—the fancy stuff comes later. [^2]: A version of this game is playable on [tiqtaqtoe.com](https://www.tiqtaqtoe.com) under the mode "Full Quantumness"; the only difference to this version is that entangling with both halves of a superposition is disallowed. [^3]: The fact that the order in which you resolve the pieces does not affect the outcome is mathematically less than obvious. Long story short, the reason is that thanks to the *renormalization* that is performed on the dice between rolls (such as changing two $1$s into $2$s after finding that a piece is *not* in the third square), any full board defines a probability distribution over classical tic-tac-toe boards, and the observation phase is simply a procedure to sample from said distribution. [^4]: The middle row is blocked for both players, since due to entanglement, the row will contain at least one of each piece, $X$ and $O$; therefore, neither player has a chance of getting a three-in-a-row there. By placing a piece in either the top-left or bottom-left corner (or both, using a superposition move!), you instead open up for three-in-a-rows on the diagonal or along the left edge, so that you have a chance regardless of where the entangled pieces end up. [^5]: Regardless of where the superposition was placed, Player 2 can now ensure that you only get a $25\%$ chance of capturing the powerful middle square, either by entangling with the middle square directly (if you placed in it), or by splitting your $2$ elsewhere and "dragging" a $1$ *into* the middle. The "no self-entanglement" rule then blocks you from further interacting with it, meaning you are stuck with a quarter probability of capturing it, whereas your opponent gets a full half! [^6]: Again, see [[TiqTaqToe Player Reference]] for a glossary explaining various terms used here. [^7]: This can be seen from the starting board having one more $X$ piece than $O$ piece ($4$ versus $3$ different colours). [^8]: These are known as *amplitudes*, in reference to classical waves, and are in a sense *the square root* of classical probabilities. For more on this, see [[The Physics of TiqTaqToe]].