When you play quantum tic-tac-toe using dice (see [TiqTaqToe with d4 dice](https://hans.heum.me/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: In the following, let the blue player be 𝑋, and the orange player be 𝑂. Recall that the number on the die represents the probability (amplitude-squared) that a piece is observed in a given square: If the number on the die is 𝑎, then the probability of observing it in the given square is 𝑎/4. If the d4 is placed on the board like a right-side up triangle (△), then the phase is positive and the value is that of the player, in this case 𝑋: ![[IMG_8296.jpg|600]] If the d4 is placed like an upside-down triangle (∇), then the phase is negative, and the value becomes that of the opposing player, in this case −𝑂: ![[IMG_8297 1.jpg|600]] Of course, global phases are unobservable, so if you observe a single downward-pointing die, the result will simply be an extra placement of the opponents symbol, e.g. 𝑂 for player 𝑋. Not a great move! However, when entangling with the opponent, the opposite phase allows the player to _cancel_ part of the opponents wavefunction. For instance, if 𝑂 makes a classical move, ![[IMG_8299.jpg]] then 𝑋 can cancel by playing a superposition, half 𝑋 and half −𝑂: ![[IMG_8300.jpg]] Observing this board would yield an empty middle square, and either an 𝑋 or an 𝑂 with equal probability in the square to the right: ![[IMG_8301.jpg]] We thus name this move the _Shiftflip_ move. If 𝑂 starts out with a superposition move, ![[IMG_8302.jpg]] 𝑋 can once again play a cancelling move on one part of the superposition, which gets halved as usual: ![[IMG_8305.jpg]] We similarly name this the _Half-shiftflip_ move. When we perform a measurement after a half-shiftflip move, we have to be a bit careful. We start with the observation that the middle square now contains $1/4\ O - 2/4\ O = - 1/4\ O$: ![[IMG_8306.jpg]] However, the probabilities for orange and blue both sum up to less than one. This is resolved in the game (as it is in nature) by *renormalizing* the probabilities, i.e. by multiplying both parts of the wavefunction with some number such that they sum to one again. Since each colour currently sum to $3/4$, this number must be equal to $1/(3/4) = 4/3$, and so we get the new probabilities $1/4 \cdot 4/3 = 4/12 = 1/3$, and $2/4 \cdot 4/3 = 8/12 = 2/3$—as we might have expected. The result is that the orange dice have a one-third chance of resolving to an $O$ in the lower square, and two-thirds chance of resolving to an $O$ in the rightmost square. Meanwhile, the blue dice have a two-thirds chance of resolving to an $X$ in the lower square, and a one-third chance of resolving to an $O$—the symbol of the opponent!—in the middle square. This makes the *Half-shiftflip* a potentially powerful move that comes at the risk of giving the opponent an extra piece on the board![^5] If the player entangles with half of a superposition while cancelling both squares, ![[IMG_8363.jpg]] after subtraction, we end up with the following situation: ![[IMG_8364.jpg]] Thus, there is now a fourth chance of observing an $O$ in either square, and half a chance of observing an $O$ in the rightmost square. Following the rule of minimal renormalization (see *Physical postulates* below), this is equivalent to the following state, ![[IMG_8365.jpg]] effectively smearing the state of the opponents piece across space without adding one of your own. We therefore name this move the *Spreading* move. The final possible move is the *Cancelling* move, in which the player entangles with a classical state while cancelling in both squares: ![[IMG_8367.jpg]] This leads to both squares being empty. ![[IMG_8368.jpg]] Or in other words, both pieces are moved off the board (back where they came from!). It is therefore possible for one player to prolong the game indefinitely, *if* there is an even number of empty squares left *and* the first player insists on only using classical moves. %% Previously this was disallowed: "... at which point the state of the opponent's piece becomes non-renormalizable; or, in other words, *electron mass is not preserved* (see Postulate 3 below). And so, the *Cancelling* move is disallowed." However, moving a piece off the board is just the time-reversal of *adding* a piece to the board. Neither move is unitary by itself, and making them unitary requires adding ancillary qubits that represents "off the board" (much like in the quantum implementation of Quantum Chess), or allow for a space off the board in the wave packet formalism, and so my argument for why it should not be allowed feels too weak. %% ### List of moves #### Original moves - *Classical* - Placing a single die on an empty square. - *Superposition* - Placing two dice on two empty squares. - *Entangling* - Placing two dice on two squares, one of which contains a die in a classical state. - Yields a fully entangled state. - *Half-entangling* - Placing two dice on two squares, one of which contains a die in a superposition state. - Yields a half-entangled state. #### Phase moves - *Shiftflip* - Entangling with a classical state while cancelling out one of the squares. - *Half-shiftflip* - Entangling with half of a superposition while cancelling one of the squares, albeit imperfectly. - Board renormalizes. - *Spreading* - Cancelling half of the opponent's superposition state. - After cancelling, the result is that half the opponent's state is spread to two fourths. - *Cancelling* - Entangling with a classical state while cancelling in *both* squares. %% - *Disallowed* due to mass preservation rule (see below). %% ### Physical interpretation The natural implementation of the game on a quantum computer would assign to each square a qu*trit* with the three basis states $\{\eket, \xket, \oket\}$. This is a very general framework, and implementing our ruleset on nine qutrits (plus nine ancilla qubits representing "off the board") would simply be a task of switching the states of the squares according to the rules. However, I find this framework *too* general, as the lack of restrictions does not inform us much about what rules might count as "physically sensible" or not, particularly when phase cancellations are concidered. Therefore, the game board is instead interpreted as a discretized pace containing a number of fermionic particles, i.e. electrons, in spin-up and spin-down states. It is tempting to interpret each colour as representing a single electron, but this does not always lead to an accurate picture. Consider e.g. the *Shiftflip* move described above; despite the presence of two colours, this is most naturally interpreted to lead to a single electron in a determined space, but in an equal superposition of spin-up and spin-down. Rather, the dice are interpreted as *terms* in a larger wavefunction, one that might describe either one, two, or more electrons. The terms are the following: For $a\in \{1, 2, 4\}$, where $a$ is the number on the die, - Player $X$ colour placed right-side up: $\sqrt{\frac{a}{4}}\ \upket$ - Player $X$ colour placed up-side down: $-\sqrt{\frac{a}{4}}\ \downket$ - Player $O$ colour placed right-side up: $\sqrt{\frac{a}{4}}\ \downket$ - Player $O$ colour placed up-side down: $-\sqrt{\frac{a}{4}}\ \upket$ Placing a colour on the board then adds these terms to the wavefunction describing the contents of the affected squares, entangling with any colour already present. This may involve placing a new electron to the board (adding mass to the space, technically by moving an electron from "off the board" to "on the board"), or simply affecting the state of the electrons that are already there (as in the case of the *Shiftflip* move). The *Classical*, *Superposition*, *Entangling*, *Half-entangling*, and *Half-shiftflip* moves all add an electron to the board, meaning the affected squares will contain one more symbol after observation than they would have were the move not made. The *Shiftflip* and *Spreading* moves do not add an electron to the board, but rather alter the state of an already present one. (Technically speaking, the electron on the board entangles with an electron off the board.) Note how single electrons can exist in superpositions in space (such as results from the *Superposition* move) as well as superpositions in spin (such as results from the *Shiftflip* move). This leads to certain configurations having several equivalent interpretations, such as the result of playing an *Entangling* move: Each colour could be said to represent an electron in an entangled superposition of two possible positions on the board; or alternatively, each square could be said to contain an electron in a determined square, but each in an entangled superposition of spin-up and spin-down. Of course, due to the indistinguishability of identical particles, the reality is that neither interpretation can be said to be capital-T "True", only that the two-particle configuration is fully described by the relevant wavefunction. Followings are the set of gameplay rules (equal to the original game) and physical postulates (new) from which the possible moves are derived. Gameplay rules take precedence, due to a desire for moves to mirror the base game: *You place dice of a colour not yet used in up to two squares, at least one of which is empty*. The only difference is that now each die can be placed either right-side up or up-side down! #### Gameplay rules 1. *Placement:* Each move involves placing either one 4-die or two 2-dice of a colour that has not yet been used, on individual squares among which at least one is empty. 2. *No self-entangling:* A player can not place two of their own dice on the same square. 3. *Observation:* The board is observed when there are no more empty squares. 4. *Observed states:* Observed $X$s and $O$s are classical and can not be affected by quantum moves. 5. *Endgame:* If observation results in one player getting a three-in-a-row *or* a full board, then the game ends. If the game ends and no player *or both players* have a three-in-a-row, the game is a draw.[^3] 6. *Continuation:* After observing, if no player has won and there are still empty squares on the board, the game continues, with the turn going to the opposing player to whomever initiated the measurement. #### Physical postulates 1. *Electron placement:* The (initially-empty) board contains electrons in spin-up and spin-down states, distributed across the nine spaces. Each move moves at most one electron from off-the-board to on-the-board. 2. *Spin:* An electron observed to be in the spin-up state $\upket$ is interpreted as $X$. An electron observed to be in a spin-down state $\downket$ is interpreted as $O$. 3. *Exclusivity:* Each square can inhabit at most one electron.[^1] 4. *Minimal renormalization:* When several physical interpretations of a move are available, the game defaults to the interpretation requiring the least amount of renormalization.[^4] %% Preivously: 3. *Mass preservation:* The number of electrons on the board is a non-decreasing integer. %% [^1]: This could be seen as an illustration of the Pauli exclusion principle, in which two indistinguishable electrons can not inhabit the same space, although the analogy is imperfect as electrons in spin-up and spin-down states *are* distinguishable, and therefore *could*, physically speaking, inhabit the same space. [^2]: This rule is a bit arbitrary, but has been found to lead to more interesting board configurations and strategies than the alternative. The rule only affects measurements following a *Half-shiftflip* move. %% Footnote leads nowhere? Handlet om at "renormalisering", slik jeg gjorde det tidligere, ble rundet til 1-3 heller enn til 2-2. %% [^3]: A fun alternative ruleset is to let each three-in-a-row count as one point, and let the game continue until the board is full, at which time points are tallied to determine if there is a winner. [^4]: This only affects the *Spreading* move, the result of which might also have been interpreted as a wavefunction describing two on-board electrons. [^5]: Open question: Is there any situation in which the Half-shiftflip is better than either Half-entangling or Spreading? I don't know!