Ny versjon: TiqTaqToe with basis change inkluderer nå også kansellering. (Playtesting viste at det ikke egentlig var nødvendig å introdusere disse hver for seg, da trekkene ikke interagerer på noen kompliserte måter.) Wip; se [[Notater baseskifte-TiqTaqToe]] ### Tidligere TiqTaqToe with spin-superpositions Når du "contester" én av motstanderens ruter, kan du velge å legge en blå toer i én av rutene. Motstanderens terning i den andre ruten blir da også blå (unntatt hvis det var en superposisjon som ble splittet; motstanderens terning blir da blå *dersom* partikkelet ikke var i den tredje ruten). Resultatet av trekket er det samme som det sammenfiltrende trekket, bortsett fra at nå er sannsynlighetene *uavhengige*. Forventningsverdien per spiller er den samme som før, men med høyere risk/reward, siden med 25% sannsynlighet hver så blir resultatet $XX$ eller $OO$. Hvis spiller $O$ gjør trekket på rute 1 og 2, er trekket representert av den følgende operasjonen $\ket{X}_1 \longrightarrow \minusket_1 \plusket_2$ hvor $\minusket$ havner i ruten hvor spiller $O$ plasserte sin blå terning, og $\plusket$ havner i ruten med motstanderens blå terning. Senere, i [[TiqTaqToe with basis change (v1)]], vil disse da få 100% sannsynlighet for å måles som $X$ eller $O$. ### Blogpost wip When playing quantum tic-tac-toe, or TiqTaqToe, [[TiqTaqToe with d4 dice|using dice]], ### 2. 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. %%