terminologi: straight vs shifted *lense*? "Observing under a straight lense, observing under a shifted/rotated/skewed lense" ## Level 3 (Expert): Complementarity Welcome to the third and final level of TiqTaqToe. In [[TiqTaqToe with Interference|Level 2]], you encountered a new kind of superposition: Now, a piece could be in a superposition over the *symbols* $\triangle$ and $\square$. Two such states are known as the *plus* and *minus* states, depending on whether the two parts of an equal symbol-superposition have the same or opposite signs.[^6] As we will see, these states behave even stranger than one may anticipate, as the probability of seeing each value will crucially depend on *which question you choose to ask*. This phenomenon, that there are questions one can ask of quantum particles that are not "compatible" in the sense that one can not *even in principle* learn the answer to both at the same time, is the phenomenon coined as "complementarity" by Niels Bohr, and it is embodied in Werner Heisenberg's famous uncertainty principle. (To learn more about the people who pieced together the strange new logic of quantum mechanics, check out [[The Phenomena Behind TiqTaqToe]].) ##### Table of Contents - Setting Up The Game - Creating Plus and Minus States - A Question of Choice - Superpositions and Entanglement - Half-plus and Half-minus States - Observation Phase Example 1 (without half-cancelling) - Observation Phase Example 2 (with half-cancelling) - Balance and Strategy - What Next? ### Setting Up The Game Pieces are distributed as in [[TiqTaqToe with Interference|Level 2]], the only difference being that now Player 2 will have to use the $-2$ numbers on their d6 dice as well when performing cancelling moves. ``` Player 1 Player 2 /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P r---r g---g b---b y---y p---p /\ /\ /\ /\ /\ | 4 | | 4 | | 4 | | 4 | | 4 | / 4\ / 4\ / 4\ / 4\ / 4\ r---r g---g b---b y---y p---p R----R G----G B----B Y----Y P----P ``` ### Making Plus and Minus States At this level, we slightly tweak the cancelling move introduced at [[TiqTaqToe with Interference|Level 2]], which will enable the main new feature of this level in the observation phase. Recall that after Player 1 performs a cancelling move, the non-cancelled square will be in a symbol superposition like the following. ``` +--------------+--------------+ +--------------+--------------+ | R---R | R---R | | | R---R | | r---r | 2 | | /\ | 2 | | | | /\ | 2 | | | |-2 | R---R | / 2\ R---R | -> | | / 2\ R---R | | r---r | r----r | | | r----r | | | | | | | +--------------+--------------+ +--------------+--------------+ ``` This is an unentangled superposition over symbols, and since the two symbols in the superposition have the same "phase" (namely positive), this is a plus ($+$) state. From now on, we will think of plus states as belonging to Player 1. Likewise, minus states will belong to Player 2; hence, whenever Player 2 performs a cancelling move, they will want to create one. This can be done using the $-2$ numbers on their d6 dice, as follows. ``` +--------------+--------------+ +--------------+--------------+ | /\ | R---R | | | R---R | | /\ /-2\ | /\ |-2 | | | | /\ |-2 | | | / 2\ R----R| / 2\ R---R | -> | | / 2\ R---R | | r----r | r----r | | | r----r | | | | | | | +--------------+--------------+ +--------------+--------------+ ``` Note how *both* the dice placed by Player 2 must now use the $-2$ number, and it is their *shapes* that define which square is cancelling ($\frac{1}{2} \triangle - \frac{1}{2} \triangle$) and which square is the minus state ($\frac{1}{2} \triangle - \frac{1}{2} \square$). ### A Question of Choice New to this level is that whoever initiates the observation phase gets to choose between two *complementary observations*, corresponding to two possible *questions* to ask of the board: 1. "Are the pieces in X or O states?" This is the standard question to ask, and the one we have been using so far; we say that we observe the board in the *standard view*. As before, classical pieces are determined and don't have to be rolled, but plus states, minus states, and entangled states must be rolled. 2. "Are the pieces in plus or minus states?" We call this observing the board in the *shifted view*. Now we won't have to roll for any plus or minus states on the board, as they answer the question directly; but *classical* states, which can be seen as a superposition of a plus and a minus state, must suddenly be rolled. Entangled states must also still be rolled for, since entanglement between pieces is an "objective" feature in the sense that it does not change just because you changed your view. In the following board segment, there are four pieces, all in well-defined positions (no superpositions over spaces), meaning that after observing the board, all four squares will end up filled. The pieces are in states classical X (top-left), classical O (top-right), plus (bottom-left), and minus (bottom-right), respectively. ``` +--------------+--------------+ | | | | /\ | R---R | | / 4\ | | 4 | | | r----r | R---R | | | | +--------------+--------------+ | G---G | B---B | | /\ | 2 | | /\ |-2 | | | / 2\ G---G | / 2\ B---B | | g----g | b----b | | | | +--------------+--------------+ ``` If we observe the board in the standard view by asking the question "are the pieces in X or O states?", the top two squares would immediately be replaced by an X and an O symbol, and then the players would take turns rolling for the two lower squares, which are each X or O with 50/50 probability. If we observe the board in the *shifted* view, by asking the question "are the pieces in plus or minus states?", then the bottom two squares would immediately be replaced by an X (for plus) and an O (for minus) symbol, and then the players would take turns rolling for the *upper* two squares. These are classical states, which in the shifted view will be observed to be plus (X) or minus (O) with 50/50 probability. So the two kinds of state *switch roles* between the standard view and the shifted view, always with one kind determined and the other rolled: - Classical X state $\leftrightarrow$ Plus state: $\triangle \leftrightarrow \frac{1}{2}\triangle + \frac{1}{2}\square$ - Classical O state $\leftrightarrow$ Minus state: $\square \leftrightarrow \frac{1}{2}\triangle - \frac{1}{2}\square$ Note how by this nature, it is impossible to know both answers to the question at once: If the probability of being observed as an X is $100\%$, then the probability of being observed as a plus is $50\%$, and if the probability of being observed as a plus is $100\%$, then the probability of being observed as an X is $50\%$. This is exactly what Niels Bohr meant when he said that certain questions one can ask of quantum particles are *complementary*, in that it is not possible *even in principle* to know everything there is to know about a particle at the same time. One well-known example is the *position* and *speed* (momentum) of a particle: If you perfectly determine one, the other becomes perfectly undetermined! This is because, at some fundamental level, the position of a particle *is* a superposition over velocities, and vice versa—just like the classical and plus/minus states above. This is what Heisenberg's famous uncertainty principle tells you then: how precisely you can know one before losing all information about the other, because by measuring one (position) you forced the particle into a superposition state of the other (velocity). (Check out [[The Phenomena Behind TiqTaqToe]] to learn more.) However, our complementary questions of "X or O" versus "plus or minus" are only well-defined if we know *where* the piece is. That is why in the observation phase we will now *first* roll for positions, *then* for symbols. (We could have made this distinction at earlier levels too, but there it wouldn't have made a difference.) Concretely, the observation phase now proceeds over three steps (repeated for your convenience in [[TiqTaqToe Player Reference]]): 1. Roll for positions, until all empty squares have been identified 2. Initating player: Select a view, standard or shifted 3. Roll for symbols, until all non-empty squares contain an X or an O #### Superpositions and Entanglement To illustrate, consider the superposition move. ``` +--------------+--------------+ | | | | /\ | /\ | | / 2\ | / 2\ | | r----r | r----r | | | | +--------------+--------------+ ``` Since this is a superposition over squares and not over symbols, we have to figure out *where* the piece is before we can ask *what* it is. You point to a square and ask, "Is the piece here?", and roll. But now, instead of drawing in an X symbol as before, simply place the piece in a classical state in the square where it was observed. ``` +--------------+--------------+ | | | | /\ | | | / 4\ | | | r----r | | | | | +--------------+--------------+ ``` Now both questions are valid questions to ask of this piece: "Are you X? Are you plus?" What about entanglement? ``` +--------------+--------------+ | R---R | R---R | | /\ | 2 | | /\ | 2 | | | / 2\ R---R | / 2\ R---R | | r----r | r----r | | | | +--------------+--------------+ ``` In this case, both squares are guaranteed to contain a piece, meaning there are no empty squares to identify. Further (as mentioned above), entanglement between pieces is a view-independent property. Hence, regardless of the choice of view, these pieces will have to be rolled for exactly like in [[TiqTaqToe with Entanglement|Level 1]]. Finally, there's the half-entangled state. ``` +--------------+--------------+--------------+ | | R---R | R---R | | /\ | /\ | 2 | | /\ | 2 | | | / 2\ | / 1\ R---R | / 1\ R---R | | r----r | r----r | r----r | | | | | +--------------+--------------+--------------+ ``` The procedure asks us to first identify the empty squares. We have three squares and two pieces, so one of them will end up empty—the question is which one. Start by rolling for the left square, since it is empty with $50\%$ probability. If it is there, then there is a superposition in the middle and right squares. ``` +--------------+--------------+--------------+ | | | | | /\ | R---R | R---R | | / 4\ | | 2 | | | 2 | | | r----r | R---R | R---R | | | | | +--------------+--------------+--------------+ ``` After rolling for this superposition too we end up with two pieces in classical states, and we have found our empty square. ``` +--------------+--------------+--------------+ | | | | | /\ | R---R | | | / 4\ | | 4 | | | | r----r | R---R | | | | | | +--------------+--------------+--------------+ ``` Both symbol-questions are well-defined, and the observation phase can proceed. If the first roll found that the leftmost square was empty, we already identified the empty square, and we get an entangled state in the middle and right squares. ``` +--------------+--------------+--------------+ | | R---R | R---R | | | /\ | 2 | | /\ | 2 | | | | / 2\ R---R | / 2\ R---R | | | r----r | r----r | | | | | +--------------+--------------+--------------+ ``` We are back to the case described above: Fully entangled pieces, which will have to be rolled regardless of the choice of view. #### Half-plus and Half-minus States Since half-cancelled states can be tricky to reason about, we again recommend that you start playing without half-cancelling moves while learning the ropes of this level. Return here once you have played a few games and feel ready for a new challenge. Ready? Ok, let's go. What about the half-cancelling move? Let's say Player 1 made the following move, cancelling one half of Player 2's superposition. ``` +--------------+--------------+--------------+ | | R---R | R---R | | R---R | r---r | 1 | | /\ | 1 | | | | 2 | | |-2 | R---R | / 2\ R---R | | R---R | r---r | r----r | | | | | +--------------+--------------+--------------+ ``` Despite all three squares containing dice, there is only one piece on the board, since the cancelling move does not add a piece. In the observation phase, we would therefore first have to which two dice will end up empty; the left square is empty with $50\%$ probability, and the middle square is empty with $97\%$ probability. Assuming this is the outcome, the result will be: ``` +--------------+--------------+--------------+ | | | R---R | | | | /\ | 1 | | | | | / 2\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ ``` Now that we know that there will be a piece in the right square, we can move on to choosing a view for our symbol-observation. If we choose the standard view, then we get the same distribution as before: $2/3$ probability that we will observe an X, and $1/3$ for O. What if we choose the shifted view? The result may surprise you:[^9] *In the shifted view, we will observe an X with $97\%$ probability, and on O with $3\%$ probability.* Hence, we can roll for this using two d6 dice, just like we did for half-cancelled states: Unless we get snake-eyes (double 1), then the observed state will be a plus state, so the symbol is X; if we *do* get snake-eyes, then luck is not on the side of Player 1, and the symbol is an O. To perhaps gain some intuition about where these numbers come from, notice that what we have is somehow an "almost-plus" state: If only the Player 2 die had been a 2 instead of a 1, then it would be a plus state, and observing it in the shifted view would lead to a fully determined outcome (X). The shift away from this determinedness is *the same* as the shift away from the half-cancelled state being an empty square, and because we are doing all our cancellations under a square root, what looks on the board like it is "a quarter" away from the state that we wanted, turns out to actually be only $3\%$ away in terms of probabilities. (For the full details of the calculation, I refer you as always to [[The Physics of TiqTaqToe]].) The same of course holds in the case of Player 2 performing the half-cancelling move, leading to an "almost-minus" state with the probabilites $2/3$ O vs. $1/3$ X in the standard view, and $97\%$ O (minus state) vs. $3\%$ X (plus state) in the shifted view. ``` +--------------+--------------+--------------+ | | | R---R | | | | /\ |-2 | | | | | / 1\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ ``` ### Observation Phase Example 1 (without half-cancelling) Let us observe the following board. This time neither player played any half-cancelling moves; we save those for the next example.[^10] ``` +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | Y---Y | | / 2\ B---B | / 2\ B---B | | 2 | | | b----b | b----b | Y---Y | | | | | +--------------+--------------+--------------+ | | /\ | R---R | | /\ | /\ /-2\ | g---g | 2 | | | / 4\ | / 2\ G----G| |-2 | R---R | | y----y | r----r | g---g | | | | | +--------------+--------------+--------------+ | | G---G | R---R | | Y---Y | /\ |-2 | | /\ | 2 | | | | 2 | | / 2\ G---G | / 2\ R---R | | Y---Y | r----r | g----g | | | | | +--------------+--------------+--------------+ ``` Recall the order of operations: First identify all empty squares, then choose a view in which to observe the board, then roll for symbols. **Step 1: Identify empty squares.** First, remove the dice from any fully cancelled squares. ``` +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | Y---Y | | / 2\ B---B | / 2\ B---B | | 2 | | | b----b | b----b | Y---Y | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 4\ | | | | y----y | | | | | | | +--------------+--------------+--------------+ | | G---G | R---R | | Y---Y | /\ |-2 | | /\ | 2 | | | | 2 | | / 2\ G---G | / 2\ R---R | | Y---Y | r----r | g----g | | | | | +--------------+--------------+--------------+ ``` Then find the remaining empty squares by resolving the superposition moves, except instead of directly replacing the die with a symbol, move the die in the square where the piece was observed to the value $4$, as if the player had made a classical move there, and remove the other one. Player 2 has one piece in a superposition state, represented by their yellow (Y) dice. They roll, and find it to be in the lower-left square. ``` +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | | | / 2\ B---B | / 2\ B---B | | | b----b | b----b | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 4\ | | | | y----y | | | | | | | +--------------+--------------+--------------+ | | G---G | R---R | | Y---Y | /\ |-2 | | /\ | 2 | | | | 4 | | / 2\ G---G | / 2\ R---R | | Y---Y | r----r | g----g | | | | | +--------------+--------------+--------------+ ``` That's all the empty squares identified, so we are to move on to the next step: The initiating player (whoever placed the last piece), which in this case is Player 2, chooses their view. **Step 2a: Observing in the Standard View** They choose the standard view. Then, observation proceeds exactly as in the previous level: First, replace classical states by their corresponding tokens, ``` +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | | | / 2\ B---B | / 2\ B---B | | | b----b | b----b | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / \ | | | | +----+ | | | | | | | +--------------+--------------+--------------+ | | G---G | R---R | | +---+ | /\ |-2 | | /\ | 2 | | | | | | / 2\ G---G | / 2\ R---R | | +---+ | r----r | g----g | | | | | +--------------+--------------+--------------+ ``` then take turns asking "Is this square mine?", and roll. The top and top-left squares are entangled, so a single roll suffices to determine their outcome, ``` +--------------+--------------+--------------+ | | | | | /\ | +---+ | | | / \ | | | | | | +----+ | +---+ | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / \ | | | | +----+ | | | | | | | +--------------+--------------+--------------+ | | G---G | R---R | | +---+ | /\ |-2 | | /\ | 2 | | | | | | / 2\ G---G | / 2\ R---R | | +---+ | r----r | g----g | | | | | +--------------+--------------+--------------+ ``` whereas the two bottom squares are in equal superpositions over symbols, so each independently have a 50/50 chance of being X or O. Perhaps Player 2 chose to observe in the standard basis, since they noticed that this would give them a $25\%$ of winning the bottom row? But the dice opt for a different story: ``` +--------------+--------------+--------------+ | | | | | /\ | +---+ | | | / \ | | | | | | +----+ | +---+ | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / \ | | | | +----+ | | | | | | | +--------------+--------------+--------------+ | | | | | +---+ | /\ | /\ | | | | | / \ | / \ | | +---+ | +----+ | +----+ | | | | | +--------------+--------------+--------------+ ``` There is still no winner, and there are empty squares remaining. The game continues, with the turn going to Player 1. What would you do next? **Step 2b: Observing in the shifted view** Here is our starting point again. ``` +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | | | / 2\ B---B | / 2\ B---B | | | b----b | b----b | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 4\ | | | | y----y | | | | | | | +--------------+--------------+--------------+ | | G---G | R---R | | Y---Y | /\ |-2 | | /\ | 2 | | | | 4 | | / 2\ G---G | / 2\ R---R | | Y---Y | r----r | g----g | | | | | +--------------+--------------+--------------+ ``` Player 2 chooses to observe in the shifted view. Start by replacing any plus states with an X symbol, and any minus states with an O symbol, being careful to leave any entangled squares alone for now. ``` +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | | | / 2\ B---B | / 2\ B---B | | | b----b | b----b | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 4\ | | | | y----y | | | | | | | +--------------+--------------+--------------+ | | | | | Y---Y | +---+ | /\ | | | 4 | | | | | / \ | | Y---Y | +---+ | +----+ | | | | | +--------------+--------------+--------------+ ``` Next, take turns to roll for each of the *classical* states. This time it is Player 2 who is in luck, as rolling for the middle-left and the bottom-left squares both yielded O: ``` +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | | | / 2\ B---B | / 2\ B---B | | | b----b | b----b | | | | | | +--------------+--------------+--------------+ | | | | | +---+ | | | | | | | | | | +---+ | | | | | | | +--------------+--------------+--------------+ | | | | | +---+ | +---+ | /\ | | | | | | | | / \ | | +---+ | +---+ | +----+ | | | | | +--------------+--------------+--------------+ ``` Finally, roll for one of the two entangled squares; as before, the second square is determined as being opposite to the first one: ``` +--------------+--------------+--------------+ | | | | | /\ | +---+ | | | / \ | | | | | | +----+ | +---+ | | | | | | +--------------+--------------+--------------+ | | | | | +---+ | | | | | | | | | | +---+ | | | | | | | +--------------+--------------+--------------+ | | | | | +---+ | +---+ | /\ | | | | | | | | / \ | | +---+ | +---+ | +----+ | | | | | +--------------+--------------+--------------+ ``` There is still no winner, and there are empty squares remaining. Once again the turn goes to Player 1. What would you do next? ### Observation Phase Example 2 (with half-cancelling) Let us observe the following board. This time around, we have a board with two half-cancelling and one half-entangling move. ``` +--------------+--------------+--------------+ | B---B | | B---B | | /\ | 1 | | /\ | y---y | 1 | | | / 2\ B---B | / 4\ | |-2 | B---B | | y----y | B----B | y---y | | | | | +--------------+--------------+--------------+ | G---G | /\ | R---R | | /\ | 2 | | /\ /-2\ | /\ |-2 | | | / 1\ G---G | / 1\ R----R| / 1\ R---R | | r----r | r----r | r----r | | | | | +--------------+--------------+--------------+ | G---G | | | | /\ | 2 | | /\ | B---B | | / 1\ G---G | / 4\ | | 2 | | | r----r | G----G | B---B | | | | | +--------------+--------------+--------------+ ``` **WIP FROM HERE** SHOULD WE MAKE THE RULE THAT WE ALWAYS FIRST RESOLVE THE SUPERPOSITION, THEN THE HALF-CANCELLING? RE-CHECK CALCULATION. **Step 1: Identify empty squares.** The piece corresponding to Player 2's blue dice has been spread **Step 2a: Observing in the Standard View** **Step 2b: Observing in the shifted view** As you can see, observing a half-cancelling move under the rotateted mode makes the move much more powerful than under standard observations—much like its full-cancelling counterpart! ### Balance and Strategy Previously, whoever did *not* initiate the observation phase gained a possible advantage, because they would then get to go first once the game continued, provided no winner was declared. Now, the pendulum swings back in the other direction, as whoever places last gets to choose the *mode* of observation. What's more, they even get to wait with their decision until *after* seeing where on the board the pieces end up! It turns out that the second player can always *force* the game in such a way that they go last, and can thus be guaranteed to gain control over the observation phase. (We leave it to you to figure out how.) Does this mean we now have a *second*-player advantage? Well, implementing that strategy typically involves performing a lot of superposition moves, foregoing the more powerful entangling and cancelling moves in the process. Player 1 will also get to place one more piece on the board than Player 2, increasing their stake on the board. And so, several possible strategies present themselves: Will you sacrifice everything for guaranteed control over the observation phase? Or do you prefer to work towards a position that is as advantageous as possible under *either* mode of observation, and take opportunities as they present themselves? ### What Next? Maybe [this](https://www4.uib.no/studier/program/kvanteteknologi-integrert-masterprogram-sivilingenior)? [^5]: The terminology of a "shifted" observation is inspired by the example of polarization filters, as explained in [[The Physics of TiqTaqToe]]. More common terms include "measurement in the Hadamard basis", or performing an "X measurement" (with standard observations known as "Z measurements"). [^6]: Only the *relative* phase matters here, so a state like $-\triangle-\square$ would still be a plus state, and $-\triangle+\square$ would be a minus state. %%Such configurations will not show up in this game, though they appeared in [[Fully Quantum TiqTaqToe (d4s)|certain earlier versions]].%% [^7]: At this point, I suspect some of you reading this must think to yourself "Now he's just making stuff up." Not so! Every rule is directly informed by quantum-mechanical calculations, which you can check for yourself in [[The Physics of TiqTaqToe]]. Reality really is stranger than fiction! [^8]: If the classical state is $\triangle$, then the two halves of this "superposition" will have the same sign (like in a plus state), and if the classical state is $\square$, then the two halves will have opposite signs (like in a minus state). As an exercise, you may try writing out this superposition-over-superpositions, and verify that the terms cancel in exactly the right way to leave just a classical state, as needed. [^9]: It certainly surprised me! [^10]: In case you want to reconstruct the moves, recall that all ASCII diagrams assume the players use dice in the order red $\to$ green $\to$ blue $\to$ yellow $\to$ purple.