## Level 2 (Advanced): Interference
This is where the negative dice numbers enter the stage. As already mentioned, negative numbers is one important sense in which quantum amplitudes differ from classical probabilities, allowing quantum probabilities to *interfere* with each other in much the same way as two colliding waves interfere upon meeting to create pockets of strengthened and cancelled-out regions.
The fact that quantum matter sometimes behaves like particles and sometimes like waves is known today as the "wave-particle duality" of quantum matter. Though the debate about whether light was made up of particles or waves had already raged for centuries (and apparently even Isaac Newton himself had briefly contemplated whether light could somehow be both at the same time), it was Louis de Broglie who first suggested in 1924 that the same must be true for *all* quantum matter, including particles known to have mass such as electrons and protons. (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
- The Cancelling Move
- Cancelling Superpositions
- Observation Phase Example
- Observing Doubly-Cancelled Superpositions
- Balance and Strategy
- Levelling Up
### Setting Up The Game
In addition to the four dice of each colour used in Level 1, each player now gets access to one die per colour of the opponent's shape: d6s for Player 1 (X), and d4s for Player 2 (O). This will allow one player to place a die representing "minus half of the opponent's shape" on the board, cancelling out their probability of being in that square.
The d6 dice also appear to allow Player 2 to place its own symbol using a negative probability. Doing so would have no affect at this point in the game, so for the sake of simplicity we disallow it. This will change in [[TiqTaqToe with Complementarity|Level 3]] though, once we introduce Heisenberg's concept of *uncertainty*.
To start, distribute the pieces as in [[TiqTaqToe with Entanglement|Level 1]], except also give each player one die of the opponent's shape per colour.
```
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
```
### The Cancelling Move
The cancelling move is an interacting move, performed exactly like the entangling move, except that you now place one $-2$ of your opponent's shape in addition to the $2$ of your own shape.
Let's say Player 1 started by playing a classical move in the middle square.
```
Move 1
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | | |
| | /\ | |
| | / 4\ | |
| | r----r | |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
```
Player 2 can now cancel Player 1's probability of ending up in the middle square.
```
Move 2, pre-adjustment
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | /\ | R---R |
| | /\ /-2\ | | 2 | |
| | / 4\ R----R| R---R |
| | r----r | |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
Move 2, post-adjustment
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | /\ | R---R |
| | /\ /-2\ | /\ | 2 | |
| | / 2\ R----R| / 2\ R---R |
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
```
The result is that once the game enters the observation phase, the middle square will be empty with $100\%$ probability. Meanwhile, the middle-right square will be either X or O with 50/50 probability.
Note how the total number of pieces on the board is unchanged after the cancelling move: There was one symbol to be observed before the move, and there remains only one symbol to be observed after the move. So the cancelling move *does not add a piece to the board!* Rather, it simply alters the state of a piece that your opponent had already placed.
You can also choose to let the cancelling happen in a different square from the one you are "invading".
```
Move 2, pre-adjustment
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | R---R | /\ |
| | /\ | 2 | | /-2\ |
| | / 4\ R---R | R----R|
| | r----r | |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
Move 2, post-adjustment
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | R---R | /\ |
| | /\ | 2 | | /\ /-2\ |
| | / 4\ R---R | / 2\ R----R|
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
```
Just like there is a "half-entangling" move, there is also a "half-cancelling" move. This move gets a bit complicated, however, and is arguably the least unintuitive part of the whole game, so we recommend trying out a few games without any superposition-cancelling before reading on.
### Cancelling Superpositions
This "half-cancelling" move is performed just like the "half-entangling" move, except that, just like above, you place one $-2$ die of your opponent's shape and one $2$ die of your own shape.
```
Move 1
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | | |
| | /\ | /\ |
| | / 2\ | / 2\ |
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
Move 2, pre-adjustment
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | /\ | |
| | /\ /-2\ | /\ |
| | / 2\ R----R| / 2\ |
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| | | 2 | | |
| | R---R | |
| | | |
| | | |
+--------------+--------------+--------------+
```
The board then adjusts itself exactly as if you were entangling, "splitting" the opponent's probability in two and "dragging" half over to the new square.
```
Move 2, post-adjustment
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | /\ | |
| | /\ /-2\ | /\ |
| | / 1\ R----R| / 2\ |
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| | /\ | 2 | | |
| | / 1\ R---R | |
| | r----r | |
| | | |
+--------------+--------------+--------------+
```
But looking at the numbers, we appear have gotten ourselves into a strange situation here: Summing the numbers, the dice are telling us is that the middle square has "a quarter $\triangle$ minus half a $\triangle
quot;. We have overshot our cancellation! What happens then? Can a square contain *minus* a quarter probability of $\triangle$?
As it turns, residue minus signs like the above are irrelevant when observing a quantum state.[^4] Does that mean that the state is the same as $1/4\ \triangle$, that is, a $25\%$ probability that cancelling fails and the square ends up with an X after all?
Well, not quite. This is because quantum amplitudes are fundamentally *square roots* of classical probabilities. So when we present a state as "half $\trianglequot; (a d4 with the number 2), the *actual* amplitude of the quantum state it represents is $1/\sqrt{2}\ \triangle$. This means that when we subtract amplitudes from each other, we have to subtract the *square roots* of the probabilities, before squaring again at the end to find the result.
This turns out to be quite fortuitous for the cancelling player: At the end of the calculation (which you can find in [[The Physics of TiqTaqToe]]), the probability that cancelling failed and the square is observed to contain X is still not zero, but rather than failing with $25\%$ probability, it fails with only $3\%$ probability! So with $97\%$ probability, the half-cancelling move was a success.
And how do we roll for a $3\%$ probability? Quite easily actually: Just roll two d6 dice, and watch for snake-eyes (double ones)! As it turns out, $3\% \approx 1/36$, the probability of rolling snake-eyes, so this gives the correct probability to within half a percentage point.
So that's the half-cancelled square. What do we get in the second square? Well, *if* the piece is not in the middle ($P = 97\%$) *and* it is not on in the middle-right square ($P = 50\%$), *then* the piece must be in the bottom-middle square.[^5] We see that it can be either X or O, but that it has twice the chance of being O than it does of being X. Since we now know that the square can not be empty, this means that there must be a one-third probability of observing X, and a two-thirds probability of observing O!
Finally, should you so wish, you could then go on to also cancel the other half of the superposition on a later turn.
```
Move 3 & 4, post-adjustment
+--------------+--------------+--------------+
| | | |
| /\ | | |
| / 4\ | | |
| g----g | | |
| | | |
+--------------+--------------+--------------+
| | /\ | G---G |
| | /\ /-2\ | /\ | 2 | |
| | / 1\ R----R| / 1\ G---G |
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | /\ |
| | /\ | 2 | | /\ /-2\ |
| | / 1\ R---R | / 1\ G----G|
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
```
Note that now Player 2 chose to place the cancelling die in the previously-empty square, rather than in the "invaded" one.
### Observation Phase Example
Let us observe the following board.
```
+--------------+--------------+--------------+
| /\ | B---B | |
| /\ /-2\ | /\ | 2 | | /\ |
| / 2\ B----B| / 2\ B---B | / 2\ |
| b----b | b----b | y----y |
| | | |
+--------------+--------------+--------------+
| | /\ | |
| /\ | /\ /-2\ | /\ |
| / 2\ | / 1\ R----R| / 2\ |
| y----y | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| G---G | /\ | 2 | | /\ |
| | 4 | | / 1\ R---R | / 4\ |
| G---G | r----r | g----g |
| | | |
+--------------+--------------+--------------+
```
Start by replacing the classical pieces with their corresponding symbols, and remove the dice the fully-cancelled square.
```
+--------------+--------------+--------------+
| | B---B | |
| | /\ | 2 | | /\ |
| | / 2\ B---B | / 2\ |
| | b----b | y----y |
| | | |
+--------------+--------------+--------------+
| | /\ | |
| /\ | /\ /-2\ | /\ |
| / 2\ | / 1\ R----R| / 2\ |
| y----y | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| +---+ | /\ | 2 | | /\ |
| | | | / 1\ R---R | / \ |
| +---+ | r----r | +----+ |
| | | |
+--------------+--------------+--------------+
```
The top-middle square is either X or O with 50/50 probability. Player 1 points to it and asks, "is this square mine?", with the roll of an even number interpreted as yes. They roll and get a $5$: The answer is *no*, and so square goes to Player 2.
```
+--------------+--------------+--------------+
| | | |
| | +---+ | /\ |
| | | | | / 2\ |
| | +---+ | y----y |
| | | |
+--------------+--------------+--------------+
| | /\ | |
| /\ | /\ /-2\ | /\ |
| / 2\ | / 1\ R----R| / 2\ |
| y----y | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| +---+ | /\ | 2 | | /\ |
| | | | / 1\ R---R | / \ |
| +---+ | r----r | +----+ |
| | | |
+--------------+--------------+--------------+
```
Notably, and unlike the entangling move, Player 1 gets *nothing* in this case. Tough luck!
Next, Player 2 wants to know whether the middle square was successfully cancelled. With $97 \%$ probability, it will end up empty, but with $3\%$ probability it will contain an X, in which case it is the middle-right and bottom-middle squares that will be empty.
```
If cancelling fails
+--------------+--------------+--------------+
| | | |
| | +---+ | /\ |
| | | | | / 2\ |
| | +---+ | y----y |
| | | |
+--------------+--------------+--------------+
| | | |
| /\ | /\ | |
| / 2\ | / \ | |
| y----y | +----+ | |
| | | |
+--------------+--------------+--------------+
| | | |
| +---+ | | /\ |
| | | | | / \ |
| +---+ | | +----+ |
| | | |
+--------------+--------------+--------------+
```
Player 2 rolls two d6 dice, hoping for anything but snake-eyes. They roll a $1$ and a $2$: Close! But it is not snake-eyes, so the square was successfully cancelled.
```
Cancelling succeeded
+--------------+--------------+--------------+
| | | |
| | +---+ | /\ |
| | | | | / 2\ |
| | +---+ | y----y |
| | | |
+--------------+--------------+--------------+
| | | |
| /\ | | /\ |
| / 2\ | | / 2\ |
| y----y | | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| +---+ | /\ | 2 | | /\ |
| | | | / 1\ R---R | / \ |
| +---+ | r----r | +----+ |
| | | |
+--------------+--------------+--------------+
```
Player 1 continues resolving the half-cancelled state by checking if their piece is in the middle-right square, which happens with the same $50\%$ probability as before, since that half of the superposition remains unaffected. They roll a $3$—a *no*, so the square is empty.
```
Cancelling succeeded
+--------------+--------------+--------------+
| | | |
| | +---+ | /\ |
| | | | | / 2\ |
| | +---+ | y----y |
| | | |
+--------------+--------------+--------------+
| | | |
| /\ | | |
| / 2\ | | |
| y----y | | |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| +---+ | /\ | 2 | | /\ |
| | | | / 1\ R---R | / \ |
| +---+ | r----r | +----+ |
| | | |
+--------------+--------------+--------------+
```
So the piece is in the bottom-middle square, which is what Player 2 hoped for, since they now have twice the probability of capturing it compared to Player 1: Two-thirds versus one-third. They decide that if they roll $3$ or higher on the d6 die ($P = 2/3$), then the square goes to them, and if they roll a one or a two ($P = 1/3$), the square goes to Player 1.
They roll a $5$. Things are starting to look good for Player 2!
```
+--------------+--------------+--------------+
| | | |
| | +---+ | /\ |
| | | | | / 2\ |
| | +---+ | y----y |
| | | |
+--------------+--------------+--------------+
| | | |
| /\ | | |
| / 2\ | | |
| y----y | | |
| | | |
+--------------+--------------+--------------+
| | | |
| +---+ | +---+ | /\ |
| | | | | | | / \ |
| +---+ | +---+ | +----+ |
| | | |
+--------------+--------------+--------------+
```
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 $6$, so the answer is yes.
```
+--------------+--------------+--------------+
| | | |
| | +---+ | /\ |
| | | | | / \ |
| | +---+ | +----+ |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | | |
| +---+ | +---+ | /\ |
| | | | | | | / \ |
| +---+ | +---+ | +----+ |
| | | |
+--------------+--------------+--------------+
```
There is still no winner, and there are now *four* empty squares remaining. The game continues! Player 1 placed the last piece, initiating the observation phase, so the turn goes to Player 2.
What would you do next?
#### Observing Doubly-Cancelled Superpositions
What if Player 2 had cancelled *both* halves of Player 1's superposition state?
```
+--------------+--------------+--------------+
| | | |
| /\ | | |
| / 4\ | | |
| y----y | | |
| | | |
+--------------+--------------+--------------+
| | /\ | G---G |
| | /\ /-2\ | /\ | 2 | |
| | / 1\ R----R| / 1\ G---G |
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
| | R---R | /\ |
| | /\ | 2 | | /\ /-2\ |
| | / 1\ R---R | / 1\ G----G|
| | r----r | r----r |
| | | |
+--------------+--------------+--------------+
```
Even though the state is now spread over four squares, since both moves were cancelling and therefore did not add a new piece to the board, only *one* of the four squares will be non-empty.
There are several ways to observe this state, but the easiest may be as follows: First, check which of the halves the original superposition ends up in: Is it the half that has been cancelled by the red Player 2 dice (middle column), or the half with the green Player 2 dice (right column)? Since each half of the superposition were independently affected, this probability is still 50/50.
Player 1 points to the left column and asks, "is my piece in this column"? They roll a $2$, so the answer is yes, and the right column is empty.
```
+--------------+--------------+--------------+
| | | |
| /\ | | |
| / 4\ | | |
| y----y | | |
| | | |
+--------------+--------------+--------------+
| | /\ | |
| | /\ /-2\ | |
| | / 1\ R----R| |
| | r----r | |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| | /\ | 2 | | |
| | / 1\ R---R | |
| | r----r | |
| | | |
+--------------+--------------+--------------+
```
Then we proceed as before: We first check whether the cancellation succeeded in the middle square by rolling two d6. If the result is double ones, then cancelling failed, and the piece is in that square. Otherwise, the middle square is empty, and the piece is in the bottom-middle square, with a $2/3$ probability of being O and a $1/3$ probability of being X.
```
+--------------+--------------+--------------+
| | | |
| /\ | | |
| / 4\ | | |
| y----y | | |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | R---R | |
| | /\ | 2 | | |
| | / 1\ R---R | |
| | r----r | |
| | | |
+--------------+--------------+--------------+
```
### Balance and Strategy
In terms of gameplay, the main feature of the cancelling move is that you now get access to an interacting move with a *certain* outcome (or near-certain, in the case of the half-cancelling move), namely *an empty square*. This means that if Player 1 attempts to capture the coveted middle square, instead of settling for a $50\%$ chance of taking it from them with an entangling move, Player 2 can cancel the square completely, and hope to be able to capture it themselves after the first observation phase.
Effective use of cancelling moves also means that more squares will end up empty at the end of 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 turn phases and observation phases, with the final board gradually revealed to both players.
We have found that this adds a refreshing layer of strategy to the game. For example, should you make sure to place in as many squares as possible, or should you prioritize *not* placing last so that you get to go first after the observation phase? And should you go for as much cancellation as possible, to lower the probability of an immediate win, or should you focus on entangling moves to ensure that you don't run out of luck and end up with *nothing* on the board?
After playing a few rounds, I think you will agree that the increased potential for strategic play is much welcome. As one playtester remarked: "Now it feels like a *game!*"
### Levelling Up
In [[TiqTaqToe with Entanglement|Level 1]], we thought of each *piece* as represented by one *colour*, with a definite *symbol* (namely that of the player who placed it), but possibly existing in a (possibly entangled) superposition over different squares on the board.
At this level, however, we saw that the cancelling move does not add to the number of pieces on the board, but rather affected the state of an existing piece, so the "one colour $=$ one piece" interpretation breaks down. Further, cancelling a classical move leads to square that will definitely end up empty, and another square that is in an equal superposition of X and O. Hence, we no longer have a superposition over *squares*, but over *symbols*.
```
+--------------+--------------+ +--------------+--------------+
| /\ | R---R | | | R---R |
| /\ /-2\ | /\ | 2 | | | | /\ | 2 | |
| / 2\ R----R| / 2\ R---R | -> | | / 2\ R---R |
| r----r | r----r | | | r----r |
| | | | | |
+--------------+--------------+ +--------------+--------------+
+--------------+--------------+ +--------------+--------------+
| | | | | |
| | /\ | | | +---+ |
-> | | / \ | or | | | | |
| | +----+ | | | +---+ |
| | | | | |
+--------------+--------------+ +--------------+--------------+
```
Just like superposition over different points in space, such superpositions over *values* also appear in nature. In fact, quantum particles can be found in superpositions not only over space, but over *any* of its observable features, including travelling speed, internal energy, and the quanta of magnetism known as "spin".[^1] It is exactly this kind of superposition that sits at the heart of quantum computing, where the fundamental elements, the *qubits*, are always in well-defined *positions* on the quantum chip,[^7] but can be in arbitrary superpositions over the bit values $0$ and $1$.
A piece like the above, which is in a well-defined position but an equal superposition over two values, is said to be in a *plus* ($+$) state: $\frac{1}{2}(\triangle + \square)$. There is also a *minus* ($-$) state, which is the same as the plus state except that the two symbols have opposite signs: $\frac{1}{2}(\triangle - \square)$. We can represent this using the $-2$ numbers on Player 2:
```
Plus state: Minus state:
+--------------+ +--------------+
| R---R | | R---R |
| /\ | 2 | | | /\ |-2 | |
| / 2\ R---R | | / 2\ R---R |
| r----r | | r----r |
| | | |
+--------------+ +--------------+
(assuming no entanglement)
```
At this level, the fact that we can make both plus and minus states is nothing but a nice observation, since plus states and minus states behave identically when observed: Both states give $\triangle$ or $\square$ with 50/50 probability.
But as we will see in the next level, these "plus" and "minus" states will allow us 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 that the probabilities can depend on what question you're asking, in such a way that *asking one question changes the probabilities of the other*. This is Heisenberg's famous uncertainty principle, also known as **complementarity**.
%%At the next level, whoever places last and thus initiates the observation phase, also gets to choose which *question* to ask when observing the board: Either the standard question, "Is the state X or O?", or the *shifted* question, "Is the state + or -?" %%
To learn what this means, and how it opens for whole new set of strategies, take the leap and enter **Level 3 (Expert): [[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 (corresponding to a counterclockwise magnetic field around the spin axis), then it is a $\triangle$, and if it is observed in a spin-down state (a clockwise magnetic field), it is a $\square$.
[^2]: The fact that probabilities adjust themselves the more we learn about a piece in such a way that they always add up to $100\%$ probability for a piece to be on the board, is known as *renormalization*, and is essential to making quantum mechanics a physically sensible theory.
[^3]: Numbers "in between" positive and negative are known as [complex numbers](https://en.wikipedia.org/wiki/Complex_number).
[^4]: As the quantum catchphrase goes: *Global phases are unobservable*.
[^5]: Recall that the cancelling move does not add a piece to the board, but rather alters the state of a piece the opponent already placed, so even though three squares now contain dice, only one of them will end up non-empty.
[^6]: You may wonder if the probabilities are really the same in this case as they were in the single-cancelling case. In fact, the probability of each half-cancelled square being non-empty is slightly smaller than before, but the difference is so small that it is swallowed by the rounding.
[^7]: This is true even in "chips" that allow qubits to be moved around in space, such as in ion-trap and neutral-atom-based quantum computers. The exception is photonic quantum computers, where the *trajectories* of the photons are an integral part of the computation.