Superposition

Quantum Tic-Tac-Toe was invented with three strategic objectives:

• Demonstrate superposition in its simplest possible form.
• Demonstrate that collapse can be forced by self-reference.
• Demonstrate that an objective measurement process is conceivable.

Spooky Marks

The key to superposition is multiple possibilities. The smallest multiple possibilities is two. Therefore, just two spooky-marks, not three, not more, not variable. The simplest possible weighting is to make the weights equal. Superposition in its simplest possible form - pure scientific reductionism.

Placement Moves

When a player places a pair of spooky-marks on the quantum tic-tac-toe board, he is not in both squares, but rather sort of half in one and half in the other. As one wit put it, “It’s the square root of a move.” This aligns well with the principle in quantum mechanics that the odds of a measurement outcome is equal to the square of the weight for that state.

Interpretation

It is a little murky in physics how to interpret superposition. The math is clear, the understaning is not. This is unnerving. We have never found another math that can predict the correct experimental outcomes. And the accuracy of the predictions is truly phenomenal. We trust the math, we trust the algorithm that converts the math into statistical outcomes, we marvel at the fact the odds we see match the odds we calcluated. But we can’t say we understand it.

The situation in quantum tic-tac-toe is radically different, the understanding is immediate, obvious, and un-controversial. A pair of spooky-marks means you are now playing two games of classical tic-tac-toe at once. In the first classical game, the move is where the first spooky-mark is at, in the second classical game, the move is where the second spooky-mark is at. Trivial. The set of classical games is called the classical ensemble.

For the additional moves, the classical ensemble is simply duplicated. Two sets, each identical to the previous state of the quantum board. In the first set, the move is where the first spooky-mark is, in the second set, the move is where the second spooky-mark is. The classical ensemble grows.

Our paradigm of reality is classical. The ‘Universe’ is just out there, matter in motion. There is just one instance of the Universe. It changes state, slowly, dynamically, in accordance with the laws of physics. But it is otherwise just there.

If the interpretation of quantum tic-tac-toe is applied to the Universe as a whole, then our paradigm falls radically short of the reality. Every superposition, every evolution of the state, every new multiple possibility, implies the sudden, discountinuous creation of another set of ‘universes.’ All the matter, all the energy, all the complexity duplicated atto-second to atto-second, everywhere.

Everytime we learn a little more the universe gets bigger in some way.

• The earth is not at the center, the sun is - the universe got bigger.
• The sun is not at the center, it is part of a galaxy - the universe got bigger.
• The universe is teaming with galaxies - the universe got bigger.
• The universe is expanding, it has an observational horizon we cannot see past - the universe got bigger, only now we can’t even tell by how much.
• The planck scale; our measurement ruler has much finer resolution than supposed - the unverse got ‘bigger’ because the scale got super refined.

Now, its not just one universe? But a whole plethora of universi? The Universe got bigger, again, in a whole new way. My head hurts.

Entanglement

Quantum mechanics has no shortage of conceptual mysteries. Entanglement is one of the most perplexing. In classical systems, given two identical random situations, the total situation is just the product of the individual situations. Two plus two equals four; obvious, logical, commonsense in action. Mathematically, the two individual situations are separable.

In quantum systems, two individual situations can become nonseparable. In a word, they become entangled. In quantum tic-tac-toe this occurs whenever two pairs of spooky-marks share a square. Instead of there being four realities in the classical ensemble there are only three - two plus two equals three? Why? Because in the missing reality, both classical moves were in the same square, and that is an illegal situation in classical tic-tac-toe - only one mark per square allowed. The missing reality was pruned from existence. Pruned because it was contradictory, its classical history violated the rules of the game.

Again, apply this interpretation to the whole Universe. Not only are new universi being created willy-nilly, out of nothing, atto-second to atto-second, but now some of them are being thrown away, atto-second to atto-second as well. Poof, here is a new universe, poof, another disappears. If this is actual reality, then the quip is accurate, “Not only is the universe queerer than we imagine, it is queerer than we can imagine.”

The entanglements which emerge in quantum tic-tac-toe are of a very particular kind: semi-entanglements. They lie halfway between separable states (classical) and the maximum possible entanglement (EPR).

Decoherence

Just to deepen the mystery, these semi-entanglements can be chained together. We have jokenly labled them as demi-entanglements, hemi-entanglements, semi-demi-hemi-entanglemts, and deliberately left unspecified exactly what these mean or how to lable them. But the upshot is this; the longer the entanglement the more some parts start to look classical. The odds begin to drift away from 50/50. So, even though the superpositions of spooky-marks are unweighted, somehow probability still sneaks in. This means quantum tic-tac-toe offers a model of decoherence. Decoherence shows up as an emergent phenomenon; as the system evolves, it begins to creep in.

Self-Reference

The endpoint of an entanglement chain is when it loops back upon itself. This is a cyclic entanglement. It is self-referential - it is causally self-referential. Attend. Consider three spooky-marks in a cyclic entanglement in squares 1, 2, & 5 (corner, side, center).

• X1(1) - X1(2)
• O2(2) - O2(5)
• X3(5) - X3(1)

The cycle should be obvious;

• square 1 is linked to square 2 by X1
• square 2 is linked to square 5 by O2
• square 5 is linked to square 1 by X3

Since there can be only one classical mark per square, if any spooky mark insists on being in a single square, it necessarily ’ejects’ any other spooky marks out of that square. A demonstration is in order.

First, assume that X1 is in square 1,

• that forces X3 to be in square 5
• which forces O2 to be in square 2
• which forces X1 to be in square 1

which is consistent with our first asumption.

Second, assume that X1 is in square 2,

• that forces O2 to be in square 5
• which forces X3 to be in square 1
• which forces X1 to be in square 2

which is consistent with our second asumption.

Both assumptions work - the cycle is indeterminate. Where one mark ends up depends on where that very mark ends up - self-reference. If X1 is in square 1, that causes X1 to be in square 1, and in perfect symmetry, if X1 is in square 2, that causes X1 to be in square 2. It is an instance of a circular argument, it assert itself. It is an Ontological Indeterminacy, not a paradox in sight. The same arguments apply to all the moves in the cyclic entanglement.

It is causally self-referential - and causally consistent.

Strategy

The game specifies that the player who did not create the cyclic entanglement gets to choose its collapse. This is not a metaphor for physics and alternative rules are possible. It is chosen simply to balance the game strategically, to make it a better game.

In the example above, it was X’s 2nd move that created the cyclic entanglement. Therefore, O gets to chose which way the cycle collapses. She has two choices; one puts her move in square 2 (side), the other puts it in square 5 (center). In a classical game, the center is unambiguously a more powerful location. She might very well then, choose the collapse that puts her 1st move in the center of the board.

After she chooses her collapse move, she still gets to chose her regularly scheduled placement move, which will be her second placement move in this example game.

Collapse Moves

A cyclic entanglement has a number of interesting properties.

First, note that all the ‘probabiities’ have returned to 50/50. While the semi-demi-hemi chain was growing, the endpoints were becoming more and more classical. But once the chain closes, that asymmetry gets broken. However, any squares not in the loop, but still connected to it (call them stems), become fully classical. It doesn’t matter how the loop collapses, their values are already determined.

Second, it doesn’t matter how long the loop is, or how convoluted the stem squares are, there are only two possible collapses. There are only two games left in the classical ensemble. The collapse move will choose one of them. In the cases where there are other moves on the board not entangled with the cyclic entanglement, there are two sets of games left in the classical ensemble. The collapse move will choose one of them.

Third, it has to collapse, now, for if it doesn’t any move where both spooky marks end up on the loop, prune all the classical games out of the classical ensemble. Without collapse, the entire ‘universe’ disappers.

Fourth, a cylic entanglement occurs during the natural evolution of the game. No outside interaction ‘measures’ it. The is no ambigous division between the quantum system under study and the external classical environment that interacts with it. No need for that pesky observer. The loop is an objective physical fact - it is an isolated quantum system that self-collapses. This totally destroys the claim that collapse is fictitous and only represents a change in our knowledge.

Quantum tic-tac-toe makes a compelling suggestion, that the quantum measurement problem has a solution. That somewhere in physics there is an objective measurement process, where an isolated quantum system can measure itself - no observer needed. Part and parcel of this metaphor is the idea that self-reference automatically generates indeterminacy - the randomness of quantum systems explained; it comes from self-reference. It comes from the indeterminacy inherent in self-consistent feedback loops.

It is still a metaphor, QT3 is still a toy universe, but it is a powerful one. It suggests what to look for in the hunt for a quantifiable solution to the quantum measurement problem.

Indistinguihsability

Because all the spooky-marks are subscripted with the number of the moves, the individual classical marks are technically distinguishable. In real quantum systems, the potential indistinguishability of identical particles signficantly impacts the probabilities.

But note that given a position on a classical board, the order in which the X’s were placed, and the order in which the O’s were placed is unknown. An ‘X’ is an ‘X’, an ‘O’ is an ‘O’ - all of each player’s moves become indistinguishable.

To explore this part of the metaphor will require investigating a variation on the rules where the moves are indistinguishable.

Perfect Play

Given, however, that the moves are distinguishable, it can be proven that X has a winning strategy in quantum tic-tac-toe, where in classical tic-tac-toe he does not. This is one more example where what is impossible in a classical reality becomes possible in a quantum reality.

Chronoblocks

There is one more implication of quantum tic-tac-toe that deserves mention. A cyclic entanglement occurs in time as well as in space. Not only are the squares entangled (square 1, 2, & 5 in the above example) but so are the moves (1, 2, & 3).

Attend. When was the cause that move one of the game ended up in square 2? It was the collapse at the end of move three. The cause of move one being here, rather than there, occured two moves later - the future impacted the past. Dogma is having a stroke.

Our classical paradigm is that the present is an infinitely thin barrier between the past and the future. What quantum tic-tac-toe is hinting at is that the present is a classical abstraction, that time does not proceed smoothly and uniformly, but suddenly, discontinuously, in chunks. Call these spatio-temporal chunks chronoblocks.

The present has thickness. Within this window around now, the concepts of past, present, and future become ambiguous. If you try to talk about causality within a chronoblock you find yourself using the language of time travel. At least we have such a language, thank you science fiction writers.

It is going to take a while to work through the philosophical implications of this. If this metaphor fully applies to reality, then even causality, one of our most basic paradigms, can no longer be trusted. The events of spacetime are a classial abstraction, in the quantum realm, causality is not point-like, it is distributed, in both space and time.

These chronoblocks can be sequential. If small and the scale large, they asymptotically approach the classical ideal of an instant present. But they can also overlap and even nest.

Now the world’s biggest IF - if causality is distributed across spacetime, then there is going to be a family of technologies which will make the invention of fire, the wheel, the industrial revolution, atomic power, transitor, computer, internet, AI, all of it, look like a dress rehearsal - tomorrow will be opening night.

“Perhaps it is self-reference that we don’t understand.”