
The Universe’s Ultimate Complexity Revealed by Simple Quantum Games - pseudolus
https://www.quantamagazine.org/the-universes-ultimate-complexity-revealed-by-simple-quantum-games-20190305/
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yetihehe
> There are barriers to actually carrying out Slofstra’s experiment. For one
> thing, it’s impossible to certify any laboratory result as occurring 100
> percent of the time.

Hmm, yeah, I thought there will be, like with all of those quantum tests
trying to say if space is continuous.

> “In the real world you’re limited by your experimental setup,” Yuen said.
> “How do you distinguish between 100 percent and 99.9999 percent?”

And that is exactly the problem with idea of discrete space. Once you try to
pin-down those pixels, it appears universe is blurry before you can go down to
that resolution. "In the real world, you are limited by real world".

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_bxg1
As a layperson, this feels like it's just passing the buck of infinity,
doesn't it? "Given infinite time you could prove whether or not the universe
is infinite" is almost a tautology. Is there some non-experimental way this
could be used instead?

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ajuc
I don't understand what they mean by N-dimensional state space. There are
bijections between R^n and R for any n. What does it even mean, then, that
state-space is N-dimensional?

If you can transmit 1 real number you can transmit any number of real numbers.

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macawfish
Where did they use the term "N-dimensional"? When you say _N-dimensional_ are
you talking about what the article refers to as "finite dimensional"? If so, I
think you're touching on something about the distinction between finite-
dimensional and infinite-dimensional spaces.

For any finite n, the cardinality of R^n and R are the same.

I'm guessing that those bijections break down once you step into infinite
dimensional spaces, but I haven't found a simple reference for this. Can
someone verify this?

~~~
ajuc
> Where did they use the term "N-dimensional"?

From the article:

> Researchers talk about a state space as having a certain number of
> dimensions, reflecting the number of independent characteristics you can
> adjust in the underlying system.

> For example, even a sock drawer has a state space. Any sock might be
> described by its color, its length, its material, and how raggedy and worn
> it is. In this case, the dimension of the sock drawer’s state space is four.

That's meaningless IMHO, 4 is the same as 1.

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philipov
Dimensionality is a type of structure that can be applied to quantities.
Dimensionality refers to being able to find independent bases within a vector
space. The study of vector spaces is a richer language than set theory. By
allowing only cardinality to be meaningful, you are denying the value of
dimensionality as an organizing principle. Therefore, its meaninglessness is
tautological.

Your mistake is to think that 4 is the same as 1. It is not. You are putting a
hand up to one eye and complaining that stereoscopy is meaningless. You are
pretending that only set theory exists, and claiming that therefore linear
algebra is meaningless. It is rhetorical nonsense!

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ajuc
> It is not.

Why? If I can describe a system with 1 number why invent 4 parameters to
describe it? And why not 100? The choice seems arbitrary.

~~~
NotAnEconomist
Because dimension is about the connectivity structure of a set of objects, not
the size of the set of objects.

Sets with the same cardinality can have wildly different topologies, which
describe the possible paths through the space, which in turn describes the
nature of functions into and out of that space, as well as transformations and
relationships within that space.

A line through the center of a sphere doesn't necessarily touch the sphere in
a "4 dimensional" connectivity structure, _must_ touch the sphere in a "3
dimensional" connectivity structure, while a sphere doesn't even exist in a "1
dimensional" connectivity structure.

The main reason that we use "4 dimensional" spacetimes for particle physics is
that there are three independent components of motion with respect to time,
and so it makes the math simpler to allow for independent variation of those
parameters to functions, instead of compressing them into a single encoded
number, eg, derivatives wouldn't function correctly (or likely even be
defined) on a single parameter model. An additional reason that we don't use a
"1 dimensional" spacetime is because the product of a "1 dimensional" time and
a "1 dimensional" space is a "2 dimensional" spacetime -- and we don't have
any models of reality that lack either space or time. (Which is good: any
model needs to explain both, if only as an emergent feature.)

Topology is about connectivity, not scale; dimension is a topological fact.

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snarfy
This is a pretty good video describing Bell's Inequality -
[https://www.youtube.com/watch?v=5_0o2fJhtSc](https://www.youtube.com/watch?v=5_0o2fJhtSc)

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tabtab
Maybe a more concise model of the universe and physics can be bred via a
genetic algorithm and other AI techniques. A couple of decades ago one
researcher would bred algorithms written in lisp: trial and error programming.
Lisp is good for that because you can slice and dice it and still get a
runnable program (if the interpreter ignores/forgives missing parentheses).

Start with a simulation using the best known but complex formulas. Then use
the genetic algorithm (mutation and cross breeding) to find the most compact
formulas/algorithms that can mirror/predict the complex one. Who knows, the
whole shebang may be reducible to something as simple as E = mc^2 or Newton's
F=ma.

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snaky
The whole shebang may be reducible to something as simple as 42, as we all
know.

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copperx
Isn't that the argument of Stephen Wolfram? That infinite complexity can arise
from deterministic processes? The details may be wrong but I think that's the
gist of it.

Many consider him to be wrong.

~~~
tabtab
If and when the Grand Formula(s) is found, we'll know.

