Ahhh! Yes, this is very helpful! Thanks for the explanation.
Question: if I'm considering an isolated system (~= "the entire universe"), you say that I can swap between state-vector-format and matrix-format via
. But later, you say...
If is uncoupled to its environment (e.g. we are studying a carefully vacuum-isolated system), then we still have to replace the old state vector picture by a (possibly rank ) density matrix ...
But if , how could it ever be rank>1?
(Perhaps more generally: what does it mean when a state is represented as a rank>1 density matrix? Or: given that the space of possible s is much larger than the space of possible s, there are sometimes (always?) multiple s that correspond to some particular ; what's the significance of choosing one versus another to represent your system's state?)
That is... a very interesting and attractive way of looking at it. I'll chew on your longer post and respond there!
I have an Anki deck in which I've half-heartedly accumulated important quantities. Here are mine! (I keep them all as log10(value in kilogram/meter/second/dollar/whatever seems natural), to make multiplication easy.)
Quantity | Value |
---|---|
Electron mass | -30 |
Electron charge | -18.8 |
Gravitational constant | -10.2 |
Reduced Planck constant | -34 |
Black body radiation peak wavelength | -2.5 |
Mass of the earth | 24.8 |
Moon-Earth distance | 8.6 |
Earth-sun distance | 11.2 |
log10( 1 ) | 0 |
log10( 2 ) | 0.3 |
log10( 3 ) | 0.5 |
log10( 4 ) | 0.6 |
log10( 5 ) | 0.7 |
log10( 6 ) | 0.8 |
log10( 7 ) | 0.85 |
log10( 8 ) | 0.9 |
log10( 9 ) | 0.95 |
Boltzmann constant | -22.9 |
1 amu | -26.8 |
1 mi | 3.2 |
1 in | -1.6 |
Earth radius | 6.8 |
1 ft | -0.5 |
1 lb | -0.3 |
world population | 10 |
US federal budget 2023 | 12.8 |
SWE wage (per sec) | -1.4 |
Seattle min wage (per sec) 2024 | -2.3 |
1 hr | 3.6 |
1 work year | 6.9 |
1 year | 7.5 |
federal min wage (per sec) | -2.7 |
1 acre | 3.6 |
I thank you for your effort! I am currently missing a lot of the mathematical background necessary to make that post make sense, but I will revisit it if I find myself with the motivation to learn!
This is a good point! I'll send you $20 if you send me your PayPal/Venmo/ETH/??? handle. (In my flailings, I'd stumbled upon this "fractional step" business, but I don't think I thought about it as hard as it deserved.)
How are you defining "basically equivalent"
Nyeeeh, unfortunately, sort of "I know it when I see it." It's kinda neat being able to take a fractional step of a classical elementary CA, but I'm dissatisfied because... ah, because the long-run behavior of the fractional-step operator is basically indistinguishable from the long-run behavior of the corresponding classical CA.
So, tentative operationalization of "basically equivalent": is "basically equivalent" to a classical elementary CA if the long-run behavior of is very close to the long-run behavior of some , i.e., uh,
...but I can already think of at least one flaw in this operationalization, so, uh, I'm not sure. (Sorry! This being so fuzzy in my head is why I'm asking for help!)
I was imagining the tape wraps around! (And hoping that whatever results fell out would port straightforwardly to infinite tapes.)
I've never been familiar enough with group-theory stuff to memorize the names (which, warning, also might mean that it will take you a lot of time to write a sufficiently-dumbed-down version), but the internet suggests is related to... the Minkowski metric? I would be flabbergasted to learn that something so specific-to-our-universe was relevant to this toy mathematical contraption.
I think compared to the literature you're using an overly restrictive and nonstandard definition of quantum cellular automata.
That makes sense! I'm searching for the simplest cellular-automaton-like thing that's still interesting to study. I may have gone too far in the "simple" direction; but I'd like to understand why this highly-restricted subset of QCAs is too simple.
Specifically, it only makes sense to me to write as a product of operators like you have if all of the terms are on spatially disjoint regions.
Hmm! That's not obvious to me; if there's some general insight like "no operator containing two ~'partially overlapping' terms like can be unitary," I'd happily pay for that!
Oh, this is genius. I love this.