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 $H_S$ 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 $|\varphi ⟩\in H$ by a (possibly rank $>1$) density matrix $\rho \in {\text{Op}}_H$...

But if $\rho :=|\varphi ⟩⟨\varphi |$, 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 $\rho$s is much larger than the space of possible $|\varphi ⟩$s, there are sometimes (always?) multiple $\rho$s that correspond to some particular $|\varphi ⟩$; 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.)

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": $U$ is "basically equivalent" to a classical elementary CA if the long-run behavior of $U$ is very close to the long-run behavior of some $U_{classical}$, 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 $\tilde{I}so(n,1)$ 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 $U$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 $\cdots \otimes |x⟩(⟨x|+⟨y|)\otimes |y⟩(⟨y|+⟨z|)\otimes \cdots$ can be unitary," I'd happily pay for that!

Things have coalesced near the amphitheater. When the music kicks off again, we'll go northeast to... approximately here. 47.6309473, -122.3165802 JMJM+99F Seattle, Washington