Ahhh! Yes, this is very helpful! Thanks for the explanation.
Optimization Process
Karma: 653
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!
Quantum without complication
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!
[Question] Why are there no interesting (1D, 2-state) quantum cellular automata?
If I ran the zoo
Two children’s stories
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
Announcement 1: I, the organizer, will be 5-10min late. Announcement 2: apparently there’s some music thing happening at the amphitheater! I’ll set up somewhere northeast of the amphitheater when I get there, and post more precise coordinates when I have.
$10 bounty for anybody coming / passing through Capitol Hill: pick up a blind would-be attendee outside the Zeek’s Pizza by 19th and Mercer. DM me your contact information, and I’ll put you in touch, and I’ll pay you on your joint arrival.
Update: the library is unexpectedly closed due to staffing issues. The event is now at Fuel Coffee, one block south and across the street.
Oh, this is genius. I love this.