Predominantly Erroneous (Exohedron nonsense blog)

"All this talk about deification and reification. Back in my day we just ificated correctly the first time and never had to undo it or redo it again."

 

It was kind of surprising to find a card game that had wandered into 4chan material while not being deliberately shock humor the way that, say, Cards Against Humanity is. I mean, yeah, there's some weird stuff in the game, but most of it was innocuous, and some of it was stuff like "Donald Trump's Hands" or "Alex Trebek with a mustache" and some of it was Goatse and Tubgirl.

For those wondering, the game was Monikers, which in addition to 4chan references has an interesting underlying mechanic which I'm not going to go into here because it's a little complicated.

 

Realizing that in addition to italics and scare quotes, I also indicate sarcasm or insincerity via Capitalization. I would say that in general emphasizing a word indicates sarcasm, but I might eventually just put a word in ALLCAPS for some reason and I want to reserve that for not-sarcasm. Similarly for brackets and bold and underline; I want to reserve those for indicating other paralinguistic nuances.

 

The jukebox inside my head has finally flushed out almost all of the Christmas music. The only remaining Christmas song is Making a List, which I find perfectly acceptable.

 

I think one of my favorite things about Tumblr is the prevalence of people using references to FullMetal Alchemist to cause each other emotional pain.

 

That thing where I start with "Speaking of" and then completely change the subject. No, we weren't speaking of that, but we are now.

 

And once again I am in the "am I hungry or bored" part of the "determining if I am sick" process.

 

I definitely prefer fire drills in the winter to fire drills in the summer, at least during the day when it's not precipitating.

 

That feel when you're trying to play pictionary but you don't have the official cards with the words on them so you just make a bash script to randomly grab words from a dictionary file, except twenty rounds in you realize that you're only getting words starting with A, B or C, and you discover that bash has a ridiculously small range for its random number generator (0 to 32767) so you have to pause to write a better random number generator.

 

The words "ship" and "shitpost" kind of feel like you should be able to mash them together into a single word that would translate roughly to "crack-pairing" with slightly different connotations but I can't get it to work.

 

At least once every few months I end up wasting a nontrivial amount of time due to my inability to reliably perform basic arithmetic.

 

I wish I had the time and energy to liveblog another math paper. That was fun.

 

You know what? Let's do this.
Homological Tools for the Quantum Mechanic, by Tom Mainiero. ArXiv link.
Part 0: the abstract and TOC
So the paper seems to be about trying to detect quantum entanglement via homological algebra techniques, namely (co)-chain complexes. There's some stuff about non-locality in the quantum mechanical sense aligning with non-local properties in the topological sense, which, um... okay? I think I'm mostly going to stick to the purely algebraic interpretation of factoring tensor products.
And then a bunch of terms that I'm not familiar with anymore because I never went that far into quantum mechanics-as-practiced. I'm more comfortable with the homological algebra aspect of things.
Also holy shit this paper is 126 pages long.
Wow, look at that table of contents.

Part 1: Introduction
So the starting motivation is two factorization questions: (C) When does a probability measure on a product space factor into a product measure, and (Q) When does a state on a tensor product of Hilbert spaces factor into a tensor product of states?
And then, because this is in some sense a math paper, we then give a single unifying description: given an algebra over a field that is the tensor product of several algebras and a linear functional on that algebra, does the functional factorize? (C), the classical case, considers commutative function algebras on the product space, while (Q) is about noncommutative algebras of endomorphisms on the Hilbert space. I'm glad that, even though entanglement is really a quantum phenomenon, the author shows that the particular question being asked here does have a classical analogue.
Okay, focusing on the finite-dimensional case. I like the finite-dimensional case, because then a lot of the topological/analytic considerations can be ignored.
Oh, so the point is that tensor factors of an algebra correspond to disjoint subsystems, which is kind of like locality.

I'm not sure how less-vague the diagram is, to be honest.
I'm having a little trouble reconciling the notion of "gluing" that I'm familiar with from topology to the notion of "gluing" being used here with tensor products of random variables. I mean, I sort of get it, but it's not coming naturally.
On the other hand, I can readily see how mutual information, and in particular the inclusion-exclusion version of it, matches up to Euler characteristic, at least formally.
Okay, so the argument is being made that if Euler characteristic is useful information, then the full complex from which the Euler characteristic is computed must have more useful information. Importantly, while mutual information determines factorizability for bipartite systems, it fails to do so for tripartite systems, which I guess is reasonable.
Not using category and homotopy techniques that haven't made their way to physicists. Hmm. I guess that's good since this paper is of interest to physicists, but disappointing for me as a mathematician.

More later, I guess.

#MathPaperLiveblog

 

I lie a lot, not as much as I did before, but I make shit up, but only if I'm sure that the person I'm talking to won't believe me. I don't like to lie for the point of deception.

I can sort of understand why people would lie to deceive people for the purpose of gaining power or money or influence.

I am baffled by people who lie to deceive in order to convert people to an ideology or a belief system; if you have to deceive people in order to get them to believe then how can you take your beliefs to be true? I mean, it might be some sort of greater-good type deal, I guess, but it's just, if your ideology can't stand up on its own merits, then why do you believe in it?

I am also kind of baffled by people who lie to deceive people for personal entertainment, but that's might be just an issue of preferences.

 

More paperblog, because I kind of want to keep reading.

Section 1.1: Provided Software
This I wasn't expecting. I mean, sure, lots of people who have a supposedly computable thing make scripts and implementations to make sure that at least in the small cases their thing is in fact computable. I guess I just didn't expect this paper to be one of those ones that comes with actual releases, although it does say that this is alpha, and hence definitely not production-ready.

Section 1.2: Summary and Key Results
Oh boy, a super high-level summary of the paper: this paper has two parts, which each read as an individual paper. Good to know. I'm being mildly sarcastic here.

Section 1.2.1 (this is always a good sign): Cohomology for Bipartite Density States
So in the last section, the author notes that mutual information can determine whether a bipartite system can be factorized. So I guess that's why we're focusing on that case first.
Ah, notation. Truly the most difficult problem in mathematics.
Okay, so they're expecting the homology of a chain complex to be the dual of the cohomology from the dual cochain complex, which I guess requires us to be working over a field because that doesn't really work over non-fields.
Did I miss the part where reduced density state is defined? I mean, I'm guessing it's some sort of restriction/projection, but it would be nice to know for sure. This is what I get for not learning about operator algebras.
Why am I worrying about that? This is a summary (I'm still in the introduction 9 pages in. I've read books with shorter intros) so of course very little is actually explained.
So the 0-cochains are built from the individual factors, and the 1-cochains are built from the 2-factor joint stuff. So I can see how it goes in higher degrees: the n-cochains are the (n+1)-factor joint stuff. Makes sense. Reminds me of simplicial cohonology, or Cech cohomology.
Okay, so we get to see how to build cochaiins from densities finally: inner products with stuff in the image of the density state and then tensor with elements of the Hilbert space.
Hey, covariance and variance forms.
Okay, so the 0-th cohomology is the set of pairs of states where the covariance of the two states is the same as the variance of each state individually, and the traces of the two states are equal, modulo the pairs with 0 covariance. I guess the 0-covariance pairs are the factorizable ones.
Okay, so another computation of the 0-th cohomology via supports of projections of stuff. I should figure out what Schmidt rank is.
Man, this is tiring. More later.
#MathPaperLiveBlog

 

A slight variation on the Trolley Problem, where track one has one guy tied to it, and tied to the other track is Five Guys Famous Burgers and Fries.

 

Back to this nonsense:
Still on section 1.2.1
The dimension of H¹(G(rho_AB)) is a nice product: (d_A - S)(d_B - S). I like things that factor.
Okay, so now we finally get to the other chain complex, which I'd honestly forgotten about. Ooh, commutants. I have a vague understanding of what that is.
Why do you say "otherwise" when you could just say "if k = 1"? I mean, aren't the only options at the moment k = 0 or k = 1?
Okay, so we've got a way to determine factorizability in the pure bipartite state case. Two ways, in fact. Apparently mixed states are more difficult, unsurprisingly.

Section 1.2.2 Cohomology for Multipartite Density States
Oh, so I was right: these are Cech complexes. The author just didn't want to get into that setup because physicists don't know about sheaves. But yeah, the degree k cochains are built from all subsystems with k+1 factors.
And now explaining simplicial cohomology.
Coboundary is projection onto oriented sums of parts, as expected.
Stuff that indicates correlations along each face modulo stuff that looks like it does but actually comes from non-factorizability along lower-dimensional faces.
I'm really hoping that I'll get to the actual meat of this paper soon, because there is so much stuff that I'm not sure if I'm not understanding it or if it's not being explained in enough detail to understand.
Okay, so the cohomology, in addition to detecting failure to factorize, indeed indicates where the obstructions are.
Ooh, a Kunneth formula. Always a good sign. The shift in degree is kind of weird, but not entirely unexpected, I guess.
Right, so for a fully factorizable multipartite state, the Kunneth formula, applied a bunch of times, tells us that the cohomology of the multipartite state is the product of the cohomology of a single part state, and hence the cohomology of the product vanishes in a bunch of places. Great.
And finally the obvious conjecture: this is an iff statement: that the cohomology vanishing in the appropriate places does in fact indicate factorizability.

So that's the end of section 1.2.2, so I'll stop there for now.
#MathPaperLiveBlog

 

Section 1.2.3: Tripartite Computational Results
Okay, now we get to see what actually happens in some nontrivial multipartite qubit cases. Also I finally find out what the GHZ and W states are: the GHZ is |000> + |111> and W is |001> + |010> + |100>, which I guess make sense. These are important because they're representative of the equivalence classes of states that aren't factorizable or bipartite.
I guess the term that the author has been using, "local invertible transformations", is something I should parse at some point. Probably transformations of individual factors?
Okay, finally some concrete justification for cohomology rather than Euler characteristic: for the GHZ and the W state, the mutual information vanishes, but the cohomology doesn't. Also, justification for the two different cohomologies: the Poincare polynomial for the GNS cohomology can't distinguish between the GHZ and W states, but the Poincare polynomial for the commutant cohomology can. This is actually kind of nice, to see that these different cases so concretely. I kind of wish these had shown up earlier, but I guess it makes sense that the author needed to define a bunch of crap before stating these results.
No, I take that back, I think the fact that one of cohomologies can distinguish GHZ from W and the other can't, and the mutual information can't even detect nonfactorizability, could have been mentioned very early on. I mean, the author did say that the multipartite case was different from the bipartite case, but an example at the time would have been nice.

Section 1.2.4: The State Index and a Path Toward Categorification of Mutual Information
The State index is kind of like a Poincare polynomial, except there's a trace involved. Strange.
Should I be surprised that the state index is multiplicative? I don't think I should be. I'm not, but I'm a little worried that I've gotten complacent about this kind of thing.
There's an identity that I guess is some sort of inclusion-exclusion thing for the state index. Sort of. The interpretation is kind of off, because there's tracing instead of subsets and merging (I'm not entirely sure what is meant here) instead of intersection.
Oh, so the state index has the mutual information and the two Euler characteristics as limiting cases.
There are some missing parentheses that I really don't like. Not in the sense that there's an open paren without a closing thesis, but rather that there is a function that is denoted without explicitly delineating where its argument ends.

Section 1.3 (finally out of 1.2!): Related work
Bunch of names, bunch of names.
Hey, I recognize some of those names.
Bunch of names, bunch of names.

Section 1.4: Future directions:
Link invariants? I guess there's some way to turn a link into a multipartite state, and then link invariants might become Poincare polynomials or something?
That reminds me of an old model of the atom, by Lord Kelvin, wherein all the different elements were actually tiny knots in the aether, and the different types of knots gave rise to different types of elements. Of course, when the aether was disproven, this idea kind of went with it.
Oh well.
Anyway, the author isn't talking about Lord Kelvin's theory, he's talking about Chern-Simons theory, which at least has a lot of mathematics behind it.
Ooh, my favorite word: noncommutative. I like noncommutative.

Hmm, the disclaimer is kind of disappointing: the setup here doesn't even really detect factorizability in mixed states; instead it detects "support factorizability", which is significantly weaker.

Section 1.5: Acknowledgements:
Bunch of names, bunch of names.

Okay, I am finally out of the introduction, so I am stopping here.
#MathPaperLiveBlog

 

I really dislike that "tired" and "sleepy" are not the same state. When I'm tired but not sleepy I just want to sit down and not do anything. When I'm sleepy but not tired I make bad decisions like walking up too many flights of stairs.

 

I try not to have much "school spirit" for my alma mater out of a contrarianism that's probably misaimed because my local social environment also has a contrarian adverse opinion of said college, and indeed there is a decently large scale perception of "it's a good school but not as good as it makes itself out to be and is therefore worthy of derision and scorn" which may or may not be proportionate to the actual gap between the school's actual and self-proclaimed merit, and may be driven by some amount of envy.
Anyway, I try not to care and generally don't say anything unless someone asks, because there's only so much dodging between humble-bragging and insincere self-deprecation that I am willing to perform.

Today someone said something about the school and I felt the need to correct them and I think I might even be offended, which is a bizarre feeling both because I didn't realize I could feel that strongly about the place but also because being offended isn't something I've felt in quite a while either, and I'm trying to figure out if it's actual offense or if it's just that I want to win the argument.
I'm not sure how happy I would be with either case being true. Being offended is not a state or a process that I find useful or wanted, but getting so worked up simply because I want to beat someone in an argument is petty. The admittedly thin illusion of detachment that I like to maintain has a hole that I wasn't previously aware of, and while I do value the awareness, the existence of the hole itself is irritating.