## What is the relationship between unitary evolution and collapse in quantum mechanics?

Last time we started discussing the measurement problem. In quantum mechanics textbooks, a quantum system is presented as undergoing two distinct time evolution:
• a unitary time evolution before measurement
• a sudden change known as the collapse of the wavefunction
The epistemic explanation of it is very simple: the collapse corresponds to information update. QBism has self consistent explanations to all quantum puzzles, so why bother seek a different solution to the measurement problem? I mean besides trying to erase from your memory the creepy picture Chris Fuchs is using in his talks about qbism:

For some reason this picture of an one-eye man with measurement dials instead of hands is like a bad song you cannot get out your brain.

The answer is that any non-unitary time evolution is fatal to quantum mechanics. Here is why:

Unitarity in the state space formulation of quantum mechanics, just like the Leibniz identity in the algebraic formalism are a consequence of the invariance of the laws of nature under time evolution and moreover each one can be derived from the otherI have showed in prior posts how to reconstruct quantum mechanics using Leibniz identity. What reconstruction of quantum mechanics shows is that breaking the Leibniz identity makes the whole quantum formalism inconsistent: no more Hilbert spaces or Hermitean operators before or after measurement.

So for pure mathematical consistency arguments, can we describe the (non-unitary) update of information using only unitary time evolution? Yes we can, and the answer is that the non-unitary collapse is nothing but a change in the GNS representation, but to show it rigorously I need to build up the required machinery.

Last time I made a strong claim: the transformation

$$(\lambda |\psi_A \rangle + \mu |\psi_B \rangle )\otimes |M_0 \rangle\ \rightarrow \lambda |\psi_A \rangle \otimes |M_A \rangle + \mu |\psi_B \rangle \otimes |M_B \rangle$$

is not correct. In foundations of quantum mechanics the justification for the transformation above stems in part from the following argument: quantum mechanics is universal, and the measurement process should be described quantum mechanically in a Hilbert space. Asher Peres had a rebuttal to this but that was only handwaving inspired by Godel incompletness theorem. This rigorous  reason the argument is faulty is because it turns out that there are many Hilbert spaces involved. I will show that $$|\psi_A \rangle \otimes |M_0 \rangle\ \rightarrow |\psi_A \rangle \otimes |M_A \rangle$$ and $$|\psi_B \rangle \otimes |M_0 \rangle\ \rightarrow |\psi_B \rangle \otimes |M_B \rangle$$ should not be understood as unitary evolution, but as a change in representation. Since changes in representation do not happen in a Hilbert space, there is no superposition and they cannot be combined.

Interestingly enough, the relationship between unitary evolution and collapse is deeply related with the relationship between the addition and subtraction in the case of integer numbers. There is a category construction called the Grothedieck group construction which is at play in both cases. The Grothendieck group construction is an universal property, meaning it is both unique and natural.

For now I only want to point a problem in the easy integer numbers setting. When you learn arithmetic in elementary school , first you learn to add and subtract. Then you learn about multiplication and division. Fast forward to say your high school or early undergraduate years, you learn to formalize those operations in the concepts of groups, rings, and fields. The natural numbers for example are an abelian monoid. Going from monoids to groups one needs to add the inverse elements. Natural numbers N become the integers Z. And in the integers abelian group Z you have two operations: $$+$$ and $$-$$. But wait a minute, a group has only one operation! So what does the subtraction operation mean? Do we have two independent operations? In a field there are two independent operations, addition and multiplication, but not in a group.

Starting from this trivial observation that subtraction must be not independent from addition, the problem is how to express one in terms of the other. The first try is to say: no problem at all, subtraction is addition by negative elements. I bet you already heard that in school. But this is mathematically sloppy and unacceptable answer because the definition of an operation should be decoupled from the nature of the elements it works with. It turns out that there is a unique way one can turn an abelian monoid into an abelian group, and this was figured out by Mr. Grothendieck.

So what does this have to do with quantum mechanics? Do you know a natural abelian monoid there? When we compose quantum systems we use the tensor product: $$\otimes$$. This is associative: it does not matter the order in which we compose Hilbert spaces. It has a unit element: compose with nothing. It is also commutative. We will see that the inverse operation $${\otimes}^{-1}$$ is deeply related with the collapse postulate, but it is not a straightforward relationship. Please stay tuned.

1. Hi, Florin.
Your last two posts inspired this 2-part question, regarding the two 'cousins' to your interpretation: Consistent Histories and Qbism. It seems the majority of CH supporters regard it as 'Copenhagen done right', yet there's also a faction that regards it as 'MWI done right', and the 'official position' is that it posits a non-unique outcome. Given all of this, I've always wondered how one arrives at the neo-Copenhagen position in CH, as they don't seem to be working on a well-defined mechanism for it as you are. Regarding Qbism, what do you think the 'orthodox' Qbist response would be to your claim that non-unitary collapse breaks quantum mechanics? For instance, what would Chris Fuchs' counterargument be? (Keep in mind, I'm not trying to imply that anybody's wrong, here--just curious.)
Best,
Eric Hamilton.

2. Hi Eric, sorry for the long delay, I was busy shoveling after "snowzilla" and I ignored anything else. It was so bad that the National Guard was activated and they were driving in Humvees to map the unpassable neighborhoods. It was like in movies where meteors are about to wipe all life on Earth :)

I had an in depth discussion with Schack about my approach and I answered his questions. I guess it is still early in the process of promoting a new paradigm and I don't yet have an "official" reaction. I guess after I'll publish a few papers on this the situation will change.

Regarding CH I will not venture to answer for them; it is best to ask them directly. On MWI my position is that I do not understand it. The root cause is that I have no idea how to make sense of probabilities in this framework.

Best,
Florin

3. Perfectly alright about the snow--I'm well west of you, so I got my snow earlier. I guess, to be more specific about Consistent Histories--I've always been under the impression it runs into the same problems as MWI if all histories are realized, and many supporters seem to insist they're not (for instance, I doubt Lubos would have at any point endorsed CH if it were essentially MWI!)--and of course, you've mentioned it as a variant of Copenhagen. So I guess what I was asking was, is it true that CH is sometimes not a many-worlds theory?
(In any case, such questions should be moot whenever you prove 'unitary collapse'. I'm obviously not an authority on any of this, but your solution sounds very elegant!)
Best,
Eric.

1. I later realized that my reply to you was only 1 day old, but I did not check my blog since last Friday...

I don't know who is defending CH like MWI. Do you have an example? MWI and CH are very different beasts. In CH there is a big Hilbert space and a history is composed of a sequence of subspaces which change when measurements take place. My approach is mathematically different, but conceptually similar with CH. I have many Hilbert spaces each corresponding to a representation and the measurement jumps the representation. If we embed all those Hilbert spaces into one large Hilbert space you get CH.

4. I think it was Jess Reidel (though it's been awhile since I've seen the reference)who was arguing they were not necessarily incompatible and that the CH formalism could be used to improve MWI (which he seemed to be interested in doing). I was also thinking of Wikipedia's tabulation of interpretations (which you adapted to describe the Elliptic Composability Int.); it lists CH as having a non-unique outcome. Of course, there are different types of non-uniqueness; traditional Qbism could be non-unique in a subjective sense, not in a parallel world sense. I guess I was always curious what CH non-uniqueness means, given that they often purport to be 'observer-free Copenhagen'.
I think some of the confusion I've noticed when people talk about CH is the study of 'counterfactual histories' that goes on there. Of course, most of the real consistent historians seem to view said histories as merely a study in logic, yet people outside the interpretation (even some quantum physicists) go on to interpret them as ontologically real.
Eric.

5. Yes, it was Jess Riedel--I looked it up. He thinks the CH formalism could provide a useful description of Everettian branch structure. I should've included this in the last post but wasn't sure I could find it (it was from over 2 years ago, a Physics Stack Exchange in May 2013 in which he was discussing preferred basis).
Eric.

6. Hi Eric,

I could not find the exact reference where Jess Riedel argues for the compatibility of MWI and CH, but in general I understand the mutual attraction between the two approaches. First, MWI may look at consistent histories as an incarnation of their worlds and hope for the respectability of Copenhagen approach to rub on them. In the other direction the Copenhagenist (is this a real word?) who in his heart of hearts are still rooting for a realist interpretation want a secret "get out of jail, free" card to introduce realism back MWI style.

7. Yeah, I think the murkiness I perceived in CH came from hearing realists describe it. As for Riedel, I have no idea if he actually wrote a paper on this (he probably hasn't); what I saw was more of an informal discussion where he referred to similarities between CH and MWI. (Personally, I've always disliked MWI, but until I read your debate of the past month, I never realized just how much of it has to be put in by hand!)
Now, one other question I've always wanted to ask a top physicist: How do you prepare electrons for the double-slit experiment?
Do you break it to them gently? :)!!
Eric.

1. Do you break it to them gently? :) LOL

If you look at an old cathode ray tube TV, in the back they have an electron gun which emit the electrons which hit the screen. The TV deflects the electrons left right and top bottom to produce the TV image pixel by pixel. If you do not deflect the electrons and place a double slit barrier in between the electron source and screen then you get the usual interference pattern. The image is made one electron hit at a time. Then you may ask the question: through which slit did the electron go? If you tag the electrons at each slit by say aligning their spins a certain way such that when you detect them you can tell the "which way" information, then the interference pattern disappears. Welcome to the world of QM.

8. Thanks for the explanation, though I wasn't actually expecting one (being familiar with the double slit experiment). I've just really, really always wanted to tell that joke to a physicist!!! Feel free to use it (if it hasn't already been independently discovered); it can be my own extremely modest contribution to quantum foundations.
I actually hadn't thought about TV sets as quantum measuring devices, though I suppose it really shouldn't have surprised me. As you have pointed out--TV's, transistors, apples--it's a quantum mechanical world we're living in.
Best,
Eric.

1. I got it, humor is sometimes hard to to get across. A good joke...

9. I just remembered--it must've been in the back of my mind this whole time--a place I heard CH compared to MWI was actually here: the 'interview with an anti-quantum zealot' post with Matt Leifer--you know, the April Fools' joke on Lubos? (the two of you were very funny, by the way). Anyway, Matt talked about how CH formalism had been used in Wallace-Saunders Many Worlds and how there doesn't seem to be consensus among the CH leaders about what it means--is it realist; is it neo-Copenhagen? Of course, since Dr. Leifer is a realist himself, that might influence his opinion on CH. But I think that's what was confusing me--though I've always assumed it was a variant of Copenhagen, I've heard several 'interpretations' of CH from a number of different sources. That seems to distinguish it from say, Qbism, or Bohmian Mechanics. For instance, I've always thought the Bohmian approach sounded kind of nuts, but there's certainly only one reading of it. (I'm assuming this is also true of CH, and the muddiness of the waters comes from what you mentioned above, about closet realists). Anyway, I'd be interested to know what you thought about Leifer's classification of Consistent Histories.
Thanks,
Eric.

1. I had to re-read what Matt wrote to refresh my memory. My take on CH is in the style of Omnes. Matt's key contention is spelled out in this sentence: "Therefore, what justifies taking a formula that applies to a necessarily invasive process and saying that it applies even without that process?" As I stated earlier, if you take my approach with many Hilbert spaces and embed them in a single Hilbert space then you get CH. And I do have a good answer to Matt's objection. This will become clear as I am presenting my approach.

But I would like to take Matt's argument and turn it on its head. If measurement is this invasive process, who was there to observe the world before the Earth was formed? QM was not applicable then? There is a famous quote by Aharonov as told by Sidney Coleman:

Aharonov to Sidney: "Tell me, did you father reduced the wavefunctions before you were born?"

10. Thanks for all the different clarifications--hope it didn't seem like I was pestering you; it's just that CH, of all the interpretations, has always been the one that's confused me (you've helped clear a lot of that up). I never got the impression, though, that Matt, being a realist, is a subjective reduction guy; at the time I thought what he was saying about measurement was more like 'unperformed experiments have no results, thus measurement has an affect'--which now that I think about it would also be very odd for him, as a realist. Maybe what he was doing was assuming there is a well-defined underlying state that a measurement can disturb--or maybe he was assuming that the only way non-realism can work is with an observer (neither of which positions, I realize, are held by neo-Copenhagenists).
Thanks again,
Eric.

1. No problem, I love to explain things. Just to prevent you from getting the wrong impression I want to say something more. Indeed unperformed experiments have no results and this is a consequence of the K-S theorem. When one considers that unperformed experiments do have results, this is called counterfactual definiteness (CDF). Local realism in Bell's theorem = locality + CDF. So in a sense CDF = realism. However realism usually refers to how we should regard the wavefunction: as something just as physical as a rock, or as a paper wave, a mathematical tool to compute probabilities.

11. Thanks. I think I had the right impression of realism; I was just speculating about what Matt, as a realist, might've been thinking about measurement. As for my own (grossly uninformed) take on QM, I think I skew non-realist, because it sounds to me like one has to work harder and harder to justify the realist position (as opposed to Copenhagen). But then, I'm a novelist by profession!!
Best,
Eric.

1. Hi Eric,

If you are a novelist then I hope you will excuse my typos, grammatical mistakes, and poor sentences. Funny you say that: "one has to work harder and harder to justify the realist position" There is a theorem by Hardy called "ontological excess baggage theorem". Also to make your head spin, in addition to CDF and realism, there is also psi-epistemic and psi-ontic.

Best,
Florin

12. You have typos. I couldn't construct a Hilbert space if my life depended on it! I try not to judge. :)
Actually, your great knowledge of and enthusiasm for QM always shines through (along with your warmth and good humor), making this blog quite eloquent.
Psi-ontic and Psi-epistemic (and what they mean) are easy enough for the English major to remember--in the case of psi-ontology, just think of Tom Cruise standing atop a couch. As for the theorems, the two that are easiest for me to remember are PBR & the Free Will Theorem. They both cut to the heart of QM and the title of the second seems kind of self-explanatory. Also PBR, for me, evokes the image of a peanut-butter-and-jelly sandwich.
Best,
Eric.