What we learned from proving a quantum postulate redundant

[@ARBB1]

Great work Gabriele, very insightful.

[@andreantoine8005]

~7:20, what does it mean if two states a and b, are not compatible? How does that imply the opposite of independence? Would it mean for two incompatible states, that they are "dependent" as opposed to "independent"?

[@gcarcassi]

"Not independent" is not the same as "dependent". Take two electrons: if I prepare the first one in a particular state (say the ground state of a hydrogen atom), I cannot prepare the second one in the same state since only one electron can occupy a particular state. Thus they are not independent, the preparation of the first tells me something about the preparation of the second. However, I can choose any other state. Thus the preparation of the first does not fully constrain the second. The preparation of the second would depend on the preparation of the first only if one case was possible. The two electron are "almost independent", but neither independent or "dependent". Does it help?

[@andreantoine8005]

@@gcarcassi a concrete example is better, but I guess I have a hard time saying that electron B is not dependent on electron A since if I were to describe the state I would have to include the information about state A., maybe some time in contemplation will elucidate the issue.

[@Achrononmaster]

@22:00 for universality I think you need more than that the map _m_ is bilinear, don't you? Since there are counterexamples. In a non-classical GPT (vector space of positive cones) there are two distinct tensor products, a ⊗_min and a ⊗_max. QM is such a GPT. These refer to different ways of completing the algebraic tensor product of C∗-algebras or von Neumann algebras. The different completions are required because operator algebras must preserve additional structures like the *-operation and norm conditions, which are not considerations in the purely classical algebraic setting. It probably does not mess up your result too much(?), but might be something you want to consider. The problem (possibly?) is that (A ⊗_min B) = (A⊗_max B) only in the classical GPT case, but your discussion is all about the non-classical case where (A ⊗_min B) ⊂ (A⊗_max B), so what gives? If you ignore entangled cones then you are ok. (Since you are then classical.) But the point about foundational axioms of QM is that you _don't want to ignore entanglement!_ So it might be that you are missing something about "dropping the tensor product axiom" since in a C*-algebra or GPT setting you do need something to distinguish separable composite systems from entangled. Otherwise all you have is axioms for a classical measurement theory.

[@gcarcassi]

I do not follow. The argument starts with two physical requirements: a composite system must describe independent preparation (what I think you call separate composite systems... which are really some particular states); a composite system must the smallest quantum system to describe those independent preparations. We do assume the composite system is a quantum system. You seem to want a different claim: that quantum composition is the only "GPT space" that has some property. If that's the case, that is not at all our goal.

[@GeoffryGifari]

I was having this thought about entanglement (related to irreducibility): the EPR thought experiment argues that it makes no sense that two measurements arbitrarily far apart can be correlated without a hidden variable, and the common answer to that (which keeps quantum mechanics as the valid) explains that it doesn't matter, because no *information* is carried faster than light anyway. Is there a way to make this notion clear? in which physical systems do information actually travel, and how fast? working this out rigorously sounds like your area

[@gcarcassi]

The argument goes like this: an entangled system is irreducible, meaning that the internal dynamics between the two system is inaccessible. Suppose you could use the entanglement to transfer information from one side to the other. Then it means you can manipulate one system with a known effect on the other system. But then it means that you know something of the dynamics between the two systems. Therefore the internal dynamics of the two systems is not inaccessible. Contradiction! Therefore you can't use an entangled system to transfer information from one side to the other. Makes sense? To do all of this rigorously, there are still bits of math I am missing... But the intuition works (i.e. you can make qualitative prediction)

[@GeoffryGifari]

@@gcarcassi hmm there are some things i get from this comment: 1. if information transfer can be done, there is a cause and effect relationship (system B is that way because we made system A this way, before) 2. to have information transfer, observer must be able to extract what happens "along the way" between two systems (locally?), implying reducibility in space now maybe.... as long as we have a large, reducible system its possible to define flow of information between the subsystems?

[@GeoffryGifari]

oh... for a large, reducible system extended in space, each subsystem must also be coupled to one another for information to flow.... i don't think a bunch of noninteracting parts can transfer information between each other

[@gcarcassi]

@@GeoffryGifari Exactly! :-D And being able to describe the dynamics between parts is exactly being able to describe how the information about each parts flows. And the state spaces of classical and quantum mechanics both allow us to define information entropy in a coordinate invariant way (such that this flow of information is something that is observer independent). All these pieces fit together really well once you look at them in the right way...

[@geoffrygifari3377]

i got lost on that commentary where the tensor product on a Hilbert space does not exist according to category theory. does that mean there is an official tensor product in math, can be examined through category theory, and the physicists' "tensor product" which we uses, and works, on quantum mechanics but not exactly fits with the math one already invented by mathematicians?

[@gcarcassi]

Right, it's a very technical thing. You essentially have two definitions. The categorical tensor product is the one that satisfies the universal property. The standard tensor product defined on vector spaces coincides with the categorical tensor product. No problem there. On Hilbert spaces, it is as you say: the standard tensor product exist (it is not just the physicists'... also mathematicians use it) but it is not the categorical tensor product because that doesn't even exist. Now, if you don't care about category theory, you are not going to care about this anyway. Some people do care about category theory, probably in relation to algebraic geometry... So we got that objection. At the beginning, I didn't even understand what they were talking about! :-)

[@Achrononmaster]

@@gcarcassi Bad language framing? The vector space tensor product is invisible to category theory, it's not that it "doesn't exist".

[@gcarcassi]

@@Achrononmaster IIRC (it was years ago) the claim was that there is no tensor product in the category of Hilbert spaces. Meaning that there is no object that satisfies the definition. Never heard the claim of "invisibility. :-) What do you mean by that?

[@geoffrygifari3377]

hmm so this is how assumptions of physics research is done... one thing i noticed: removing one postulate (the tensor product) by mathematically defining a composite system, don't you think that definition of composite system now becomes a postulate, replacing the tensor product? like... is it even possible to do quantum mechanics of many particles (real world situation) without examining preparation independence first etc

[@gcarcassi]

If by "postulate" you simply mean the starting point of the formal system, yes. But that's not what I said a postulate was, and the role they play in quantum mechanics, so: no. The definition of a composite system is not a postulate... is just a definition. It's a physical object, it is clear what we mean physically by it, we do not need to "interpret it". The issue of whether one needs preparation independence, my take is that you need it for defining even a single system. This is something we are looking more in detail and it a general assumption on any physical system, not just a quantum one.

[@davidhand9721]

Fun. Now try to make the Born rule and the Schrodinger equation compatible. They are both postulated to describe time evolution, yet they are totally contradictory. I'm not talking about assigning probabilities to outcomes via the projective space, I'm talking about a superposition of states evolving into a single state instantaneously. This can only arise through the Schrodinger equation in the most trivial, contrived examples; a discontinuous transition, such as what occurs during a measurement as predicted by the Born rule, is simply not compatible. One or both of those postulates must be incorrect, or we are forced to introduce contrivances to differentiate which of the two rules is applicable in order to make QM a complete theory. Copenhagen, for example, gives us the constructs of a measurement and observers, but fails to explain why those entities are exempt from QM's continuous evolution or how they extend that property to normally quantum systems. Everett covers all objects, including measurement devices and observers, and explains why the states of those objects should each observe the Born rule while the Born rule is not physical. However, while mathematically more logical and clean on paper, Everett asks us to suspend the judgment of our intuition to an extreme degree. I would love to see a serious, technical drill down on the conditions it would take to reconcile the two time evolution postulates. From what I've heard, it might involve non-linear terms added to the Schrodinger equation, or some kind of differential form of the Born rule. I think both would be detectable, in theory.

[@gcarcassi]

We take a different approach. I posted a "state of the project" video last month, and I go through some of the details. Roughly, unitary evolution is a series of infinitesimal collapses. I'll be hopefully writing these things down this year... in both paper and video form.

[@Achrononmaster]

They are not contradictory. If you assume Schrödinger holds _for all time_ then they are contradictory. But why would you assume so? It'd be tantamount to presuming the Schrödinger equation is a Law of Nature, but clearly it is not. It is a tool for accounting for a spacetime cobordism from a preparation to a record, or alternatively a steady state system, nothing more.