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 other. I 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.