Thanks for taking the time to let me know you found this useful! Keeps me motivated.

]]>For my research project, I have recently struggled to understand the condition named **Conditional Completely Positivity** of Lindbladian you mentioned, but due to the lack of knowledge of algebra, I’ve been very confused about it. But luckily now I find your blog, so I just need to figure out the relationship between Choi matrix and superoperator. Thanks so much to your instructive article!

I do address your first sentence when I say “Note that we have not lost the generality of our argument by assuming that the various components of the apparatus end up in pure states…”. However, even if we take the tensor structure to be fixed and we allow for the different subsystems to become entangled, one can check that we get the same conclusion in the end so long as the different subsystems have distinct reduced states conditional on the index , i.e., that for . That is, all that is necessary is that some useful information about the outcome appears in the many subsystems indexed by .

I agree with your second sentence, but it’s important to emphasize that the overlaps between the states must be *exponentially* small in . For any reasonable amplifier, this is extremely tiny.

Even if we assume (4), it only implies that the system is *approximately* orthonormal, with bigger, more amplifying measurement apparatuses leading to smaller values of for .

an established blog. Is it very difficult

to set up your own blog? I’m not very techincal but I can figure things out pretty fast.

I’m thinking about creating my own but I’m not sure where to begin. Do you have

any ideas or suggestions? Cheers ]]>

Thanks much. Typo fixed.

]]>There’s a small typo under Eq. (1) an $=0$ is missing in the assumption to be satisfied. ]]>

I think — as you perhaps do also — that the same physical intuitions that are usually brought to bear to find the difficulties in many worlds, or objective branching, interpretations, will also show that it is very easy to engineer situations in which the assumptions of this approach do not hold. e.g. the Wigner friend scenario with a reversible friend. That particular topic is one I have thought a lot about recently with respect to quantum interpretations (paper in preparation). I think this idea of Weingarten will just not cope with this (supposedly measure-zero) situation.

My response:

Yes, I agree this is unlikely to be a literal measure-zero type situation, as Weingarten claims. (I’m in the process of writing another post about some of the problem’s with his proposal, of which this is one.) But I think it’s plausible something similar to Weingarten’s proposal could work. The aspect of this proposal that is particularly promising to me, and that may help evade the situations you raise, is that it is trying to directly bake in *irreversibility* (without modifying quantum mechanics).

There are (at least) three key issues:

1. Is it actually feasible to reverse Wigner’s friend in the real universe? It seems likely to me that the resources necessary for this grow exponentially in some measure of the relative complexity between the two versions of the friend, and thus the chance that any Wigner anywhere in the universe accomplishes this task is negligible. One could still object that FAPP isn’t good enough for a fundamental theory, but I would still say massive progress had been made in understanding quantum mechanics without measurement.

2. Will we accept a theory that decides whether a candidate branching event “really happened” at a certain time based on whether those branches recombine in the (perhaps distant) future? It seems distasteful…but hard to reject out of hand given our limitations as physical agents who have access only to our physically encoded memories.

3. How the heck do we think about the post-heat-death universe when FAPP irreversibility breaks down?

]]>Yep, I agree with that.

]]>One other thought about your footnote “a”, regarding degenerate observables. Instead of saying that a generic (possibly degenerate) measurement apparatus represents a commuting subalgebra, can’t you instead say that it represents a particular decomposition of the Hilbert space into an (internal) direct sum of orthogonal subspaces? Each measurement outcome corresponds to a different subspace, and the nondegenerate case corresponds to a maximal decomposition into a direct sum of only one-dimensional subspaces.

This seems (to me) conceptually simpler than thinking in terms of commuting algebras, although that’s perhaps a matter of opinion. And more importantly, it seems closer to the spirit of the rest of your discussion, where your point is that a particular measurement apparatus is better thought of as being associated with an intrinsic property of the Hilbert space itself (e.g. a particular direct-sum decomposition or choice of basis for the space) than with specific operators on that Hilbert space.

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