@glS I think your right. Sorry it took some time to assimilate

@Mithrandir24601 Is this question written clearly?

I'm afraid my communication skills might be getting in the way

@MoreAnonymous I just skimmed it, but I get what you're asking, yeah

@MoreAnonymous I think it's mostly fine (though there's a few points that could help some clarification). Though I've got to say, that looks like the kind of question that fits more on physics.SE than here

when you start using position/momentum observables and throw in some relativity, you're probably going to get better served in physics I think

I do not understand the relation between the observables/possible outcomes you write and the "detector clicks". What is the detector measuring?

also, generally speaking, the "is the above correct?" question format is never great IMO. It's almost always better to reframe a question to ask about something, rather than to ask to check one own's calculation.

Yea Maybe after correcting it I'll transfer to physics SE

How do I vote to migrate?

@glS

I think I'll try Physics SE

@MoreAnonymous so in practice you are doing some measurement of the state, but the time at which the measurement is performed is uncertain?

I'd probably write in geneneral that the probability of getting some outcome $b$ would be $\int dt \, p(t) \langle \mu(b),\Phi_t(\rho)\rangle$, where $\rho$ is the initial state, $\Phi_t$ the channel describing time evolution (you can think of it as just a way to write $e^{-iHt}\rho e^{iHt}$ here), $\mu(b)$ is the measurement operator corresponding to the outcome (e.g. the eigenvector of the observable corresponding to the outcome for you), and $\langle A,B\rangle\equiv {\rm Tr}(A^\dagger B$)

and $p(t)$ is the probability density of the measurement taking place at time $t$

@MoreAnonymous in general you just raise a flag and say you want to migrate (I don't think we have migration paths for physics set). Or you can just tell me here and I'll do it

@glS Yes

And $p(t)$ is the probability distribution right?

@glS In this

How would you do uniform probability?

@MoreAnonymous well, probability density if you consider a continuum of possible outcomes

@MoreAnonymous just $p(t)=1/I$ with $I$ the length of the time interval under consideration

@glS Yes but that fails in special relativity where we have another observer with velocity $v$

Because the time-interval will be different

sure. I've got no idea what's the proper way to generalise this to a relativistic framework.

@glS Same

Hence the question

@glS Raised a flag :P

@glS I'm not 100% sure this is right, which is why I think it's actually an interesting question - I think that what's actually happening is that, because you're not sure of the measurement time, you're not sure of the frame in the sense that you probably need to integrate over all possible unitaries/channels corresponding to the uncertainty in measurement time

So in practise, this would cause some kind of decoherence effect

In principle, making this relativistic is a matter of making the formalism relativistic, but then it gets a bit more tricky to actually do

@Mithrandir24601 in the lab I doubt one actually knows the exact time they perform the measurement (else they would probably be using irrational numbers). Do they in practice just take the most probable time and use the Born Rule?

@MoreAnonymous In the lab, it's the other way round - what you get (assuming a lot of repeats/a large sample) are noisy probabilities

But detectors do have a property called 'jitter', which is what you're talking about, yeah

@Mithrandir24601 hmmm ... interesting.

@Mithrandir24601 didn't know about this

Funnily enough, if your detector jitter is too big, then you've essentially got a mixed state, because your photons (I'm assuming photons, because that's what I'm familiar with) are essentially too far apart, which is why I'm reasonably sure that what I said above about causing decoherence is likely the way to go about it

(OK, not strictly true about 'if jitter is too big', rather 'if a large jitter causes the photons to be detected at completely different times' is more accurate)

@Mithrandir24601 Do you think Physics SE is the right place for this then?

@MoreAnonymous Physics SE is definitely a suitable place, but I dunno if here is the 'wrong' place or not

@Mithrandir24601 Ah okay. I would have thought this falls under quantum information

@Mithrandir24601 what do you mean with "frame" here?

@MoreAnonymous I mean, I think it ultimately does, but it's not explicity written into the question :P

@glS This sort of thing: arxiv.org/abs/quant-ph/0610030

@MoreAnonymous to be clear, I didn't say this is off-topic here. Lines are quite blurred for these things. Just that when these issues of relativity are involved, I think the people at physics tend to have more experience with that. Or at least that's my impression

@Mithrandir24601 mh.. so essentially, relativistic frames? So, if you consider relativity issues, I'm with you that that expression is likely incorrect. But in a nonrelativistic setting I don't quite see it

@glS Ah no worries :)

@MoreAnonymous done

@glS No, not at all relativistic frames! I mean, I might not be right anyway, but I think the key difference lies between 'probability of [event] happening between time t and t+dt' and 'the event happened but we don't know when'

@MoreAnonymous wow, the retagging was quick. For a second I thought there was some kind of automatic retagging taking place when migrating

@glS Haha ... No I wish that was the case :P

It's effectively the continuous versions of 'we know that if [event] happens at t1, the probability is p1' and 'we don't know if the event happened at t1 or t2, so it's a mixed sum of the state at t1 with prob p1 and the state at t2 with prob p2'

@Mithrandir24601 uhm. Isn't that pretty much what I wrote though? The probability is a mixture of the probabilities corresponding to evolving and then measuring at different times

@glS I dunno, it might end up being the same, but I'm not 100% sure

(also, if they're different, I don't know which would actually be right)

like, the fact that $t$ is time isn't really important here. You can think of this as saying that the state is a mixture of the results of different evolutions. Here $\int dt p(t) \Phi_t(\rho)$, and you measure on this state

@glS Yeah, it might end up being the same thing, but the difference is that you're integrating over some parameter instead of integrating over the unitaries with that parameter, so I don't actually know if one becomes the other or not

Hmm, yeah, I think it is the same?

@Mithrandir24601 not sure. I'm not familiar with the context of that paper. When you say to integrate over unitaries, that sounds like you are saying "I don't know on which direction I'm measuring", assuming the unitaries connect different measurement bases. Otherwise, the $\Phi_t$ in the above is also unitary if the time-evolution is, so that also counts as "integrating over some unitaries"

@glS Yeah, my problems are that it's been a few years since I sat down and actually did any calculations with this sort of thing and that I remember it being very easy to mess up, so looking at it, I don't see why they would or should be any different, but as soon as I say that, I'll be wrong :P

@Mithrandir24601 From schrodinger's cat to Schrodinger's answer :P