I'm re-reading your answer and I'm wondering why doesnt "effective description of a unitary process" increase the scope of applicability of this when we use "Heisenberg's equation to obtain the result

I'm guessing in the last paragraph of the answer means we have to talk about different systems. I'm getting the impression from here: "time $t+ϵ$ (and by this I mean that you repeat the measurement without measuring at $t$)" In light of the previous comment why can't I talk about the same system?

@MoreAnonymous because even if you have an "effective description of a unitary process", the only thing that changes is how you describe what happens in between having $|\psi\rangle$ and having $|\psi_i\rangle$. However, the calculation will apply unchanged when you only consider the post-measurement state (here $|\psi_i\rangle$).The way you will explain obtaining the measurement results will be different, but not the measurement results themselves ,which is what this calculation uses.

@MoreAnonymous I'm still talking about the same system, but here you have to be careful in how you measure things. What I mean is that you need to measure $\langle \Omega\rangle(t)$, which is obtained by evolving unitarily $|\psi\rangle$ from $\tau=t_0$ (whatever $t_0$ is here) to $\tau=t$, and then you need to measure $\langle\Omega\rangle(t+\epsilon)$, which you do evolving unitarily $|\psi\rangle$ from $\tau=t_0$ to $\tau=t+\epsilon$. However, the calculation does not apply if you first collapse the system at $\tau=t$, then evolve unitarily from $t$ to $t+\epsilon$ and then measure again

@MoreAnonymous or if you like, yes you are considering different copies of the same system, which is what I mean when I talk of the need of repeated measurements to actually observe this. Each system can only be made to collapse once before being irreversibly changed, so to measure expectation values and variances of operators you need to prepare the same system $|\psi\rangle$ at time $\tau=t_0$ multiple times and have each collapse at $\tau=t$ or $\tau=t+\epsilon$, in order to collect enough statistics to compute the quantities involved in the relation

yes this is what I'm referring "if you like, yes you are considering different copies of the same system" to when I said different systems ... The derivation above makes no reference to unitarity (or non-unitarity) of the measurement ... So my question can be rephrased as "if I assume the measurement is unitarity what does this derivation mean then? (where I'm talking about the same physical system with $2$ measurements)"

... Not in the case where you restrict the "the calculation does not apply if you first collapse the system at 𝜏=𝑡, then evolve unitarily from 𝑡 to 𝑡+𝜖 and then measure again" (I mean there is no reason to restrict if everything is unitary)

@MoreAnonymous I'm not sure what you mean. You cannot assume that measurement is unitary, because that is not true. The relationship between pre- and post-measurement state is always non-unitary, and everything else is unitary. This is the only thing we use in the derivation (not caring about how the process $|\psi\rangle\mapsto|\psi_i\rangle$ actually takes place). Being able to explain what happens in between might give you more insight, but won't change the result.

"if one adheres to the view that the non-unitarity of measurement is an effective description of a unitary process in which some of the information is neglected, à la Zurek)." then one can talk about successive measurements even if the price will be applying the uncertainty principle = "physical system (we're interested in)" + "experimental apparatus" (?) @glS

the last line may be unclear so: even if the price of applying the uncertainty principle is applying it to "physical system (we're interested in)" + "experimental apparatus" (?) @glS

@MoreAnonymous I don't really understand what you are saying here

*2. However, "way 4" seems to circumvent that by saying even in the situation in 1. may be applicable since (it may be plausible) that the measurement is a "unitarity of measurement is an effective description of a unitary process in which some of the information is neglected, à la Zurek."