10:16 AM
@MoreAnonymous hi, sorry, busy week. Thing is, I don't really understand the need of a "way out". The scope of the result is as general as it gets. It holds at any point in time. You can apply it immediately after a measurement, the only caveat is that the post-measurement result will be different then the state before measurement, and therefore the quantities (averages and variances) will be different
I think the problem is that you are somehow thinking of it as a statement about dynamics, while it is a statement about the state of the system at any point of time. The only part of the derivation that might not hold in all cases is to interpret $\Delta t$ as josh is doing in the answer (see also my comments to his answer), but if you forget this particular interpretation of what $\Delta t$ means physically, the rest holds
in other words, I agree with the overall sentiment of needing to ponder about your different ways out. It absolutely makes no different how you describe the measurement process, as long as you stick with the prescription that a state with amplitudes $c_k$ (somehow) corresponds to observed probabilities $|c_k|^2$. There is not a single interpretation of QM that disagrees with this (nor that could be one)
note that you can also describe formally the "measurement process" as a map $\mathcal E$ defined as $\mathcal E(\rho)=\sum_k \mathrm{tr}(\rho P_k)P_k$ with $P_k$ projectors onto the measurement basis. In this notation, the process of performing a measurement is described as simply applying $\mathcal E$ to the state.
10:41 AM
Would it be okay to mention "note that you can also describe formally the "measurement process" as a map $\mathcal E$ defined as $\mathcal E(\rho)=\sum_k \mathrm{tr}(\rho P_k)P_k$ with $P_k$ projectors onto the measurement basis. In this notation, the process of performing a measurement is described as simply applying $\mathcal E$ to the state.
The uncertainty formula still holds at all times (with the additional caveat that you are renouncing the purity of the states, being $\mathcal E$ non-unitary, so josh's interpretation of $\Delta t$ is even harder to justify in this formalism)" in you…
The uncertainty formula still holds at all times (with the additional caveat that you are renouncing the purity of the states, being $\mathcal E$ non-unitary, so josh's interpretation of $\Delta t$ is even harder to justify in this formalism)" in you…
@glS I'm really confused about the "particular interpretation" of $\delta t$. I'm glad that your asking josh :) Also someone told be to be wary of physicists intuition of the time-energy uncertainty principle and cited this paper: arxiv.org/abs/1610.09619 However, to construct a quantum circuit to disprove an uncertainty principle involved is beyond my skill level :P
@glS I'll probably accept your answer. However, is it possible to edit an answer after a bounty is awarded? I'm not sure :/ Also please conside adding this point: chat.stackexchange.com/transcript/message/51758133#51758133
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