I would need to think about this more to be sure, but from my first read it seems as if your idea can be mapped to decoherence.
The maths you are using looks a bit different than what I am used to, but I am somewhat confident that your uncalibrated experiment is equivalent to a suitably defined decohering quantum channel. The amplitudes that you are calculating would be transition amplitudes from the prepared initial state to the measured final state (Denoting the initial state as |i>, the final state as |f> and the time evolution operator as U, your amplitudes would be <f|U|i> in the notation of the linked wikipedia article). The go-to method for describing statistical mixtures over quantum states or transition amplitudes is to switch from wave-functions and operators to density matrices and quantum channels (physics lectures about open quantum systems or quantum computing will introduce these concepts) - they should be equivalent to (more accurately: a super-set of) your averaging over s and t for the uncalibrated experiment, as one can just define a time evolution operator for fixed values of s and t and then get the corresponding channel by taking the probability weighted integral (compare the Operator-sum representation in the Wikipedia article) to arrive at the corresponding channel.
Regarding all the interesting aspects regarding the Born rule, I cannot contribute at the moment.
I was also surprised to learn about this formalism at my university, as it wasn’t mentioned in either the introductory nor the advanced lecture on QM, but turns out to be very helpful for understanding how/when classical mechanics can be a good approximation in a QM universe.
I would need to think about this more to be sure, but from my first read it seems as if your idea can be mapped to decoherence.
The maths you are using looks a bit different than what I am used to, but I am somewhat confident that your uncalibrated experiment is equivalent to a suitably defined decohering quantum channel. The amplitudes that you are calculating would be transition amplitudes from the prepared initial state to the measured final state (Denoting the initial state as |i>, the final state as |f> and the time evolution operator as U, your amplitudes would be <f|U|i> in the notation of the linked wikipedia article). The go-to method for describing statistical mixtures over quantum states or transition amplitudes is to switch from wave-functions and operators to density matrices and quantum channels (physics lectures about open quantum systems or quantum computing will introduce these concepts) - they should be equivalent to (more accurately: a super-set of) your averaging over s and t for the uncalibrated experiment, as one can just define a time evolution operator for fixed values of s and t and then get the corresponding channel by taking the probability weighted integral (compare the Operator-sum representation in the Wikipedia article) to arrive at the corresponding channel.
Regarding all the interesting aspects regarding the Born rule, I cannot contribute at the moment.
Thanks for all the pointers! I was, somewhat embarrassingly, unaware of the existence of that whole field.
I’m glad if this was helpful.
I was also surprised to learn about this formalism at my university, as it wasn’t mentioned in either the introductory nor the advanced lecture on QM, but turns out to be very helpful for understanding how/when classical mechanics can be a good approximation in a QM universe.