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- TL;DR Summary
- An experimenter can use QM to compute probabilities for possible experimental outcomes (events) so long as the experimenter chooses a measurement basis. Decoherence will select a preferred basis for the experiment, but does not explain why events associated with that basis occur vs events associated with other bases. I believe the resolution to this issue can be found in the subtle difference between "complementary" and "mutually exclusive".

This thread picks up a discussion between myself and @A. Neumaier in another thread. In particular, this comment:

First some basic background. If we consider the usual presentation of a destructive measurement of an observable ##A = \sum_i a_i|a_i\rangle\langle a_i|## of system ##s## by detector ##D##, we have the evolution $$U(t)|\psi_0,D_\Omega\rangle = \sum_ic_iU(t)|a_i,D_\Omega\rangle = \sum_ic_i|D_i\rangle$$where ##|\psi_0\rangle## and ##|D_\Omega\rangle## are initial states of the system of measurement and detector respectively, and ##|D_i\rangle## are orthonormal states on ##\mathcal{H}_s\otimes\mathcal{H}_D##.

We can then build an event algebra from a sample space of outcomes built in turn from a projective decomposition of the identity $$I_{s+D} = \sum_i|D_i\rangle\langle D_i|$$The probability for the occurrence of outcome ##D_i## is $$p(D_i) = \mathrm{tr}(|D_i\rangle\langle D_i|U(t)|\psi_0,D_\Omega\rangle\langle \psi_0,D_\Omega|U^\dagger(t))$$But QM doesn't insist upon the decomposition. Take for example some complementary observable ##B=\sum_ib_i|b_i\rangle\langle b_i|## that the device doesn't measure. We can write the same evolution as $$U(t)|\psi_0,D_\Omega\rangle = \sum_ic_iU(t)|b_i,D_\Omega\rangle = \sum_ic_i|D'_i\rangle$$We then consider the projective decomposition

$$I_{s+D}=\sum_i|D'_i \rangle \langle D'_i|$$and QM will return probabilities for these "outcomes" occurring. Interactions with the environment will explain why this decomposition is not a decomposition into pointer states of the detector, but it doesn't explain why events constructed from this decomposition don't occur.

Normally this is where a Heisenberg cut is introduced. The device is classical, and the projective decomposition ##\sum_i|D_i\rangle\langle D_i|## reproduces the classical characteristics of the device that the experimenter reads off as data. I think this is fine, as I am not aware of any instance where such a procedure is ambiguous. But there is an alternative approach. The outcomes ##\{D'_i\}## are complementary to ##\{D_i\}##, but complementary is subtly different from mutually exclusive. An example of mutually exclusive outcomes would be ##D_i## and ##D_j## where ##j\neq i##, an example

Reference: https://www.physicsforums.com/threads/nobody-understands-quantum-physics.1049370/page-8A. Neumaier said:What does it mean in quantum terms for a detector to produce an event? To give your POVM a meaning you need to refer to the classical description of the experiment done to identify the projectors with real events. This is what I mean with classical.

First some basic background. If we consider the usual presentation of a destructive measurement of an observable ##A = \sum_i a_i|a_i\rangle\langle a_i|## of system ##s## by detector ##D##, we have the evolution $$U(t)|\psi_0,D_\Omega\rangle = \sum_ic_iU(t)|a_i,D_\Omega\rangle = \sum_ic_i|D_i\rangle$$where ##|\psi_0\rangle## and ##|D_\Omega\rangle## are initial states of the system of measurement and detector respectively, and ##|D_i\rangle## are orthonormal states on ##\mathcal{H}_s\otimes\mathcal{H}_D##.

We can then build an event algebra from a sample space of outcomes built in turn from a projective decomposition of the identity $$I_{s+D} = \sum_i|D_i\rangle\langle D_i|$$The probability for the occurrence of outcome ##D_i## is $$p(D_i) = \mathrm{tr}(|D_i\rangle\langle D_i|U(t)|\psi_0,D_\Omega\rangle\langle \psi_0,D_\Omega|U^\dagger(t))$$But QM doesn't insist upon the decomposition. Take for example some complementary observable ##B=\sum_ib_i|b_i\rangle\langle b_i|## that the device doesn't measure. We can write the same evolution as $$U(t)|\psi_0,D_\Omega\rangle = \sum_ic_iU(t)|b_i,D_\Omega\rangle = \sum_ic_i|D'_i\rangle$$We then consider the projective decomposition

$$I_{s+D}=\sum_i|D'_i \rangle \langle D'_i|$$and QM will return probabilities for these "outcomes" occurring. Interactions with the environment will explain why this decomposition is not a decomposition into pointer states of the detector, but it doesn't explain why events constructed from this decomposition don't occur.

Normally this is where a Heisenberg cut is introduced. The device is classical, and the projective decomposition ##\sum_i|D_i\rangle\langle D_i|## reproduces the classical characteristics of the device that the experimenter reads off as data. I think this is fine, as I am not aware of any instance where such a procedure is ambiguous. But there is an alternative approach. The outcomes ##\{D'_i\}## are complementary to ##\{D_i\}##, but complementary is subtly different from mutually exclusive. An example of mutually exclusive outcomes would be ##D_i## and ##D_j## where ##j\neq i##, an example

*wouldn't*be ##D_i## and ##D'_j##. An observer making note of one of the events ##\{D_i\}## doesn't preclude any of the events ##\{D'_i\}##. Instead we say a**description**that includes an observer making note of one of the events ##\{D_i\}## preculdes a**description**that includes an event from ##\{D'_i\}##. QM does not select one description as more correct than the other. Instead, the experimenter selects one description as more useful than the other: namely, the description that includes the pointer observable they are capable of reading.
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