- #1
Morbert
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[Moderator's note: Thread spun off from previous discussion as it is more technical.]
Adding to this: the formalism of QM is quite flexible when it comes to applying the projection postulate. We can even apply it before the time of measurement.
Consider a preparation (at time ##t_0##) of a quantum system ##\psi## and detector ##D## that performs a nondestructive, projective measurement of observable ##A = \sum_i a_i |a_i\rangle\langle a_i|## at time ##t_1##. The outcome is perceived by the experimenter at ##t_2##. This scenario implies the unitary evolution $$U(t_1,t_0)|\psi,D\rangle = \sum_i c_i |a_i,D_i\rangle$$ We might consider the collapse to eigenvalue ##a_i## occur at time ##t_2 > t > t_1## E.g. The non-unitary evolution $$C^\dagger_{i}|\psi,D\rangle = U(t_2,t)|a_i\rangle\langle a_i|U(t,t_1)U(t_1,t_0)|\psi,D\rangle$$ with probability $$p(a_i) = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C_i\}$$ or we can consider the collapse to occur before the measurement at ##t_1 > t > t_0## E.g. $$C'^\dagger_{i}|\psi,D\rangle = U(t_2,t_1)U(t_1,t)|a_i\rangle\langle a_i|U(t,t_0)|\psi,D\rangle$$ and the probability will be $$p(a_i) = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C'_i\} = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C_i\}$$ We can liberally construct these "collapsing" non-unitary evolutions so long as it is always the case that $$\mathrm{Re}\left[\mathrm{Tr}\{C^\dagger_i|\psi,D\rangle\langle\psi,D|C_j\}\right] \approx 0, i\neq j$$ (see equation 4.5 and 4.6 in [1])
Of course, if we commit ourselves to a particular interpretation that says the projection postulate is an actual physical process, then additional constraints will come into play. But so long as the above condition is met, the formalism will return consistent and reliable probabilities for whatever and whenever we project.
[1] https://arxiv.org/pdf/gr-qc/9210010.pdf
Adding to this: the formalism of QM is quite flexible when it comes to applying the projection postulate. We can even apply it before the time of measurement.
Consider a preparation (at time ##t_0##) of a quantum system ##\psi## and detector ##D## that performs a nondestructive, projective measurement of observable ##A = \sum_i a_i |a_i\rangle\langle a_i|## at time ##t_1##. The outcome is perceived by the experimenter at ##t_2##. This scenario implies the unitary evolution $$U(t_1,t_0)|\psi,D\rangle = \sum_i c_i |a_i,D_i\rangle$$ We might consider the collapse to eigenvalue ##a_i## occur at time ##t_2 > t > t_1## E.g. The non-unitary evolution $$C^\dagger_{i}|\psi,D\rangle = U(t_2,t)|a_i\rangle\langle a_i|U(t,t_1)U(t_1,t_0)|\psi,D\rangle$$ with probability $$p(a_i) = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C_i\}$$ or we can consider the collapse to occur before the measurement at ##t_1 > t > t_0## E.g. $$C'^\dagger_{i}|\psi,D\rangle = U(t_2,t_1)U(t_1,t)|a_i\rangle\langle a_i|U(t,t_0)|\psi,D\rangle$$ and the probability will be $$p(a_i) = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C'_i\} = \mathrm{Tr}\{|\psi,D\rangle\langle\psi,D|C_i\}$$ We can liberally construct these "collapsing" non-unitary evolutions so long as it is always the case that $$\mathrm{Re}\left[\mathrm{Tr}\{C^\dagger_i|\psi,D\rangle\langle\psi,D|C_j\}\right] \approx 0, i\neq j$$ (see equation 4.5 and 4.6 in [1])
Of course, if we commit ourselves to a particular interpretation that says the projection postulate is an actual physical process, then additional constraints will come into play. But so long as the above condition is met, the formalism will return consistent and reliable probabilities for whatever and whenever we project.
[1] https://arxiv.org/pdf/gr-qc/9210010.pdf
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