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Morbert
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- TL;DR Summary
- While time asymmetry of a measurement process restricts retrodiction (the inference of previous measurement outcomes, counterfactual or otherwise, based on current measurement outcomes), it does not rule them out. It only rules out retrodiction asymptotically into the past, and we can readily construct a time-symmetric theory of measurement and identify where and when retrodictive inferences are valid. The meaning of these inferences are interpretation-dependent.
Take a simple case: A system is prepared in state ##\rho_i## at time ##t_0##, and a projective measurement is performed at time ##t_2## with an outcome ##b##. We can retrodict a projective measurement outcome ##a## at time ##t_1## where ##t_0<t_1<t_2##$$p(a|b) = \frac{\mathrm{Tr}\left[\Pi_b(t_2)\Pi_a(t_1)\rho_i\Pi_a(t_1)\right]}{\sum_{a'}\mathrm{Tr}\left[\Pi_b(t_2)\Pi_{a'}(t_1)\rho_i\Pi_{a'}(t_1)\right]}$$More generally, we can retrodict a sequence of measurement outcomes ##\alpha = i,j,\dots m## $$p(\alpha|b) = \frac{\mathrm{Tr}\left[\Pi_b(t_2)C^\dagger_\alpha\rho_iC_\alpha\right]}{\sum_{\alpha'}\mathrm{Tr}\left[\Pi_b(t_2)C^\dagger_{\alpha'}\rho_iC_{\alpha'}\right]}$$where ##C_\alpha = \Pi_i\Pi_j\dots\Pi_m##. It can be helpful to interpret the ##b## measurement as part of the preparation process: a post-selection of a final characteristic, such that e.g. the ensemble of experimental runs to be interrogated is characterised by ##\rho_i,\rho_f## and $$p(\alpha;\rho_i,\rho_f) = \mathrm{Tr}\left[\rho_fC^\dagger_\alpha\rho_iC_\alpha\right]$$This expression is time-symmetric. We can reverse the sequence of observations and preparation procedures and the expression won't change.
These computations sometimes let us seemingly violate the Heisenberg uncertainty principle. Take for example the case where a particle is prepared by preselecting for spin-x = ##\uparrow## at ##t_0## and post-selecting for spin-z = ##\uparrow## at ##t_2##. We can see that a measurement for either spin-x or spin-z at ##t_1## will have yielded the outcome ##\uparrow## with probability $$p(x_\uparrow;\rho_i,\rho_f) = p(z_\uparrow;\rho_i,\rho_f) = 1$$ Certainty about spin-x and spin-z at the same time?? These probabilities are reliable, because we aren't explicitly constructing some outcome ##c## where ##\Pi_c = \Pi_{x_\uparrow}(t_1)\Pi_{z_\uparrow}(t_1)## which has no Hilbert subspace. The complementary measurements can never be performed on the same system.
The meaning of these retrodictions are interpretation-dependent. Ensemble interpretations are the most straightforward. If you present me with the ensemble in the last paragraph, I can know that for each member of the ensemble, if you have measured spin-x at ##t_1## you will have observed ##\uparrow## and if you have measured spin-z you will have observed ##\uparrow##.
These computations sometimes let us seemingly violate the Heisenberg uncertainty principle. Take for example the case where a particle is prepared by preselecting for spin-x = ##\uparrow## at ##t_0## and post-selecting for spin-z = ##\uparrow## at ##t_2##. We can see that a measurement for either spin-x or spin-z at ##t_1## will have yielded the outcome ##\uparrow## with probability $$p(x_\uparrow;\rho_i,\rho_f) = p(z_\uparrow;\rho_i,\rho_f) = 1$$ Certainty about spin-x and spin-z at the same time?? These probabilities are reliable, because we aren't explicitly constructing some outcome ##c## where ##\Pi_c = \Pi_{x_\uparrow}(t_1)\Pi_{z_\uparrow}(t_1)## which has no Hilbert subspace. The complementary measurements can never be performed on the same system.
The meaning of these retrodictions are interpretation-dependent. Ensemble interpretations are the most straightforward. If you present me with the ensemble in the last paragraph, I can know that for each member of the ensemble, if you have measured spin-x at ##t_1## you will have observed ##\uparrow## and if you have measured spin-z you will have observed ##\uparrow##.
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