We study how well we can retrodict results of measurements made on a quantum system if we can make measurements on its final state. We know what measurements were made, but not their results. An initial examination shows that we can gain anywhere from no information to perfect information about the results of previous measurements, depending on the measurements and the initial state of the system. The case of two two-outcome measurements, the second of which is a projective measurement, is examined in some detail. We then look at a model of a qubit interferometer in which measurements are made in order to determine the path the qubit followed. The measurement made on the final state of the qubit depends on the information about previous measurement results that we are trying to determine. One can attempt to find the result of just one of the measurements or all of them or to find a measurement sequence that was not realized. We study all three possibilities.
A basic time asymmetry is said to exist in the conventional quantum theory of measurement because the initial state, determined by the initial measurement, suffers a reduction of the wave packet when a succeeding measurement is made. It is shown, completely within the conventional framework, that the sense of this time asymmetry can be reversed and a retroductive formalism introduced which is based upon the fact that only the final state is known. Moreover, if both the initial and final states of a set of measurements are fixed, a completely time symmetric formalism can be constructed for the time interval between these given states. The selection of the predictive scheme can be made with the help of macroscopic considerations and consequently it appears that the basic time asymmetry is not a consequence of conventional quantum physics including measurement.
Both in quantum theory and in general probabilistic theories, measurements with
n outcomes are modelled as
n-tuples of \emph{effects} summing up to the unit effect. Why is this the case, and can this assumption be meaningfully relaxed? Here we develop \emph{generalized measurement theories (GMTs)} as a mathematical framework for physical theories that is complementary to general probabilistic theories, and where this kind of question can be made precise and answered. We then give a definition of \emph{probabilistic state} on a GMT, prove that measurements are made of effects in every GMT in which the probabilistic states separate the measurements, and also argue that this separation condition is physically well-motivated. Finally, we also discuss when a GMT should be considered classical and characterize GMTs corresponding to Boolean algebras as those that are strongly classical and projective.