From Quantum Probabilities to Quantum Amplitudes

The task of reconstructing the system’s state from the measurements results, known as the Pauli problem, usually requires repetition of two successive steps. Preparation in an initial state to be determined is followed by an accurate measurement of one of the several chosen operators in order to provide the necessary “Pauli data”. We consider a similar yet more general problem of recovering Feynman’s transition (path) amplitudes from the results of at least three consecutive measurements. The three-step histories of a pre- and post-selected quantum system are subjected to a type of interference not available to their two-step counterparts. We show that this interference can be exploited, and if the intermediate measurement is “fuzzy”, the path amplitudes can be successfully recovered. The simplest case of a two-level system is analysed in detail. The “weak measurement” limit and the usefulness of the path amplitudes are also discussed.