Quantum systems usually travel a multitude of different paths when evolvingthrough time from an initial to a final state. In general, the possible pathswill depend on the future and past boundary conditions, as well as the system'sdynamics. We present a gedanken experiment where a single system apparentlyfollows mutually exclusive paths simultaneously, each with probability one,depending on which measurement was performed. This experiment involves themeasurement of observables that do not correspond to Hermitian operators. Ourmain result is a scheme for measuring these operators. The scheme is based onthe erasure protocol [Phys. Rev. Lett. 116, 070404 (2016), arXiv:1409.1575] andallows a wide range of sequential measurements at both the weak and stronglimits. At the weak limit the back action of the measurement cannot be used toaccount for the surprising behavior and the resulting weak values provide aconsistent yet strange account of the system's past.
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