We consider all pure or mixed states of a quantum many-body system which exhibit the same, arbitrary but fixed measurement outcome statistics for several commuting observables. Taking those states as initial conditions, which are then propagated by the pertinent Schrodinger or von Neumann equation up to some later time point, and invoking a few additional, fairly weak and realistic assumptions, we show that most of them still entail very similar expectation values for any given observable. This so-called dynamical typicality property thus corroborates the widespread observation that a few macroscopic features are sufficient to ensure the reproducibility of experimental measurements despite many unknown and uncontrollable microscopic details of the system. We also discuss and exemplify the usefulness of our general analytical result as a powerful numerical tool.
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