Timed demolition measurements

Abstract

Picture an experimental scenario where a closed quantum system, evolving through a time-independent Hamiltonian, is subject to a demolition measurement at a chosen time. The Hamiltonian, the measured observables, the initial state of the physical system and even its Hilbert space dimension are unknown; we nonetheless assume a promise or constraint on the energy distribution of the state. In this context we find that, for many natural energy constraints, the set of feasible time series or datasets can be characterized efficiently. Furthermore, under the assumption of a bounded energy spectrum, we prove that there exist ``self-testing'' datasets, whose approximate realization singles out specific Hamiltonians, states and measurement operators. Investigating to what extent the extrapolation of past measurement data is possible in this framework, we identify energy-constrained physical systems for which a non-trivial prediction at time $\tau$ requires a precision in the measurement data superexponential in $\tau$. We also discover two extrapolation phenomena: ``aha! datasets", which drastically increase the predictability of the future statistics of an unrelated measurement; and ``fog banks": fairly simple datasets that exhibit complete unpredictability at some future time $\tau$, but full predictability at a later time $\tau'>\tau$. Besides their relevance for quantum foundations, our results have applications in semi-device independent quantum communication, the simulation of complex quantum systems, and the design of optimal atomic clocks.