Measuring Energy, Estimating Hamiltonians, and the Time-Energy Uncertainty relation, Y. Aharonov, S. Massar and S. Popescu,
Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine what is the Hamiltonian. We show that in general this requires a time Dt which obeys the uncertainty relation D t DH > 1 where DH is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring energy of an unknown quantum state. It has previously been shown that if the Hamiltonian is known, then the energy can in principle be measured with an arbitrarily large precision in an arbitrarily short time. On the other hand we show that if the Hamiltonian is not known then the energy measurement necessarily takes a minimum time Dt which obeys the uncertainty relation D t DE > 1 where DE is the precision of the energy measurement., Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.