Energy-time uncertainty plays an important role in quantum foundations and
technologies, and it was even discussed by the founders of quantum mechanics.
However, standard approaches (e.g., Robertson's uncertainty relation) do not
apply to energy-time uncertainty because, in general, there is no Hermitian
operator associated with time. Following previous approaches, we quantify time
uncertainty by how well one can read off the time from a quantum clock. We then
use entropy to quantify the information-theoretic distinguishability of the
various time states of the clock. Our main result is an entropic energy-time
uncertainty relation for general time-independent Hamiltonians, stated for both
the discrete-time and continuous-time cases. Our uncertainty relation is
strong, in the sense that it allows for a quantum memory to help reduce the
uncertainty, and this formulation leads us to reinterpret it as a bound on the
relative entropy of asymmetry. Due to the operational relevance of entropy, we
anticipate that our uncertainty relation will have information-processing
applications.Comment: 6 + 9 pages, 2 figure
