  A quantum system can be described and characterized by at least two different
concepts, namely, its physical and informational properties. Here, we
explicitly connect these two concepts, by equating the time-energy cost which
is the product of the largest energy of a Hamiltonian of quantum dynamics and
the evolution time, and the entanglement fidelity which is the informational
difference between an input state and the corresponding output state produced
by a quantum channel characterized by the Hamiltonian. Specifically, the
worst-case entanglement fidelity between the input and output states is exactly
the cosine of the channel's time-energy cost (except when the fidelity is
zero). The exactness of our relation makes a strong statement about the
intimate connection between information and physics. Our exact result may also
be regarded as a time-energy uncertainty relation for the fastest state that
achieves a certain fidelity.
