  The question of how fast a quantum state can evolve has attracted a
considerable attention in connection with quantum measurement, metrology, and
information processing. Since only orthogonal states can be unambiguously
distinguished, a transition from a state to an orthogonal one can be taken as
the elementary step of a computational process. Therefore, such a transition
can be interpreted as the operation of "flipping a qubit", and the number of
orthogonal states visited by the system per unit time can be viewed as the
maximum rate of operation.
  A lower bound on the orthogonalization time, based on the energy spread
DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average
energy E, was established by Margolus and Levitin. The bounds coincide, and can
be exactly attained by certain initial states if DeltaE=E; however, the problem
remained open of what the situation is otherwise.
  Here we consider the unified bound that takes into account both DeltaE and E.
We prove that there exist no initial states that saturate the bound if DeltaE
is not equal to E. However, the bound remains tight: for any given values of
DeltaE and E, there exists a one-parameter family of initial states that can
approach the bound arbitrarily close when the parameter approaches its limit
value. The relation between the largest energy level, the average energy, and
the orthogonalization time is also discussed. These results establish the
fundamental quantum limit on the rate of operation of any
information-processing system.

Quantum Physics