  It is remarkable that Heisenberg's position-momentum uncertainty relation
leads to the existence of a maximal acceleration for a physical particle in the
context of a geometric reformulation of quantum mechanics. It is also known
that the maximal acceleration of a quantum particle is related to the magnitude
of the speed of transportation in projective Hilbert space. In this paper,
inspired by the study of geometric aspects of quantum evolution by means of the
notions of curvature and torsion, we derive an upper bound for the rate of
change of the speed of transportation in an arbitrary finite-dimensional
projective Hilbert space. The evolution of the physical system being in a pure
quantum state is assumed to be governed by an arbitrary time-varying Hermitian
Hamiltonian operator. Our derivation, in analogy to the inequalities obtained
by L. D. Landau in the theory of fluctuations by means of general commutation
relations of quantum-mechanical origin, relies upon a generalization of
Heisenberg's uncertainty relation. We show that the acceleration squared of a
quantum evolution in projective space is upper bounded by the variance of the
temporal rate of change of the Hamiltonian operator. Moreover, focusing for
illustrative purposes on the lower-dimensional case of a single spin qubit
immersed in an arbitrarily time-varying magnetic field, we discuss the optimal
geometric configuration of the magnetic field that yields maximal acceleration
along with vanishing curvature and unit geodesic efficiency in projective
Hilbert space. Finally, we comment on the consequences that our upper bound
imposes on the limit at which one can perform fast manipulations of quantum
systems to mitigate dissipative effects and/or obtain a target state in a
shorter time.
