  The Bhatia-Davis theorem provides a useful upper bound for the variance in
mathematics, and in quantum mechanics, the variance of a Hamiltonian is
naturally connected to the quantum speed limit due to the Mandelstam-Tamm
bound. Inspired by this connection, we construct a formula, referred to as the
Bhatia-Davis formula, for the characterization of the quantum speed limit in
the Bloch representation. We first prove that the Bhatia-Davis formula is an
upper bound for a recently proposed operational definition of the quantum speed
limit, which means it can be used to reveal the closeness between the timescale
of certain chosen states to the systematic minimum timescale. In the case of
the largest target angle, the Bhatia-Davis formula is proved to be a valid
lower bound for the evolution time to reach the target when the energy
structure is symmetric. Regarding few-level systems, it is also proved to be a
valid lower bound for any state in two-level systems with any target, and for
most mixed states with large target angles in equally spaced three-level
systems.
