  We provide an experimentally measurable local gauge $U(1)$ invariant
Fubini-Study (FS) metric for mixed states. Like the FS metric for pure states,
it also captures only the quantum part of the uncertainty in the evolution
Hamiltonian. We show that this satisfies the quantum Cramer-Rao bound and thus
arrive at a more general and measurable bound. Upon imposing the monotonicity
condition, it reduces to the square-root derivative quantum Fisher Information.
We show that on the Fisher information metric space dynamical phase is zero. A
relation between square root derivative and logarithmic derivative is
formulated such that both give the same Fisher information. We generalize the
Fubini-Study metric for mixed states further and arrive at a set of
Fubini-Study metric---called $\alpha$ metric. This newly defined $\alpha$
metric also satisfies the Cramer-Rao bound. Again by imposing the monotonicity
condition on this metric, we derive the monotone $\alpha$ metric. It reduces to
the Fisher information metric for $\alpha=1$.
