  State estimation is a classical problem in quantum information. In
optimization of estimation scheme, to find a lower bound to the error of the
estimator is a very important step. So far, all the proposed tractable lower
bounds use derivative of density matrix. However, sometimes, we are interested
in quantities with singularity, e.g. concurrence etc. In the paper, lower
bounds to a Mean Square Error (MSE) of an estimator are derived for a quantum
estimation problem without smoothness assumptions. Our main idea is to replace
the derivative by difference, as is done in classical estimation theory. We
applied the inequalities to several examples, and derived optimal estimator for
some of them.
