Quantum estimation for non-differentiable models

State estimation is a classical problem in quantum information. In optimization of an 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 a derivative of the density matrix. However, sometimes, we are interested in quantities with singularity, e.g. concurrence etc. In this paper, lower bounds to a mean square error 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 an optimal estimator for some of them.