# How does the quantum Fisher information provide bounds for the estimation of output states?

Assume you have some quantum process $$Q$$ (e.g. quantum state tomography) that intakes initialised states $$\rho_{i}$$, $$i=1,\ldots,n$$ and gives some output $$\rho'_i$$. $$\rho_1 \to Q \to \rho'_1 \\ \rho_3 \to Q \to \rho'_2 \\ \vdots \\ \rho_n \to Q \to \rho'_n$$ Then, assume we can estimate these outputs using some method and we produce estimates $$\hat \rho_i$$. $$\hat \rho_1 \text{ estimates } \rho'_1 \\ \hat \rho_2 \text{ estimates } \rho'_2 \\ \vdots \\ \hat \rho_n \text{ estimates } \rho'_n \\$$ How can I use the notion of the Quantum Fisher Information for finding bounds on how well these output states $$\rho'_i$$ can be estimated by my methods? E.g. if the density matrix formed by the initial states corresponds to that of a qubit ($$n=4)$$: $$\rho = \sum_{i=1}^{4} \lambda_i |i\rangle \langle i | = \sum_{i=1}^{4} \lambda_i \rho_i.$$ In this case, do I need to compute $$F_Q[\rho,\hat\rho]$$? If so, how is this done? System can be closed for simplicity so that evolution is unitary.