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.



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