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I was studying the multi-parameter quantum metrology these days. And I was confused about the saturability of the quantum Cramer-Rao bound for the multiparameter problem. If all of the generators are commuting with each others, then we could always saturate the quantum Cramer-Rao bound given by the quantum Fisher information matrix. If the generators do not commute with each others, a necessary and sufficient condition for the saturability for pure probe state would be the weak commutativity as pointed out in https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.130504 https://iopscience.iop.org/article/10.1088/0305-4470/35/13/307

Now I was wondering if I only care about one particular parameter out of all the unknown parameters, could I always saturate the minimal variance bounded by the diagonal element of the inverse of quantum Fisher information matrix? For example, if I am looking at the first parameter $\theta_1$, its variance is bounded by $Var(\hat{\theta_1})\geq [F^{-1}]_{11}$, where $F$ is the quantum Fisher information matrix, $[F^{-1}]_{11}$ is the first diagonal element of it inverse. Could I always find the proper measurement to achieve $Var(\hat{\theta_1})= [F^{-1}]_{11}$? Would this depend on the commutativity of the generators?

I found that in Sec. 4.3 of the following paper

https://arxiv.org/pdf/0804.2981.pdf

They mentioned that it is always possible to saturate $Var(\hat{\theta_1})= [F^{-1}]_{11}$ with all other parameters fixed. I am not sure why this is obvious. Is that because if we fix all other parameters, this problem simply degrades into a single parameter one so that the quantum Cramer-Rao bound is always achievable? If all other parameters are unknown as in my above question, then this conclusion won't hold?

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