On the existence of unbiased resilient estimators in discrete quantum systems

Cram\'er-Rao constitutes a crucial lower bound for the mean squared error of an estimator in frequentist parameter estimation, albeit paradoxically demanding highly accurate prior knowledge of the parameter to be estimated. Indeed, this information is needed to construct the optimal unbiased estimator, which is highly dependent on the parameter. Conversely, Bhattacharyya bounds result in a more resilient estimation about prior accuracy by imposing additional constraints on the estimator. Initially, we conduct a quantitative comparison of the performance between Cram\'er-Rao and Bhattacharyya bounds when faced with less-than-ideal prior knowledge of the parameter. Furthermore, we demonstrate that the $n^{th}$order classical and quantum Bhattacharyya bounds cannot be computed -- given the absence of estimators satisfying the constraints -- under specific conditions tied to the dimension $m$ of the discrete system. Intriguingly, for a system with the same dimension $m$, the maximum non-trivial order $n$ is $m-1$ in the classical case, while in the quantum realm, it extends to $m(m+1)/2-1$. Consequently, for a given system dimension, one can construct estimators in quantum systems that exhibit increased robustness to prior ignorance.