Nuisance parameter problem in quantum estimation theory: tradeoff relation and qubit examples

In this paper, we analyze quantum-state estimation problems when some of the parameters are of no interest to be estimated. In classical statistics, these irrelevant parameters are called nuisance parameters, and this problem is of great importance in many practical applications of statistics. However, little is known regarding the effects of nuisance parameters in quantum-state estimation problems. The main contribution of this paper is first to formulate the nuisance parameter problem in the quantum-state estimation theory for the finite sample size case. We then derive the precision limit for the parameters of interest based on the concept of locally unbiasedness for the parameters of interest. We also propose a method of how to eliminate the nuisance parameters to obtain an upper bound for estimation error of the parameters of interest. Based on the proposed method, we clarify an intrinsic tradeoff relation to estimate the nuisance parameters and parameters of interest. The general qubit models are examined in detail to emphasize that we cannot ignore the effects of the nuisance parameters in general. Several examples are worked out to illustrate our findings. (paper)