Nuisance parameter problem in quantum estimation theory: General formulation 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 for the quantum-state estimation theory, then to propose a method of how to eliminate the nuisance parameters to obtain an estimation error bound for the parameters of interest. We also develop useful methods of dealing with the nuisance parameter problem in the quantum case and reveal the significant difference from the classical case. In particular, we clarify an intrinsic tradeoff relation to estimate the nuisance parameters and parameters of interest. The general qubit model is examined in detail to emphasize that we cannot ignore the effects of the nuisance parameters in general. Several examples in qubit systems are worked out to illustrate our findings.