# Nuisance parameter problem in quantum estimation theory: General formulation and qubit examples

## TL;DR

This paper formulates the nuisance parameter problem in quantum estimation, proposing methods to eliminate their effects and revealing fundamental differences from classical estimation, with detailed qubit examples.

## Contribution

It introduces a quantum formulation of nuisance parameters, proposes a method to handle them, and highlights intrinsic tradeoffs in quantum estimation.

## Key findings

- Nuisance parameters significantly affect quantum estimation accuracy.
- A method to eliminate nuisance parameters in quantum estimation is proposed.
- Tradeoff relations between nuisance and parameters of interest are identified.

## Abstract

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.