An upper bound on the number of compatible parameters in simultaneous quantum estimation

TL;DR

This paper investigates the maximum number of parameters that can be simultaneously estimated in quantum systems while satisfying conditions for optimal estimation, using linear algebraic methods to derive bounds for specific quantum states.

Contribution

It introduces an upper bound on the number of compatible parameters in quantum estimation based on the algebraic structure of the system, with explicit calculations for qubit states.

Findings

Derived an algebraic upper bound for parameter compatibility.

Calculated bounds for single qubit and two-qubit X-states.

Provided insights into the structure of quantum systems for estimation.

Abstract

Simultaneous estimation of multiple parameters is required in many practical applications. A lower bound on the variance of simultaneous estimation is given by the quantum Fisher information matrix. This lower bound is, however, not necessarily achievable. There exists a necessary and sufficient condition for its achievability. It is unknown how many parameters can be estimated while satisfying this condition. In this paper, we analyse an upper bound on the number of such parameters through linear-algebraic techniques. This upper bound depends on the algebraic structure of the quantum system used as a probe. We explicitly calculate this bound for two quantum systems: single qubit and two-qubit X-states.