Quantum Fisher information matrix for unitary processes: closed relation for $SU(2)$

Abstract

Quantum Fisher information plays a central role in the field of quantum metrology. In this paper we study the problem of quantum Fisher information of unitary processes. Associated to each parameter $\theta_i$ of unitary process $U(\boldsymbol{\theta})$, there exists a unique Hermitian matrix $M_{\theta_i}=i(U^\dagger\partial_{\theta_i} U)$. Except for some simple cases, such as when the parameter under estimation is an overall multiplicative factor in the Hamiltonian, calculation of these matrices is not an easy task to treat even for estimating a single parameter of qubit systems. Using the Bloch vector $\boldsymbol{m}_{\theta_i}$, corresponding to each matrix $M_{\theta_i}$, we find a closed relation for the quantum Fisher information matrix of the $SU(2)$ processes for an arbitrary number of estimation parameters and an arbitrary initial state. We extend our results and present an explicit relation for each vector $\boldsymbol{m}_{\theta_i}$ for a general Hamiltonian with arbitrary parametrization. We illustrate our results by obtaining the quantum Fisher information matrix of the so-called angle-axis parameters of a general $SU(2)$ process. Using a linear transformation between two different parameter spaces of a unitary process, we provide a way to move from quantum Fisher information of a unitary process in a given parametrization to the one of the other parametrization. Knowing this linear transformation enables one to calculate the quantum Fisher information of a composite unitary process, i.e. a unitary process resulted from successive action of some simple unitary processes. We apply this method for a spin-half system and obtain the quantum Fisher matrix of the coset parameters in terms of the one of the angle-axis parameters.