Fisher information under decoherence in Bloch representation

TL;DR

This paper analyzes the behavior of quantum Fisher information under decoherence for single qubits and qudits, providing explicit formulas and numerical comparisons to understand how decoherence affects quantum parameter estimation.

Contribution

It derives explicit formulas for quantum Fisher information in the Bloch representation and compares analytical and numerical results under various decoherence channels.

Findings

Decoherence channels cause significant reduction in quantum Fisher information.

Analytical formulas match numerical results under the rotating-wave approximation.

Collisional dephasing notably decreases phase estimation precision.

Abstract

The dynamics of two variants of quantum Fisher information under decoherence are investigated from a geometrical point of view. We first derive the explicit formulas of these two quantities for a single qubit in terms of the Bloch vector. Moreover, we obtain analytical results for them under three different decoherence channels, which are expressed as affine transformation matrices. Using the hierarchy equation method, we numerically study the dynamics of both the two information in a dissipative model and compare the numerical results with the analytical ones obtained by applying the rotating-wave approximation. We further express the two information quantities in terms of the Bloch vector for a qudit, by expanding the density matrix and Hermitian operators in a common set of generators of the Lie algebra $\mathfrak{su}(d)$. By calculating the dynamical quantum Fisher information, we find that the collisional dephasing significantly diminishes the precision of phase parameter with the Ramsey interferometry.