Explicit formula for the Holevo bound for two-parameter qubit estimation problem

TL;DR

This paper derives an explicit formula for the Holevo bound in two-parameter qubit state estimation, clarifying when it matches classical bounds and how it varies with different weighting, aiding quantum statistical analysis.

Contribution

It provides a new explicit formula for the Holevo bound in two-parameter qubit estimation, linking it to quantum Fisher information and revealing its behavior across models.

Findings

The formula depends only on SLD and RLD Fisher information and a weight matrix.

The Holevo bound coincides with SLD or RLD Cramer-Rao bounds in specific models.

The structure of the Holevo bound varies smoothly with the weight matrix.

Abstract

The main contribution of this paper is to derive an explicit expression for the fundamental precision bound, the Holevo bound, for estimating any two-parameter family of qubit mixed-states in terms of quantum versions of Fisher information. The obtained formula depends solely on the symmetric logarithmic derivative (SLD), the right logarithmic derivative (RLD) Fisher information, and a given weight matrix. This result immediately provides necessary and sufficient conditions for the following two important classes of quantum statistical models; the Holevo bound coincides with the SLD Cramer-Rao bound and it does with the RLD Cramer-Rao bound. One of the important results of this paper is that a general model other than these two special cases exhibits an unexpected property: The structure of the Holevo bound changes smoothly when the weight matrix varies. In particular, it always coincides with the RLD Cramer-Rao bound for a certain choice of the weight matrix. Several examples illustrate these findings.