Weight-dependent and weight-independent measures of quantum incompatibility in multiparameter estimation

TL;DR

This paper analyzes quantum incompatibility in multiparameter estimation, introducing weight-dependent and independent measures to better approximate the fundamental precision bounds, with implications for quantum metrology.

Contribution

It introduces a weight-dependent measure $T[W]$ that tightens bounds on quantum incompatibility, improving estimation precision analysis in multiparameter quantum metrology.

Findings

The weight-dependent bound $C_{T}[W]$ often outperforms the weight-independent $R$-bound.

Analytical methods for computing the Holevo bound $C_{H}[W]$ using $C_{T}[W]$ are developed.

Higher-dimensional systems show greater benefits from the weight-dependent measure.

Abstract

Multiparameter quantum estimation faces a fundamental challenge due to the inherent incompatibility of optimal measurements for different parameters, a direct consequence of quantum non-commutativity. This incompatibility is quantified by the gap between the symmetric logarithmic derivative (SLD) quantum Cram\'er-Rao bound, which is not always attainable, and the asymptotically achievable Holevo bound. This work provides a comprehensive analysis of this gap by introducing and contrasting two scalar measures. The first is the weight-independent quantumness measure $R$, which captures the intrinsic incompatibility of the estimation model. The second is a tighter, weight-dependent measure $T[W]$ which explicitly incorporates the cost matrix $W$ assigning relative importance to different parameters. We establish a hierarchy of bounds based on these two measures and derive necessary and sufficient conditions for their saturation. Through analytical and numerical studies of tunable qubit and qutrit models with SU(2) unitary encoding, we demonstrate that the weight-dependent bound $C_{T}[W]$ often provides a significantly tighter approximation to the Holevo bound $C_{H}[W]$ than the $R$-dependent bound, especially in higher-dimensional systems. We also develop an approach based on $C_{T}[W]$ to compute the Holevo bound $C_{H}[W]$ analytically. Our results highlight the critical role of the weight matrix's structure in determining the precision limits of multiparameter quantum metrology.