Higher-order uncertainty bounds for mixed states

TL;DR

This paper develops higher-order uncertainty bounds for mixed quantum states by embedding density matrices into a Hilbert space, utilizing skew information and quantum skew moments to refine uncertainty estimates.

Contribution

It introduces a novel Hilbert-space embedding approach to derive higher-order uncertainty bounds for mixed states, extending existing bounds with explicit formulas for quantum skew moments.

Findings

Derived higher-order uncertainty bounds using skew information.

Closed-form expressions for higher-order quantum skew moments.

Enhanced understanding of uncertainty in mixed quantum states.

Abstract

Uncertainty lower bounds for parameter estimations associated with a unitary family of mixed-state density matrices are obtained by embedding the space of density matrices in the Hilbert space of square-root density matrices. In the Hilbert-space setup the measure of uncertainty is given by the skew information of the second kind, while the uncertainty lower bound is given by the Wigner-Yanase skew information associated with the conjugate observable. Higher-order corrections to the uncertainty lower bound are determined by higher-order quantum skew moments; expressions for these moments are worked out in closed form.