Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices

TL;DR

This paper develops new uncertainty relations using convex and concave decompositions of density matrices, improving bounds on quantum variances and linking them to quantum metrology and entanglement criteria.

Contribution

It introduces novel uncertainty bounds based on convex and concave roofs, enhancing the Robertson-Schrödinger relation and connecting it to quantum metrological usefulness and entanglement detection.

Findings

Improved uncertainty relations via convex and concave decompositions.

Stronger bounds on the metrological usefulness of bipartite states.

Violation of entanglement criteria implies enhanced metrological performance.

Abstract

We present several inequalities related to the Robertson-Schr\"odinger uncertainty relation. In all these inequalities, we consider a decomposition of the density matrix into a mixture of states, and use the fact that the Robertson-Schr\"odinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in Fr\"owis et al. [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relation. We present a formulation of the Cram\'er-Rao bound involving the convex roof of the variance. By considering a concave roof of the bound in the Robertson-Schr\"odinger uncertainty relation over decompositions to mixed states, we obtain an improvement of the Robertson-Schr\"odinger uncertainty relation. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of the canonical position and momentum operators for two-mode continuous variable systems. We show that the violation of well-known entanglement conditions in these systems discussed in Duan et al., [Phys. Rev. Lett. 84, 2722 (2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] implies that the state is more useful metrologically than certain relevant subsets of separable states. We present similar results concerning entanglement conditions with angular momentum operators for spin systems.