# Regensburg 2022 – wissenschaftliches Programm

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# QI: Fachverband Quanteninformation

## QI 14: Quantum Foundations

### QI 14.7: Vortrag

### Freitag, 9. September 2022, 11:30–11:45, H9

**Uncertainty relations with the variance and the quantum Fisher information** — •Géza Tóth^{1,2,3,4} and Florian Fröwis^{5} — ^{1}Theoretical Physics and EHU Quantum Center, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain — ^{2}Donostia International Physics Center (DIPC), E-20080 San Sebastián, Spain — ^{3}IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain — ^{4}Wigner Research Centre for Physics, H-1525 Budapest, Hungary — ^{5}Group of Applied Physics, University of Geneva, CH-1211 Geneva, Switzerland

We present several inequalities related to the Robertson-Schrödinger 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-Schrödinger 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 Fröwis 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 Cramér-Rao bound involving the convex roof of the variance. By considering a concave roof of the bound in the Robertson-Schrödinger uncertainty relation over decompositions to mixed states, we obtain an improvement of the Robertson-Schrödinger 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 in two-mode and two-spin systems.