A Note on the Strong Unitary Uncertainty Relations

Uncertainty relations satisfied by the product of variances of arbitrary n observables have attracted much attention. In a recent article [Phys. Rev. Lett. 120, 230402 (2018)], the authors provided so called strong unitary uncertainty relations for a set of unitary matrices by using the positivity property of the Gram matrix, from which uncertainty relations satisfied by two quantum mechanical observables are obtained. We derive the explicit uncertainty relations satisfied by n quantum mechanical observables from such Gram matrix approach. By some algebraic transformations, we show that these uncertainty relations are just the same as the ones derived from a positive semi-definite Hermitian matrix generated by the mean values of n observables in [Scientific Reports 6, 31192(2016)].