Uncertainty relations for triples of observables and the experimental demonstrations

Uncertainty relations are of profound significance in quantum mechanics and quantum information theory. The well-known Heisenberg-Robertson uncertainty relation presents the constraints on the spread of measurement outcomes caused by the non-commutability of a pair of observables. In this article, we study the uncertainty relation of triple observables to explore the relationship between the standard deviations and the commutators of the observables. We derive and tighten the multiplicative form and weighted summation form uncertainty relations, which are found to be dependent not only on the commutation relations of each pair of the observables but also on a newly defined commutator in terms of all the three observables. We experimentally test the uncertainty relations in a linear optical setup. The experimental and numerical results agree well and show that the uncertainty relations derived by us successfully present tight lower bounds in the cases of high-dimensional observables and the cases of mixed states. Our method of deriving the uncertainty relation can be extended to more than three observables.