Experimental entropic uncertainty relations in dimensions three to five

We provide experimental validation of tight entropic uncertainty relations for the Shannon entropies of observables with mutually unbiased eigenstates in high dimensions. In particular, we address the cases of dimensions <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>d</a:mi> <a:mo>=</a:mo> <a:mn>3</a:mn> </a:mrow> </a:math> , 4, and 5 and consider from 2 to <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>d</b:mi> <b:mo>+</b:mo> <b:mn>1</b:mn> </b:mrow> </b:math> mutually unbiased bases. The experiment is based on pulsed frequency bins measured with a multioutput quantum pulse gate, which can perform projective measurements on a complete high-dimensional basis in the time-frequency domain. Our results fit the theoretical predictions: the bound on the sum of the entropies is never violated and is saturated by the states that minimize the uncertainty relations.