Complementarity-based complementarity: the choice of mutually unbiased observables shapes quantum uncertainty relations

Quantum uncertainty relations impose fundamental limits on the joint knowledge that can be acquired from complementary observables: perfect knowledge of a quantum state in one basis implies maximal indetermination in all other mutually unbiased bases (MUBs). Uncertainty relations derived from joint properties of the MUBs are generally assumed to be uniform, irrespective of the specific observables chosen within a set. In this work, we demonstrate instead that the uncertainty relations can depend on the choice of observables. Through both experimental observation and numerical methods, we show that selecting different sets of three MUBs in a 5-dimensional quantum system results in distinct uncertainty bounds, i.e. in varying degrees of complementarity, in terms of both entropy and variance.