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Two observables are called complementary if preparing a physical object in aneigenstate of one of them yields a completely random result in a measurement ofthe other. We investigate small sets of complementary observables that cannotbe extended by yet another complementary observable. We construct explicitexamples of the unextendible sets up to dimension $16$ and conjecture certainsmall sets to be unextendible in higher dimensions. Our constructions providethree complementary measurements, only one observable away from the ultimateminimum of two observables in the set. Almost all of our examples in finitedimension allow to discriminate pure states from some mixed states, and shedlight on the complex topology of the Bloch space of higher-dimensional quantumsystems.

Small sets of complementary observables