$k$-fold unbiased measurements and maximal incompatibility — E8 Intelligence Research

Mutually unbiased bases capture perfect complementarity between two quantum measurements. Extensions beyond pairwise unbiasedness have been proposed, but essentially no non-trivial higher-order constructions are known. We introduce $k$-fold unbiased measurements ($k$-UMs), extending the $k$-fold unbiased bases notion of [arXiv:1706.04446] from rank-one basis measurements to arbitrary-rank projective measurements, and show that this higher-rank setting supports a much richer theory. We develop the notions of algebraic and spectral $k$-UMs and prove that they coincide for rank-one measurements and triples of measurements (3-UMs). We establish strong no-go results for higher-order rank-one constructions and three-outcome 3-UMs, but obtain infinitely many higher-rank triples using Hadamard matrices and Clifford algebras. We then give these structures an exact operational interpretation in terms of measurement incompatibility. For 3-UMs with any number of outcomes, we determine their genera Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com