Sprecher
Beschreibung
Mutually unbiased bases (MUBs) are the prototypical example of a pair of incompatible/complementary measurements. I have tried in the past to extend the MUB to more measurements, beyond the pairwise MUB condition, but the resulting n-fold unbiased bases (n-UBs) condition turned out to be too rigid to satisfy [1]. In this talk, I will introduce a higher-rank extension of n-UBs: similarly to how dropping the rank-1 condition on MUBs leads to mutually unbiased measurements (MUMs) [2], we obtain the notion of n-fold unbiased measurements (n-UMs) by dropping the rank-1 condition on n-UBs [3]. We obtain non-trivial constructions, in particular a family of 3-UMs with 4 outcomes based on Hadamard matrices and Clifford algebras.
Under some measures of incompatibility, MUBs are the most incompatible pairs of measurements in dimension d [4], and MUMs are the most incompatible pairs of n-outcome measurements [2]. Using the sum-of-squares hierarchy for finding maximally incompatible measurements recently developed by one of us [5], we numerically prove that 4-outcome 3-UMs are the most incompatible triples of 4-outcome measurements. We conjecture that n-UMs could govern maximal incompatibility in many other cases and we prove various structural results on n-UMs, expecting to find use for them across quantum information.
[1] Farkas, arXiv:1706.04446 (2017)
[2] Tavakoli, Farkas, Rosset, Bancal, Kaniewski, Science Advances 7(7), eabc3847 (2021)
[3] Designolle, Farkas, arXiv:2609.20728 (2026)
[4] Designolle, Farkas, Kaniewski, New Journal of Physics 21 113053 (2019)
[5] Designolle, New Journal of Physics 28, 064509 (2026)