Connection between coarse-grained MUM behavior and the N=6 MUB problem

Investigate whether the even-versus-odd dimensional behavior of mutually unbiased measurements (MUMs) in coarse-grained continuous-variable systems is connected to the unresolved question of the existence and number of mutually unbiased bases in the discrete six-dimensional Hilbert space.

Background

In continuous-variable systems, at most three mutually unbiased measurements exist, while in coarse-grained scenarios the structure depends on parity: even dimensions behave like the continuous case, and only odd dimensions allow for more MUMs.

Whether this parity-dependent behavior relates to the special difficulties encountered in dimension six for discrete MUBs is an open question with potential implications for understanding the discrete–continuous interplay in quantum measurement theory.