  Mutual unbiasedness of the eigenstates of phase-space operators-such as
position and momentum, or their standard coarse grained versions-exists only in
the limiting case of infinite squeezing. In [Phys. Rev. Lett. 120, 040403
(2018)] it was shown that mutual unbiasedness can be recovered for periodic
coarse grainings of these two operators. Here we investigate mutual
unbiasedness of coarse-grained measurements for more than two phase-space
variables. We show that mutual unbiasedness can be recovered between periodic
coarse graining of any two non-parallel phase-space operators. We illustrate
these results through optics experiments using the fractional Fourier transform
to prepare and measure mutually unbiased phase-space variables. The differences
between two and three mutually unbiased measurements is discussed. Our results
contribute to bridging the gap between continuous and discrete quantum
mechanics and could be useful in quantum information protocols.
