  We revisit the notion of nonclassical distance of states of bosonic quantum
systems introduced in [M. Hillery, Phys. Rev. A 35, 725 (1987)] in a general
multimode setting. After reviewing its definition, we establish some of its
general properties. We obtain new upper and lower bounds on the nonclassical
distance in terms of the supremum of the Husimi function of the state.
Considering several examples, we elucidate the cases for which our lower bound
is tight, which include the multimode number states and a class of multimode
N00N states. The latter provide examples of states of definite photon number $n
\geq 2$ whose nonclassical distance can be made arbitrarily close to the upper
limit of $1$ by increasing the number of modes. We show that the nonclassical
distance of the even and odd Schr\"odinger cat states is bounded away from
unity regardless of how macroscopic the superpositions are, and that the
nonclassical distance is not necessarily monotonically increasing with respect
to macroscopicity.
