Nonclassical distance in multimode bosonic systems

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.