  Beam splitters are routinely used for generating entanglement between modes
in the optical and microwave domains, requiring input states that are not
convex combinations of coherent states. This leads to the ability to generate
entanglement at a beam splitter as a notion of quantumness. A similar, yet
distinct, notion of quantumness is the amount of entanglement generated by
two-mode squeezers (i.e., four-wave mixers). We show that squeezed-vacuum
states, paradoxically, both minimize and maximize these notions of quantumness,
with the crucial resolution of the paradox hinging upon the relative phases
between the input states and the devices. Our notion of quantumness is
intrinsically related to eigenvalue equations involving creation and
annihilation operators, governed by a set of inequalities that leads to
generalized cat and squeezed-vacuum states.
