Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement

In quantum optics a pure state is considered classical, relative to the statistics of photodetection, if and only if it is a coherent state. A different and newer notion of nonclassicality is based on modal entanglement. One example that relates these two notions is the Hong-Ou-Mandel effect, where modal entanglement is generated by a beamsplitter from the nonclassical photon-number state $\vert 1 \rangle \otimes\vert 1\rangle$. This suggests the beamsplitter or, more generally, linear-optical networks as a mediator of the two notions of nonclassicality. We show the following: Given a nonclassical pure product state input to an $N$-port linear-optical network, the output is almost always mode entangled; the only exception is a product of squeezed states, all with the same squeezing strength, input to a network that does not mix the squeezed and anti-squeezed quadratures. Our work thus gives a necessary and sufficient condition for a linear network to generate modal entanglement from pure product inputs, a result that is of immediate relevance to the boson sampling problem.