  Pairs of entangled photons are important for applications in quantum
nanophotonics, where their theoretical description must accommodate their
bipartite character. Such character is shared at the other end of the intensity
range by, for example, the two degenerate instances of the pump field involved
in second-harmonic generation. Describing the interaction of nanophotonic
structures with bipartite states of light is, regardless of their intensity, a
challenge with important technological applications. Here, we develop a
theoretical framework for studying the interaction of material structures with
bipartite states of light. The basic element is the symmetrized tensor product
space of two copies of an electromagnetic Hilbert space. One of the benefits
inherited from the single Hilbert space is that consequences of material
symmetries are readily deduced. We derive selection rules for second-order
non-linear processes in objects with rotational and/or mirror symmetries. We
numerically verify several selection rules by combining quantum-chemical
calculations with a Maxwell solver to simulate second-harmonic generation in
two different MoS$_2$ clusters. The computationally convenient scattering
matrix method is also extended to the tensor product space when the response of
the object to one part of the state is independent of the other. For such a
case, we obtain the relation between the scattering matrix in the single
Hilbert space and the scattering matrix for bipartite states. Such a separable
case is relevant for the entanglement evolution of biphoton states interacting
with nanostructures. We discuss some possibilities for accommodating the
computations of non-linear effects in the framework, for example, through a
non-separable scattering operator, where the response of the object to one part
of the state depends on the other part.
