  We propose a photonics-based, continuous-variable (CV) form of remote
entanglement utilizing strictly second-order nonlinear optical interactions
that does not require the implementation of a state-projective measurement
(i.e. remote entanglement without conditioning). This scheme makes use of two
separate down-converters, wherein the corresponding nonlinear crystals are
driven by strong classical fields as prescribed by the parametric
approximation, as well as a fully quantum mechanical model of nondegenerate
second harmonic generation (SHG) whose evolution is described by the trilinear
Hamiltonian of the form $\hat{H}_{\text{shg}} =
i\hbar\kappa\big(\hat{a}\hat{b}\hat{c}^{\dagger} -
\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{c}\big)$. By driving the SHG process
with the signal modes of the two down-converters, we show entanglement
formation between the generated second-harmonic mode (SH-mode) and the
non-interacting joint-idler subsystem without the need for any state-reductive
measurements on the interacting modes.
