  We develop a non-equilibrium Green's function formalism to study magnonic
spin transport through a strongly anisotropic ferromagnetic insulator contacted
by metallic leads. We model the ferromagnetic insulator as a finite-sized
one-dimensional spin chain, with metallic contacts at the first and last sites
that inject and detect spin in the form of magnons. In the presence of
anisotropy, these ferromagnetic magnons become elliptically polarized, and spin
conservation is broken. We show that this gives rise to a novel parasitic spin
conductance, which becomes dominant at high anisotropy. Moreover, the spin
state of the ferromagnet becomes squeezed in the high-anisotropy regime. We
show that the squeezing may be globally reduced by the application of a local
spin bias.
