  We study the quantum entanglement of the spin and orbital degrees of freedom
in the one- dimensional Kugel-Khomskii model, which includes both gapless and
gapped phases, using analytical techniques and exact diagonalization with up to
16 sites. We compute the entanglement entropy, and the entanglement spectra
using a variety of partitions or "cuts" of the Hilbert space, including two
distinct real-space cuts and a momentum-space cut. Our results show the
Kugel-Khomski model possesses a number of new features not previously
encountered in studies of the entanglement spectra. Notably, we find robust
gaps in the entanglement spectra for both gapped and gapless phases with the
orbital partition, and show these are not connected to each other. We observe
the counting of the low-lying entanglement eigenvalues shows that the "virtual
edge" picture which equates the low-energy Hamiltonian of a virtual edge, here
one gapless leg of a two-leg ladder, to the "low-energy" entanglement
Hamiltonian breaks down for this model, even though the equivalence has been
shown to hold for similar cut in a large class of closely related models. In
addition, we show that a momentum space cut in the gapless phase leads to
qualitative differences in the entanglement spectrum when compared with the
same cut in the gapless spin-1/2 Heisenberg spin chain. We emphasize the new
information content in the entanglement spectra compared to the entanglement
entropy, and using quantum entanglement present a refined phase diagram of the
model. Using analytical arguments, exploiting various symmetries of the model,
and applying arguments of adiabatic continuity from two exactly solvable points
of the model, we are also able to prove several results regarding the structure
of the low-lying entanglement eigenvalues.
