  We use complementary optical spectroscopy methods to directly measure the
lowest crystal-field energies of the rare-earth quantum magnet
LiY$_{1-x}$Ho$_{x}$F$_{4}$, including their hyperfine splittings, with more
than 10 times higher resolution than previous work. We are able to observe
energy level splittings due to the $^6\mathrm{Li}$ and $^7\mathrm{Li}$
isotopes, as well as non-equidistantly spaced hyperfine transitions originating
from dipolar and quadrupolar hyperfine interactions. We provide refined crystal
field parameters and extract the dipolar and quadrupolar hyperfine constants
${A_J=0.02703\pm0.00003}$ $\textrm{cm}^{-1}$ and ${B= 0.04 \pm0.01}$
$\textrm{cm}^{-1}$, respectively. Thereupon we determine all crystal-field
energy levels and magnetic moments of the $^5I_8$ ground state manifold,
including the (non-linear) hyperfine corrections. The latter match the
measurement-based estimates. The scale of the non-linear hyperfine corrections
sets an upper bound for the inhomogeneous line widths that would still allow
for unique addressing of a selected hyperfine transition. e.g. for quantum
information applications. Additionally, we establish the far-infrared,
low-temperature refractive index of LiY$_{1-x}$Ho$_{x}$F$_{4}$.
