  Polarization transfer to a bound proton in polarized electron knock-out
reactions, $\mathrm{A}(\vec{e},e^{\prime}\vec{p})$, is a powerful tool to look
for in-medium modification of the bound proton. It requires comparison to
calculations which consider the many-body effects accompanying the quasi-free
process. We report here measured components $P_x^{\prime}$, $P_z^{\prime}$, and
their ratio $P_x^{\prime}/P_z^{\prime}$, of polarization transfer to protons
bound in $^{40}\mathrm{Ca}$, which is described well by the shell model and for
which reliable calculations are available. While the calculations capture the
essence of the data, our statistical precision allows us to observe deviations
which cannot be explained by simple scaling, including by varying the proton
electromagnetic form factor ratio $G_E/G_M$. We further explore the deviations
of the ratio of the polarization transfer components from that of a free
proton, $(P_x^{\prime}/P_z^{\prime})_{\rm A}/(P_x^{\prime}/P_z^{\prime})_{\rm
H}$, and its dependence on the bound-proton virtuality.
