  The proximity effect from a spin-triplet $p_x$-wave superconductor to a dirty
normal-metal has been shown to result in various unusual electromagnetic
properties, reflecting a cooperative relation between topologically protected
zero-energy quasiparticles and odd-frequency Cooper pairs. However, because of
a lack of candidate materials for spin-triplet $p_x$-wave superconductors,
observing this effect has been difficult. In this paper, we demonstrate that
the anomalous proximity effect, which is essentially equivalent to that of a
spin-triplet $p_x$-wave superconductor, can occur in a semiconductor/high-$T_c$
cuprate superconductor hybrid device in which two potentials coexist: a
spin-singlet $d$-wave pair potential and a spin--orbit coupling potential
sustaining the persistent spin-helix state. As a result, we propose an
alternative and promising route to observe the anomalous proximity effect
related to the profound nature of topologically protected quasiparticles and
odd-frequency Cooper pairs.
