  We study the perfect transfer of information in 1+1D continuous quantum
systems. This includes effective descriptions of inhomogeneous spin chains, for
which the notion of perfect state transfer in quantum information was
introduced, and here phrased in terms of waves. We show that reflection
symmetry is necessary for perfect wave transfer (PWT) in any inhomogeneous
conformal field theory, and even sufficient when restricted to one-particle
excitations. To determine if or when it is sufficient more generally, we first
break conformal invariance and study a broad class of 1+1D bosonic theories. We
show that the question can then be posed as an inverse Sturm-Liouville problem
that determines when the bosonic theory exhibits PWT. We demonstrate how to
uniquely solve this problem, which also shows that reflection symmetry is
sufficient for the special case with conformal invariance. Using bosonization,
our continuum results extend these notions to interacting quantum field
theories.
