  We consider quantum state transfer on finite graphs which are attached to
infinite paths. The finite graph represents an operational quantum system for
performing useful quantum information tasks. In contrast, the infinite paths
represent external infinite-dimensional systems which have limited (but
nontrivial) interaction with the finite quantum system. We show that {\em
perfect} state transfer can surprisingly still occur on the finite graph even
in the presence of the infinite tails. Our techniques are based on a decoupling
theorem for eventually-free Jacobi matrices, equitable partitions, and standard
Lie theoretic arguments. Through these methods, we rehabilitate the notion of a
dark subspace which had been so far viewed in an unflattering light.
