  Information recycling has been shown to improve the sensitivity of atom
interferometers by exploiting atom-light entanglement. In this paper, we apply
information recycling to an interferometer where the input quantum state has
been partially transferred from some donor system. We demonstrate that when the
quantum state of this donor system is from a particular class of
number-correlated Heisenberg-limited states, information recycling yields a
Heisenberg-limited phase measurement. Crucially, this result holds irrespective
of the fraction of the quantum state transferred to the interferometer input
and also for a general class of number-conserving quantum-state-transfer
processes, including ones that destroy the first-order phase coherence between
the branches of the interferometer. This result could have significant
applications in Heisenberg-limited atom interferometry, where the quantum state
is transferred from a Heisenberg-limited photon source, and in optical
interferometry where the loss can be monitored.
