  We experimentally investigate Andreev bound states (ABSs) in a carbon
nanotube quantum dot (QD) connected to a superconducting Nb lead (S). A weakly
coupled normal metal contact acts as a tunnel probe that measures the energy
dispersion of the ABSs. Moreover we study the response of the ABS to non-local
transport processes, namely Cooper pair splitting and elastic co-tunnelling,
that are enabled by a second QD fabricated on the same nanotube on the opposite
side of S. We find an appreciable non-local conductance with a rich structure,
including a sign reversal at the ground state transition from the ABS singlet
to a degenerate magnetic doublet. We describe our device by a simple rate
equation model that captures the key features of our observations and
demonstrates that the sign of the non-local conductance is a measure for the
charge distribution of the ABS, given by the respective Bogoliubov-de Gennes
amplitudes $u$ and $v$.
