  Quantum mechanical wave-particle duality is quantified in terms of a
trade-off relation between the fringe visibility and the which-way
distinguishability in an interference experiment. This relation is recently
generalized by Banaszek et. al., (Nature Communications 4, 2594, 2013) when the
particle is equipped with an internal degree of freedom such as spin. Here, we
extend the visibility-distinguishability trade-off relation to quantum
interference of a particle possessing an internal degree of freedom, when the
which-way detector state is entangled with an ancillary system. We introduce an
extended which-way distinguishability and the associated extended fringe
visibility satisfying the duality inequality in this scenario. We illustrate,
with the help of three specific examples, that while the which-way information
inferred solely from the detector state (without ancilla) vanishes, the
extended distinguishability retrievable via measurements on the
detector-ancilla entangled state is non-zero. Furthermore, in all the three
examples, the extended visibility and the generalized visibility (which was
introduced by Banaszek et. al., Nature Communications 4, 2594, 2013)) match
identically with each other.
