  One of the fundamental features of quantum mechanics is the superposition
principle, a manifestation of which is embodied in quantum coherence. Coherence
of a quantum state is invariably defined with respect to a preferred set of
pointer states, and there exist quantum coherence measures with respect to
deterministically as well as probabilistically distinguishable sets of quantum
state vectors. Here we study the resource theory of quantum coherence with
respect to an arbitrary set of quantum state vectors, that may not even be
probabilistically distinguishable. Geometrically, a probabilistically
indistinguishable set of quantum state vectors forms a linearly dependent set.
We find the free states of the resource theory, and analyze the corresponding
free operations, obtaining a necessary condition for an arbitrary quantum
operation to be free. We identify a class of measures of the quantum coherence,
and in particular establish a monotonicity property of the measures. We find a
connection of an arbitrary set of quantum state vectors with positive operator
valued measurements with respect to the resource theory being considered, which
paves the way for an alternate definition of the free states. We notice that
the resource theory of magic can be looked upon as a resource theory of quantum
coherence with respect to a set of quantum state vectors that are
probabilistically indistinguishable. We subsequently examine the wave-particle
duality in a double-slit set-up in which superposition of probabilistically
indistinguishable quantum state vectors is possible. Specifically, we report a
complementary relation between quantum coherence and path distinguishability in
such a set-up.
