Resource theory of quantum coherence with probabilistically nondistinguishable pointers and corresponding wave-particle duality

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