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Quantum mechanical wave-particle duality is quantified in terms of atrade-off relation between the fringe visibility and the which-waydistinguishability in an interference experiment. This relation is recentlygeneralized by Banaszek et. al., (Nature Communications 4, 2594, 2013) when theparticle is equipped with an internal degree of freedom such as spin. Here, weextend the visibility-distinguishability trade-off relation to quantuminterference of a particle possessing an internal degree of freedom, when thewhich-way detector state is entangled with an ancillary system. We introduce anextended which-way distinguishability and the associated extended fringevisibility satisfying the duality inequality in this scenario. We illustrate,with the help of three specific examples, that while the which-way informationinferred solely from the detector state (without ancilla) vanishes, theextended distinguishability retrievable via measurements on thedetector-ancilla entangled state is non-zero. Furthermore, in all the threeexamples, the extended visibility and the generalized visibility (which wasintroduced by Banaszek et. al., Nature Communications 4, 2594, 2013)) matchidentically with each other.

Quantum which-way information and fringe visibility when the detector is entangled with an ancilla