Duality relation for a generalized interferometer

It is well known that the Mach-Zender interferometer exhibits a trade-off between the a priori which-path knowledge and the visibility of its interference pattern. This trade-off is expressed by the inequality $\mathcal{P}^2 + \mathcal{V}^2 \leq 1$, constraining the predictability $\mathcal{P}$ and visibility $\mathcal{V}$ of the interferometer. In this paper we extend the Mach-Zender scheme to a setup where the central phase shifter is substituted by a generic unitary operator. We find that the sum $\mathcal{P}^2 + \mathcal{V}^2$ is in general no longer upper bounded by $1$, and that there exists a whole class of interferometers such that the full fringe visibility and the full which-way information are not mutually exclusive. We show that $\mathcal{P}^2 + \mathcal{V}^2 \leq L_U$, with $1 \leq L_U \leq 2$, and we illustrate how the tight bound $L_U$ depends on the choice of the unitary operation $U$ replacing the central phase shifter.