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 ${\mathcal{P}}^2+{\mathcal{V}}^2\le 1$, 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$isingeneralnolongerupperboundedby$1$, 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 ${\mathcal{P}}^2+{\mathcal{V}}^2\le L_U$, with $1\le L_U\le 2$, and we illustrate how the tight bound $L_U$ depends on the choice of the unitary operation $U$ replacing the central phase shifter.