Quantum Coherence and Path-Distinguishability of Two Entangled Particles

An interference experiment with entangled particles is theoretically analyzed, where one of the entangled pair (particle 1) goes through a multi-slit before being detected at a fixed detector. In addition, one introduces a mechanism for finding out which of the n slits did particle 1 go through. The other particle of the entangled pair (particle 2) goes in a different direction, and is detected at a variable, spatially separated location. In coincident counting, particle 2 shows n-slit interference. It is shown that the normalized quantum coherence of particle 2, $\mathcal{C}_2$, and the path-distinguishability of particle 1, $\mathcal{D}_{Q1}$, are bounded by an inequality $\mathcal{D}_{Q1} + \mathcal{C}_2 \le 1$. This is a kind of {\em nonlocal} duality relation, which connects the path distinguishability of one particle to the quantum coherence of the other.