Entropic framework for wave-particle duality in multipath interferometers

An interferometer—no matter how clever the design—cannot reveal both the wave and the particle behavior of a quantum system. This fundamental idea has been captured by inequalities, so-called wave-particle duality relations (WPDRs), that upper bound the sum of the fringe visibility (wave behavior) and path distinguishability (particle behavior). Another fundamental idea is Heisenberg's uncertainty principle, stating that some pairs of observables cannot be known simultaneously. Recent work has unified these two principles for two-path interferometers. Here we extend this unification to $n$-path interferometers, showing that WPDRs correspond to a modern formulation of the uncertainty principle stated in terms of entropies. Furthermore, our unification provides a framework for solving an outstanding problem of how to formulate universally valid WPDRs for interferometers with more than two paths, and we employ this framework to derive some novel WPDRs.

Figure 1

(a) $n$-path interferometer for single photons. A source injects a photon into an optical fiber. The photon approaches a fiber coupler, ${\mathsf{\text{FC}}}_1$, which allows the photon to leak into the other $n-1$ fibers, creating a superposition of which-path states. Then the photon interacts with some environment $E$, which may obtain some information, e.g., about the photon's path. Then a phase shift is applied to each arm (${\varphi }_z$ to the $z\mathrm{th}$ arm), and the arms are brought together again at a second fiber coupler, ${\mathsf{\text{FC}}}_2$. Finally, the photon is detected at some detector. (b) Guessing-game view of WPD. We think of “particles” as having a well-defined location, so the particle game asks the experimenter to guess (given access to a subsystem $E_1$ of $E$) which path the photon will take inside the interferometer. We think of “waves” as exhibiting interference—with the most extreme interference pattern corresponding to only one detector clicking all the time and no interference corresponding to a uniform distribution over all detectors. Hence the wave game asks the experimenter to guess (given access to a subsystem $E_2$ of $E$ and given the freedom to adjust the phases $\left\{{\varphi }_z\right\}$) which detector will click. WPDRs give quantitative trade-offs for the probabilities of winning these two games.