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In this article we show that, in a two-arm interferometer, pure quantumstates of perfect path distinguishability (particles) are geometricallyequidistant from all states with constant path distinguishability D. Thisproperty is not shared by other states, such as perfect fringe-visibility(waves) or maximally entangled quantum states (entanglon). Indeed, the Buresdistance between a particle and any other state depends only thedistinguishability of the latter. On the contrary, the Bures distance between awave or an entanglon, and any other single photon state depends on other set ofparameters.

Particle states are equidistant to wave and fully-entangled states in an interferometer