Geometric Coherence and Path Distinguishability

Abstract Based on the concept of geometric coherence, we propose a wave-particle duality relation that establishes an equation between Distinguishability and Visibility for any pure state in the composite system of a general n-path interferometer and a which-path detector. This relation has the desired property that any increase of Distinguishability will inevitably cause a decrease in Visibility, thereby quantitatively capturing the essence of complementarity. Additionally, we reveal a connection between the fidelity function and discrimination of pure states. Specifically, we generalize the previous result in J. Phys. A 51,414005 (2018) by showing that for any quantum state ρ, the geometric coherence can be exactly expressed as the optimal guessing probability in the discrimination of the corresponding ensemble of pure states.