Which-crystal information and wave-particle duality in induced-coherence interferometry

Abstract

We provide an operational reinterpretation of wave-particle complementarity in the low-gain Zou-Wang-Mandel (ZWM) induced-coherence interferometer. In the low gain limit, each photon pair is emitted by either one of two nonlinear crystals. Preparing nonorthogonal conditional idler states that encode which-crystal information. While previous studies inferred distinguishability indirectly from signal visibility with undetected idler photons. We show that the idler states naturally define a binary quantum hypothesis-testing problem. By performing optimal measurements on the idler, we analyze this task using both zero-error measurement unambiguous discrimination (Ivanovic-Dieks-Peres (IDP)) and minimum-error discrimination (Helstrom bound). We show that the signal visibility equals the optimal inconclusive probability of unambiguous discrimination. The Helstrom bound gives the optimal probability of identifying the emitting crystal. While signal visibility is an ensemble-averaged expectation value, the IDP and Helstrom strategies correspond to optimal single-photon decision measurements on the idler. The decision problem concerns inferring a past source event from a present measurement outcome This establishes wave-particle duality in induced coherence as a manifestation of optimal quantum decision strategies rather than a purely geometric constraint. We further extend the analysis to the presence of thermal photons introduced in the object arm, which render the conditional idler states mixed. In this case, both the visibility and the achievable distinguishability are reduced, reflecting the fundamental limitations imposed by mixed-state discrimination. The approach is model-independent and applies to general two-path interferometers with markers.