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.

Figures (3)

• Figure 1: Implementation of optimal unambiguous state discrimination (USD) BarnettClarke2001IDPLen2018IDP in the induced-coherence (Zou--Wang--Mandel) interferometer Torres2024. Two coherently pumped nonlinear crystals (A and B) generate signal--idler photon pairs in the low-gain regime. Each emission event originates from either crystal A or crystal B but never both. The signal modes interfere at a balanced beam splitter $S_1$, producing single-photon fringes with visibility $V$. The idler modes ($\hat{a}_{3}^{2},\hat{a}_{3}^{3}$) form the two nonorthogonal marker states $|\phi_A\rangle$ and $|\phi_B\rangle$, whose overlap determines $V$. The output of the first idler ($\hat{a}_{3}^{2}$) beam splitter is directed to detector $D_0$, realizing the inconclusive IDP outcome $\Pi_{inc}$. The detectors $D_1$ and $D_2$ project onto the orthogonal complement states $|\phi_B^{\perp}\rangle$ and $|\phi_A^{\perp}\rangle$, yielding the conclusive IDP outcomes $\Pi_A$ and $\Pi_B$.
• Figure 2: Visibility $V(T)$, distinguishability $D(T)=\sqrt{1-V(T)^2}$, and the Helstrom minimum error probability $P_{\mathrm{err}}^{\min}(T)=\tfrac{1}{2}[1-D(T)]$ as functions of the object transmittance $T$ in the induced-coherence interferometer. Increasing transmittance enhances signal interference visibility while reducing which-crystal distinguishability encoded in the idler. The Helstrom bound quantifies the optimal minimum-error probability for inferring which nonlinear crystal emitted the photon pair from a single idler measurement. Illustrating the operational complementarity between interference and source discrimination.
• Figure 3: (a) Operational complementarity between visibility $V$, distinguishability $D$, and the optimal IDP failure probability $P_{inc}^{\mathrm{opt}}$ for pure marker states with overlap $s = |\langle\phi_A|\phi_B\rangle|$ (see Appendix \ref{['six']}). The curves satisfy $D^2+V^2=1$ and $V=P_{inc}^{\mathrm{opt}}$, identifying wave--particle duality with the optimality condition of unambiguous state discrimination. (b) Comparison between distinguishability obtained via the coincidence-based method of Torres et al.Torres2024 (solid) and via optimal IDP discrimination on the idler (dashed). The coincidence method saturates at $D_{\mathrm{coinc}}(T=1)\approx0.52$, whereas IDP extracts greater distinguishability for intermediate $0<T<1$. Both approaches correctly give $D=0$ at $T=1$, where the marker states become identical. The advantage of IDP therefore appears only for partially overlapping markers.