Generalized matching condition for unity efficiency quantum transduction

Coherently converting quantum states between distinct elements via quantum transducers remains a crucial yet challenging task in quantum science. Especially in demand is quantum transduction between optical frequencies, which are ideal for low-loss transmission across long distances, and microwave frequencies, which admit high-fidelity quantum operations. We present a generic formalism for $N$-stage quantum transduction that covers various leading microwave-to-optical, microwave-to-microwave, and optical-to-optical linear conversion approaches. We then identify effective circuit models and the resulting generalized matching conditions for achieving maximum conversion efficiency. The generalized matching condition requires resistance matching as well as frequency matching beyond the usual resonant assumption, with a simple impedance-matched transmission interpretation. Our formalism provides a generic toolbox for determining experimental parameters to realize efficient quantum transduction and suggests different regimes of non-resonant conversions that might outperform all-resonant ones.