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Input-output theory is invaluable for treating superconducting and photoniccircuits connected by transmission lines or waveguides. However, this theorycannot in general handle situations in which retro-reflections from circuitcomponents or configurations of beam-splitters create loops for thetraveling-wave fields that connect the systems. Here, building upon thenetwork-contraction theory of Gough and James [Commun. Math. Phys. 287, 1109(2009)], we provide a compact and powerful method to treat any circuit thatcontains such loops so long as the effective cavities formed by the loops aresufficiently weak. Essentially all present-day on-chip superconducting andphotonic circuits will satisfy this weakness condition so long as thereflectors that form the loops are not especially highly reflecting. As anexample we analyze the problem of transmitting entanglement between two qubitsconnected by a transmission line with imperfect circulators, a problem forwhich the new method is essential. We obtain a full solution for the optimalreceiver given that the sender employs a simple turn on/turn off. This solutionshows that near-perfect transmission is possible even with significantretro-reflections.

Input-output theory for superconducting and photonic circuits that contain weak retro-reflections and other weak pseudo-cavities