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Modeling quantum interference in the presence of dissipation is a criticalaspect of quantum technologies. Including dissipation into the model of alinear device enables for assesing the detrimental impact of photon loss, aswell as for studying dissipation-driven quantum state transformations. However,establishing the input-output relations characterizing quantum interference ata general lossy N-port network poses important theoretical challenges. Here, wepropose a general procedure based on the singular value decomposition (SVD),which allows for the efficient calculation of the input-output relations forany arbitrary lossy linear device. In addition, we show how the SVD provides anintuitive description of the principle of operation of linear optical devices.We illustrate the applicability of our method by evaluating the input-outputrelations of popular reciprocal and nonreciprocal lossy linear devices,including devices with singular and nilpotent scattering matrices. We expectthat our procedure will motivate future research on quantum interference incomplex devices, as well as the realistic modelling of photon loss in linearlossy devices.

A generalized approach to quantum interference in lossy N-port devices via a singular value decomposition