Beyond the designed subspace: A complete operator theory for multi-plane light conversion

Coherent light manipulation in high-dimensional modal spaces is fundamental to applications ranging from optical computing to quantum entanglement. Multi-Plane Light Conversion (MPLC) systems, though capable of arbitrary unitary transformations, are limited by structured imperfections—phase errors, pixelation, and misalignments—that induce modal leakage beyond the designed subspace. Conventional random matrix theory (RMT) fails to capture this mechanism, overlooking the MPLC operator’s modal optimization and deterministic constraints. We develop a hybrid RMT framework encompassing both designed and undesigned modes, yielding a universal dimensionless leakage parameter epsilon that quantifies coupling into undesigned modes and scales predictively with hardware parameters, establishing rigorous fidelity metrics that reveal fundamental performance limits tied to the target mode family.