Scalable photonic quantum processing via isometric energy-conserving transformations: bridging hybrid optical modes

The precise manipulation of quantum states of light lies at the heart of advancing photonic technologies, yet scalable solutions for hybrid photonic systems—where disparate optical modes such as polarization, time-bin, and frequency must coherently interact—remain a critical challenge. While current literature demonstrates various energy-conserving devices that effectively map between Hilbert spaces of unequal dimensions, there is a growing need for frameworks that address variable-dimensional transformations in a more unified manner. In this paper, we introduce an operator-level framework that systematically integrates both unitary and rectangular isometric energy-conserving transformations in multi-dimensional photonic systems. Unlike prior mode-specific approaches, our formalism enables modular modeling of hybrid optical modes within effective signal spaces, offering practical tools for scalable photonic quantum processors. By classifying these transformations into fixed- and variable-dimensional regimes, we derive universal design criteria for energy-conserving photonic signal processors. This work not only advances the fundamental understanding of photonic quantum processing but also delivers actionable tools for engineering next-generation scalable optical technologies, including multi-purpose, programmable photonic circuits, energy-efficient optical signal processing in reconfigurable networks, and high-dimensional quantum key distribution with hybrid mode encoding.