How many channels can a photonic system support?

We develop a general method to bound the ordered singular values (channel amplitudes) of the electromagnetic Green function for arbitrarily structured linear photonic systems. The approach yields computable, quantitatively predictive, upper bounds on the $n^{th}$ singular value that capture the complexity of multi-channel tradeoffs from the device perspective. As an illustration of the practical value of the framework, indexed channel bounds are obtained for multi-wavelength scale three-dimensional volumes (up to $64\,\lambda^3$) and applied to common application classes related to waveguides, metasurfaces, and planewave detection. These results are immediately applicable to the calculation of information theoretic objectives such as Shannon capacity and Fisher information.