Learned Intensity Sensors Trade Separability Across Orientations

A learned sensor is frozen in hardware, so whatever it fails to record is lost to every decoder. We ask what such a sensor trades away, for intensity sensors, which record squared magnitudes of a coherent field under additive Gaussian noise. How well one separates two draws from a signal distribution, measured by the average Kullback-Leibler divergence between their measurement laws, is an inner product of two fourth moments, one of the sensor and one of the data, and equals the variance the data induce in its noiseless measurements. Averaged over rotations of the distribution, this separability is the same for every sensor with $m$ unit-norm measurements: learning redistributes it and cannot create it, although on one distribution a sensor may hold far more than isotropy. The excess is an exact overlap with the sensor's fourth-order defect, which the distance to a complex projective 2-design bounds only uniformly over a distribution class, not on the task at hand. Under a controlled rotation, a learned sensor that beats a quadratic-mask design by 2.6 dB in distribution loses to it by 0.8 dB at $90^{\circ }$ with retrained decoders, and a decoder-free overlap ranks this loss across $12$ configurations. In a matched family where only the fourth-order direction varies, the effect is graded with the alignment but small, and its sign follows the local Cram\'er-Rao bound rather than the separation. Two boundaries are as informative: the ordering reverses at low noise, and in coded diffraction imaging of phase objects no learned mask set loses more than $0.11$ dB of separability.