Free Multiplicative Convolution and Erlang Moments in Monitored Quantum Transport

What the math gives to ML

The paper gives an explicit free-probability law for the singular-value spectrum of long products of independently mixed, weakly lossy operators. Its transferable asset is not the quantum-transport application, but the closed S-transform and Lagrange-inversion formula, which predict how products accumulate attenuation, spectral spread, and a mass of nearly undamped directions as depth grows. A practical neural-network adaptation is to use the predicted law as a Jacobian-spectrum target during initialization or training, rather than relying only on scalar gradient-norm control. This yields a falsifiable spectral regularizer for deep stacks and a depth-dependent calibration of gains or residual scales.