The Born Representation Theorem and the Unistochastic Theorem

What the math gives to ML

The paper gives a constructive way to realize every conditional-probability matrix as a Born-rule measurement, and to realize it as a marginal of a higher-dimensional unitary transition. The transferable asset is not the POVM notation itself, but the exact normalization guarantee obtained by representing probabilities as squared amplitudes and marginalizing an orthogonal latent output space. This suggests a neural routing or state-transition module whose probabilities are induced by an isometry or unitary rather than independently normalized logits, with a controllable latent ancilla that can preserve information lost by ordinary stochastic mixing. The main practical caveat is dimensional overhead, so experiments should first target small routers or low-rank/shared dilations.