Theorem: Classical and quantum cut factorizations

Under the complete network, freshness, and cut-accounting hypotheses:

1. every wholly classical final decoder-output law has the form

   $$p(o\mid x,j)=\int \mu (d\lambda ){\sum _{m=1}}^K{a}_{\lambda }(m\mid x)d_{\lambda }(o\mid m,j);$$

2. every quantum output law has the form

   $$p(o\mid x,j)=\int \mu (d\lambda )\mathrm{Tr}\left[{{\mathcal{D}}^{\lambda }}_{o\mid j}({\mathcal{S}}_{\lambda }({\rho }_{x,\lambda }))\right],$$

   where ${\mathcal{S}}_{\lambda }$ is the admitted storage channel and $\{{{\mathcal{D}}^{\lambda }}_{o\mid j}{\}}_o$ is a complete decoder instrument.

All persistent input-bearing systems must be included in $M$ or in the quantum cut system.

Proof. Condition on $\lambda$. Compose the operations before the frontier. They produce a stochastic law $a_{\lambda }(m\mid x)$, or a density operator ${\rho }_{x,\lambda }$. Proposition prop:late excludes dependence on $j$.

By completeness of the cut, the remainder has no access to the encoding inputs except through that state. Compose storage and the decoder, integrating their fresh randomness or adjoining their product ancillary states. In the classical case this produces a stochastic kernel $d_{\lambda }(o\mid m,j)$. In the quantum case it produces the displayed complete instrument after the storage channel.

Systems exported to an inaccessible archive and never returned to the decoder can be summed out in the decoder marginal. No accessible correlation is removed by that step. Finally average over $\lambda$, whose law is independent of the fresh inputs.