Finite participating reductions

For a chosen capture envelope $f$, the memory output depends only on $a(\hat{f})$. Together with the target $a(h)$, it therefore requires at most two canonical input modes. Define their plane by

The normal state on its Fock space is justified by Section subsec:completion whenever its occupation matrix is finite.

The declared operational input is a normal state

Here $g$ will be the orthogonal participating mode. The ideal capture target identifies $h$ with $c$ and traces out $g$:

It does not assume ${\rho }_{hgSR}$ factorizes.

The Hawking input class used for the model theorem consists of all such normal extensions with the specified two-mode Gaussian marginal. External finite references are unrestricted. A general occupation-constrained class is treated separately below.

No normal density representation of the entire continuum is assumed. The capture model defines its reduced channel through its output Weyl operators on these finite participating modes. A physical continuum interaction with further participating degrees of freedom would require a larger input interface.