Proposition: Commuting dynamical sanity check

The radiation channel is

It has

although the complete output is exactly recoverable. Its energy ledger is

Proof. For a fixed source eigenvalue, Weyl displacement completes the square in $H_B+g{a}_i\Phi (f)$. Vacuum evolves, up to a common phase, to the coherent amplitude ${\eta }_i$. Tracing the orthogonal source labels gives the displayed measure-and-prepare channel.

The two collected pure states have trace distance $z_R$. Any decoder's Bell overlap is at most $(1+z_R)/4$: its two logical diagonal adjoint effects define a binary discrimination measurement, whose equal-prior success probability is at most $(1+z_R)/2$, obtained from the positive part of the state difference. The Bell test gives error at least $(3-z_R)/4$. Parity interchanges the two states, so an optimal discrimination measurement may be symmetrized to have equal conditional errors. Measuring it and preparing the inferred basis state yields

Its Pauli identity weight is $(1+z_R)/4$; orthogonal Pauli Bell outputs give the matching error.

The residual retains orthogonal source labels on the two basis inputs, so Lemma lem:response gives leakage at least $1/2$. Complete preservation follows from Lemma lem:H and Theorem thm:exact. The coherent bath energy is ${\Vert h^{1/2}{\eta }_i\Vert }^2$, giving the integral. The initial total energy is zero, and conservation gives $E_I=-E_B$.