Theorem: Named entropy and information trajectories

For pure message input $|0⟩$, radiation entropies are

For the maximally entangled $RM$ input,

These are entropy trajectories of the declared finite model.

Proof. Occupation outputs contribute independent entropies $H_i$. The emitted key contributes two bits. For input $|0⟩$, each occupied ciphertext conditioned on the key is pure, and vacuum and occupied sectors are orthogonal. Their mixture contributes $h_2(p)$.

For the Bell input the message marginal is $I_2/2$. Each successful ciphertext marginal is $I_{C_2,\mathrm{o}}/2$, contributing one bit with weight $p$. Thus

The successful reference–ciphertext state conditioned on the key is a mixture of two orthogonal Bell states with probabilities $(1+\eta )/2,(1-\eta )/2$, of entropy $h_{\eta }$. The failed reference state is $I_R/2$, of entropy one. Therefore

Using $S(R)=1$ gives $I(R:{\mathrm{rad}}_2)=p(2-h_{\eta })$. At the earlier prefixes the output is independent of $R$. The optimal fidelities are Theorem thm:recovery.