Corollary: Conditional fresh-pair entropy growth

Suppose radiation grows by independent tensor factors, $E_t=E_{t-1}{B}_t$. At each step there is a joint state on $A_t{B}_t{E}_{t-1}$ satisfying the hypotheses of Theorem thm:amps, with pair entropy $s_t$, outgoing dimension $d_t$, and distance allowance ${\epsilon }_t$. Then

Proof. At step $t$, $S(E_t)-S(E_{t-1})=S(B_t|E_{t-1})$. Apply (eq:ampsbound) and sum.