Assemblage relative entropy does not increase on average

Let $d,r,s,d_f,r_f,s_f,m$ be natural numbers, and let $P$ be a protocol whose six dimensions $d,r,s,d_f,r_f,s_f$ are positive. It supplies matrices $K_w:\mathbb C^d\to\mathbb C^{d_f}$, $w<m$, with $I-\sum_wK_w^\dagger K_w$ positive semidefinite, probability vectors $i_{w,x_f,x}$ over $x<s$, and probability vectors $o_{w,a_f,a,x,x_f}$ over $a_f<r_f$. Assume also that it is complete, $\sum_wK_w^\dagger K_w=I$. Let $\rho,\sigma$ be normalized input assemblages: their blocks are positive semidefinite, their sums over outcomes are independent of the setting, and each has mass one, where for a family $A$ of dimensions $D,R,S$ one sets $T_A=S^{-1}\sum_{x<S,a<R}A_{a|x}$ and $M_A=\operatorname{Re}\operatorname{tr}T_A$; an assemblage additionally requires positive dimensions and $0\le M_A\le1$. To specify the entropy used in the conclusion, define $\mathcal D(A\Vert B)\in[0,\infty]$ to be infinity unless both matrices are positive semidefinite with trace one and $\ker B\subseteq\ker A$; otherwise it is the nonnegative part of $\operatorname{Re}\operatorname{tr}(f(A)-A\log B)$, with $f(t)=t\log t$, using matrix functional calculus and the convention $\log0=0$. For probability-sized real vectors $p,q$, define $\mathcal D_{\mathrm{cl}}(p\Vert q)=\mathcal D(\operatorname{diag}p\Vert\operatorname{diag}q)$, using this same definition even if the vectors fail to define density matrices. An entropy strategy on dimensions $D,S$ consists of a natural number $h$, square matrices $L_g$ with $\sum_{g<h}L_g^\dagger L_g=I$, and nonnegative numbers $u_{g,x}$ with $\sum_{x<S}u_{g,x}=1$. For a family $A$, put $p_g^A=\operatorname{Re}\operatorname{tr}(L_gT_AL_g^\dagger)$ and let $F^A_{g,x}$ be the block diagonal matrix whose outcome block $a$ is $L_gA_{a|x}L_g^\dagger$. Define $C(p,q;A,B)$ to be $p\,\mathcal D(A/p\Vert B/q)$ if $p>0,q>0$, infinity if $p>0,q\le0$, and zero if $p\le0$, where nonnegative real factors are embedded into $[0,\infty]$. Define

For each protocol branch put $B_w(A)_{a_f|x_f}=\sum_{x,a}i_{w,x_f,x}o_{w,a_f,a,x,x_f}K_wA_{a|x}K_w^\dagger$, $p_w=\operatorname{Re}\operatorname{tr}(K_wT_\rho K_w^\dagger)$, and $q_w=\operatorname{Re}\operatorname{tr}(K_wT_\sigma K_w^\dagger)$, and normalize a branch family $B$ as $B/M_B$, using inverse zero equal to zero. The conclusion is

All sums, suprema, and inequalities involving $\mathcal E$ are in the extended nonnegative reals, with $0\cdot\infty=0$; no finiteness assumption is imposed on either entropy.