Forgetting a complete instrument’s outcome preserves monotonicity

Let $S$ assign a real number $S_{d,r,s}(F)$ to every family $F=(F_{a,x})_{a<r,\ x<s}$ of complex $d\times d$ matrices, for every natural $d,r,s$. Call such a family normalized when $d,r,s>0$, its blocks are positive semidefinite, its outcome sums are setting-independent, and $\mu(F)=\operatorname{Re}\operatorname{tr}(s^{-1}\sum_{x,a}F_{a,x})=1$. Assume the following two properties of $S$: for every normalized $F$ and every complete protocol in any dimensions, with branch families $R^{(w)}$ and branch probabilities $p_w$, one has $\sum_w b_w\leq S_{d,r,s}(F)$, where $b_w=p_wS_{d_f,r_f,s_f}(R^{(w)}/\mu(R^{(w)}))$ if $p_w>0$ and $b_w=0$ otherwise; and for all normalized $F,H$ of the same dimensions and every real $t\in[0,1]$, $S_{d,r,s}(tF+(1-t)H)\leq tS_{d,r,s}(F)+(1-t)S_{d,r,s}(H)$. Here a protocol has positive input and output dimensions, finitely many matrices $K_w$ with $I-\sum_wK_w^\dagger K_w\succeq0$, probability distributions $\alpha_w(x\mid x_f)$ and $\beta_w(a_f\mid a,x,x_f)$ over $x$ and $a_f$, respectively, and

Completeness means $\sum_wK_w^\dagger K_w=I$. Then for every such complete protocol $P$ with natural parameters $d,r,s,d_f,r_f,s_f,m$ and every normalized input $\rho$, its outcome-forgotten output $R=\sum_{w<m}R^{(w)}$ satisfies $S_{d_f,r_f,s_f}(R)\leq S_{d,r,s}(\rho)$. No nonnegativity or hidden-state vanishing property of $S$ is assumed.