Steerable weight and steering robustness are convex steering monotones

For natural numbers $d,r,s$, write an assemblage as a family $\rho=(\rho_{a|x})_{a<r,x<s}$ of complex $d\times d$ matrices, and put $T_\rho=s^{-1}\sum_{x<s}\sum_{a<r}\rho_{a|x}$ and $M_\rho=\operatorname{Re}\operatorname{tr}T_\rho$, with inverse zero interpreted as zero. It is an assemblage when $d,r,s>0$, every block is positive semidefinite, $\sum_a\rho_{a|x}$ is independent of $x$, and $0\le M_\rho\le1$; it is normalized when additionally $M_\rho=1$. A finite local hidden-state model means that it is an assemblage and there exist a natural number $h$, a probability vector $(p_\ell)_{\ell<h}$, density matrices $\xi_\ell$, and probability vectors $(q_{\ell,x,a})_{a<r}$ for every $\ell,x$, such that $\rho_{a|x}=\sum_{\ell<h}M_\rho p_\ell q_{\ell,x,a}\xi_\ell$; here a probability vector has nonnegative entries summing to one, and a density matrix is positive semidefinite with trace one. Define $W(\rho)$ to be the real infimum of all $\nu\ge0$ for which there exist normalized assemblages $\tau,\sigma$, with $\sigma$ admitting such a finite model, satisfying $\rho=\nu\tau+(1-\nu)\sigma$; this candidate condition does not separately impose $\nu\le1$. Define $R(\rho)$ to be the real infimum of all $\nu\ge0$ for which some normalized assemblage $\tau$ makes $(1+\nu)^{-1}\rho+\nu(1+\nu)^{-1}\tau$ admit a finite local hidden-state model. The declaration asserts that, for every $d,r,s$ and every normalized $\rho$, each of these two infima is a least element of its candidate set: it is itself a candidate and is at most every candidate. It also asserts the following properties separately for $S=W$ and $S=R$: $S(\rho)\ge0$ on every normalized assemblage; $S(\rho)=0$ on every normalized assemblage admitting a finite local hidden-state model; $S(t\rho+(1-t)\sigma)\le tS(\rho)+(1-t)S(\sigma)$ for normalized $\rho,\sigma$ of the same dimensions and every $0\le t\le1$; and the following average monotonicity holds for every finite protocol and every normalized input $\rho$. Such a protocol has positive input and output dimensions $d,r,s,d_f,r_f,s_f$, a natural number $m$ of branches, matrices $K_w:\mathbb C^d\to\mathbb C^{d_f}$ 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 $a_f<r_f$. Its branch assemblage is $B^w_{a_f|x_f}=\sum_{x,a}i_{w,x_f,x}o_{w,a_f,a,x,x_f}K_w\rho_{a|x}K_w^\dagger$, its branch probability is $p_w=\operatorname{Re}\operatorname{tr}(K_wT_\rho K_w^\dagger)$, and normalization of any family $B$ means multiplying every block by $M_B^{-1}$. Whenever the protocol is complete, meaning $\sum_wK_w^\dagger K_w=I$, the assertion is $\sum_{w<m}b_w\le S(\rho)$, where $b_w=p_wS(B^w/M_{B^w})$ if $p_w>0$, and $b_w=0$ otherwise. Quantification over natural dimensions includes zero, but the normalized-assemblage and protocol hypotheses exclude zero dimensions; branch counts are not separately required to be positive.