Relative entropy of steering is a convex steering monotone

For a family $F=(F_{a,x})_{a<r,\ x<s}$ of complex $d\times d$ matrices, write $B(F)=s^{-1}\sum_{x,a}F_{a,x}$ and $\mu(F)=\operatorname{Re}\operatorname{tr}B(F)$. A normalized family means $d,r,s>0$, positive semidefinite blocks, setting-independent outcome sums, and $\mu(F)=1$; a finite hidden-state model additionally means that, for some natural $h$, probability distributions $p_\ell$ and $q_{\ell,x}(a)$ and positive semidefinite trace-one matrices $\xi_\ell$ satisfy $F_{a,x}=\mu(F)\sum_{\ell<h}p_\ell q_{\ell,x}(a)\xi_\ell$. Define an extended nonnegative matrix divergence $\mathcal D(A,B)$ to be $+\infty$ unless both matrices are positive semidefinite of trace $1$ and $\ker B\subseteq\ker A$; otherwise it is

regarded as an extended nonnegative real, with matrix functions interpreted by the continuous functional calculus and real logarithms used. An entropy strategy for dimensions $d,s$ consists of a natural number $h$ of outcomes, matrices $K_g\in\mathbb C^{d\times d}$ with $\sum_{g<h}K_g^\dagger K_g=I_d$, and probability distributions $u_g(x)$ on $x<s$. For such a strategy $T$, define $p_g(F)=\operatorname{Re}\operatorname{tr}(K_gB(F)K_g^\dagger)$ and the matrix $A_{g,x}(F)$ indexed by $(a,i),(b,j)\in\{0,\ldots,r-1\}\times\{0,\ldots,d-1\}$ to have entry $(K_gF_{a,x}K_g^\dagger)_{ij}$ when $a=b$ and $0$ otherwise. Define $C(p,q;A,B)$ as $0$ if $p\leq0$, as $+\infty$ if $p>0$ and $q\leq0$, and as $\operatorname{ofReal}(p)\mathcal D(A/p,B/q)$ if $p,q>0$, where $\operatorname{ofReal}(z)=\max\{z,0\}$ in the extended nonnegative reals. Set

All sums, suprema, and infima defining $\mathcal A,\mathcal E$ use extended nonnegative arithmetic, including $0\cdot(+\infty)=0$, and $\operatorname{toReal}(+\infty)=0$. The declaration asserts three things: for every natural $d,r,s$ and every pair of normalized families $\rho,\sigma$ of those dimensions, some entropy strategy $T$ attains $\mathcal A(\rho,\sigma)$; for every normalized $\rho$, $\mathcal E(\rho)\ne+\infty$ and there exists a normalized finite-hidden-state family $\sigma$ attaining $\mathcal A(\rho,\sigma)=\mathcal E(\rho)$; and the real-valued $S$ just defined is nonnegative on all normalized families, equals $0$ on every normalized family with a finite hidden-state model, is convex on normalized families, and is strongly monotone in the following precise sense. For every choice of natural $d,r,s,d_f,r_f,s_f,m$, every protocol with positive input and output dimensions, matrices $L_w\in\mathbb C^{d_f\times d}$ satisfying $I_d-\sum_wL_w^\dagger L_w\succeq0$, input distributions $\alpha_w(x\mid x_f)$ and output distributions $\beta_w(a_f\mid a,x,x_f)$, and every normalized $\rho$, completeness $\sum_wL_w^\dagger L_w=I_d$ implies $\sum_w b_w\leq S_{d,r,s}(\rho)$, where

Convexity here requires, for every natural $d,r,s$, normalized $\rho,\sigma$, and real $0\leq t\leq1$, the inequality $S_{d,r,s}(t\rho+(1-t)\sigma)\leq tS_{d,r,s}(\rho)+(1-t)S_{d,r,s}(\sigma)$. The assertions conditioned on normalization are vacuous whenever any of $d,r,s$ is zero.