Application Eq. (21): continuity-based R and R_H bounds

For all positive integers $a,b$, every complex matrix $\rho$ on $\mathbb C^a\otimes\mathbb C^b$, and every real $0<\epsilon<\tfrac12$, assume that $\rho^\Gamma\succeq0$ and that $\rho$ is positive semidefinite of trace $1$ and admits a separable density matrix $\sigma$ satisfying $\|\rho^\Gamma-\sigma^\Gamma\|_1\le\epsilon$. Here $(X^\Gamma)_{(i,j),(k,l)}=X_{(i,l),(k,j)}$, $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$, and a separable density matrix is a finite convex combination of tensor products of local positive semidefinite trace-$1$ matrices. Then

where $h_2(t)=[-t\log t-(1-t)\log(1-t)]/\log2$, with zero contributions at the endpoints, and the bounds are regarded as elements of $[0,+\infty]$. The quantities $R,R_H$ are defined as follows. A Bell experiment applies a map $\Phi$ in the closure of finite LOCC maps to positive finite output dimensions, then measures local POVMs in any scenario with positive finite input and output counts; finite LOCC maps are generated by identities and finite complete local Kraus operations with outcome-dependent continuations, summing over the outcomes. Its box is $P_\theta(x,y,u,v)=\operatorname{Re}\operatorname{tr}[(A_{x,u}\otimes B_{y,v})\Phi(\theta)]$, where each local POVM has positive semidefinite effects summing to the identity. For any box $P$, define $\mathcal N(P)=\sup_\pi\inf_Q\sum_{x,y}\pi(x,y)D_{\mathrm{cl},2}(P_{xy}\|Q_{xy})$, where $\pi$ ranges over nonnegative input distributions summing to $1$, and $Q$ ranges over finite convex combinations of deterministic boxes $\mathbf1_{\{u=f(x),\,v=g(y)\}}$. The classical divergence is $+\infty$ if some first-distribution entry is positive while the corresponding second-distribution entry is zero; otherwise it is the nonnegative part of $\sum_{i:p_i\ne0}p_i\log(p_i/q_i)/\log2$. Let $R_n(\rho)$ be the supremum of $\mathcal N(P_{\rho^{[n]}})$ over Bell experiments, where $\rho^{[n]}$ is the grouped $n$-fold tensor power. Let $H_n(\rho)$ instead take the supremum over an experiment and a selected branch of complete finite local Kraus families after $\Phi$: for the branch $\omega=K\Phi(\rho^{[n]})K^\dagger$, with $K$ the tensor product of the selected local operators and $p=\operatorname{Re}\operatorname{tr}\omega$, its value is $0$ if $p=0$, and $p\,\mathcal N(P_{\omega/p})$ otherwise, with the experiment’s measurements applied to the normalized branch. Finally, $R(\rho)=\limsup_{n\to\infty}R_n(\rho)/n$ and $R_H(\rho)=\limsup_{n\to\infty}H_n(\rho)/n$, using positive block lengths and limits in $[0,+\infty]$.