Appendix: uniform-input versions of Theorems 2 and 3

For all positive integers $a,b$ and every positive semidefinite trace-$1$ complex matrix $\rho$ on $\mathbb C^a\otimes\mathbb C^b$,

and, if $\rho^\Gamma\succeq0$, then

Here $(X^\Gamma)_{(i,j),(k,l)}=X_{(i,l),(k,j)}$. A separable density matrix is a finite sum $\sum_t w_t A_t\otimes B_t$ with nonnegative weights summing to $1$ and positive semidefinite trace-$1$ local matrices. Define $E_{\mathrm{rel}}(\theta)=\inf_{\sigma\text{ separable density}}D_2(\theta\|\sigma)$, where $D_2=+\infty$ unless both arguments are density matrices and $\ker\sigma\subseteq\ker\theta$; otherwise $D_2$ is the nonnegative part of $\operatorname{Re}\operatorname{tr}[f(\theta)-\theta\log\sigma]/\log2$, with $f(x)=x\log x$, matrix functions given by continuous functional calculus, and the real logarithm equal to $0$ at $0$. A Bell experiment applies a map $\Phi$ in the closure of finite LOCC maps to positive finite output dimensions and then local POVMs in a 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 outcomes. Its measured box is $P_\theta(x,y,u,v)=\operatorname{Re}\operatorname{tr}[(A_{x,u}\otimes B_{y,v})\Phi(\theta)]$, with positive semidefinite POVM effects summing to the identity for each input. For its $n_x,n_y$ inputs, define

A finite local box is a finite convex combination of deterministic boxes $\mathbf1_{\{u=f(x),\,v=g(y)\}}$. The classical divergence is $+\infty$ if some first-distribution entry is positive and its corresponding second-distribution entry is zero; otherwise it is the nonnegative part of $\sum_{i:p_i\ne0}p_i\log(p_i/q_i)/\log2$. For positive $n$, let $\rho^{[n]}$ be the grouped tensor power with entries $\prod_{t=1}^n\rho_{(i_t,j_t),(k_t,l_t)}$. Define $R_{\mathrm u,n}(\rho)$ as the supremum over Bell experiments of $\mathcal N_{\mathrm u}(P_{\rho^{[n]}})$, and $T_{\mathrm u,n}(\rho)$ as the supremum over those experiments of $\inf_{\sigma\text{ separable density on }\mathbb C^a\otimes\mathbb C^b}D_{\mathrm u}(P_{\rho^{[n]}}\|P_{\sigma^{[n]}})$. Define $H_{\mathrm u,n}(\rho)$ by additionally choosing complete finite local Kraus families and one selected operator on each side after $\Phi$: for $\omega=K\Phi(\rho^{[n]})K^\dagger$, with $K$ the tensor product of the selected operators and $p=\operatorname{Re}\operatorname{tr}\omega$, the contribution is $0$ if $p=0$, and otherwise $p\,\mathcal N_{\mathrm u}(P_{\omega/p})$, measured with the experiment’s POVMs. Finally, $R_{\mathrm u},T_{\mathrm u},R_{H,\mathrm u}$ are respectively the $\limsup$ of these three block quantities divided by $n$ as positive $n\to\infty$, all valued in $[0,+\infty]$.