Theorem 2, Eqs. (15)–(16): asymptotic relative entropy bounds

For every pair of positive integers $a,b$ and every positive semidefinite trace-$1$ matrix $\rho$ on $\mathbb C^a\otimes\mathbb C^b$, the extended-nonnegative-real quantities defined below satisfy

To define these quantities, a Bell experiment on a given input space consists of a map in the ambient topological closure of finite LOCC maps to some positive finite local output dimensions, a finite Bell scenario with positive input and output sizes, and local positive semidefinite measurement effects summing to the identity for each input. Finite LOCC maps are generated by the identity and finite recursive local Kraus measurements satisfying $\sum_iE_i^\dagger E_i=I$, followed by branch-dependent finite LOCC maps and summation of their outputs. The resulting box is $P(a,b|x,y)=\operatorname{Re}\operatorname{tr}((M_{a|x}\otimes N_{b|y})L(\omega))$. For any such scenario, let $\mathcal P$ be all nonnegative input weights $p(x,y)$ summing to $1$, and let $\mathcal L$ be all boxes $Q(a,b|x,y)=\sum_iw_i\mathbf1_{a=f_i(x)}\mathbf1_{b=g_i(y)}$ with finitely many nonnegative weights summing to $1$. Define $D_p(P\|Q)=\sum_{x,y}p(x,y)D_{\mathrm{KL},2}(P(\cdot,\cdot|x,y)\|Q(\cdot,\cdot|x,y))$, where the classical divergence is $+\infty$ if some positive numerator probability has zero denominator, and otherwise is the nonnegative truncation of $\sum_{a,b:P(a,b)>0}P(a,b)\log_2(P(a,b)/Q(a,b))$. Then $R_n(\rho)=\sup_{\mathcal E}\sup_{p\in\mathcal P}\inf_{Q\in\mathcal L}D_p(P_{\mathcal E,\rho^{\otimes n}}\|Q)$, whereas $T_n(\rho)=\sup_{\mathcal E}\sup_{p\in\mathcal P}\inf_{\sigma\in\mathrm{Sep}_{a,b}}D_p(P_{\mathcal E,\rho^{\otimes n}}\|P_{\mathcal E,\sigma^{\otimes n}})$; in the latter infimum, a single separable state $\sigma$ on the original $a\times b$ space is repeated $n$ times. Tensor powers have entries $(\rho^{\otimes n})_{(u,v),(u',v')}=\prod_{i=0}^{n-1}\rho_{(u_i,v_i),(u'_i,v'_i)}$, relabeled by finite index sets. The two asymptotic quantities are $R_\infty=\limsup_{j\to\infty}R_{j+1}/(j+1)$ and $T_\infty=\limsup_{j\to\infty}T_{j+1}/(j+1)$. Finally, $\mathrm{Sep}_{a,b}$ consists of finite convex combinations of products of local density matrices, and $E_{\mathrm R}(\omega)=\inf_{\sigma\in\mathrm{Sep}_{a,b}}D_2(\omega\|\sigma)$, where $D_2=+\infty$ unless both arguments are density matrices and $\ker\sigma\subseteq\ker\omega$; otherwise $D_2(\omega\|\sigma)=\max\{\operatorname{Re}\operatorname{tr}(\omega\log\omega-\omega\log\sigma),0\}/\log2$, with matrix functions defined by functional calculus and the real logarithm assigned its total value $0$ at zero. The transpose is $\omega^{T_2}_{(i,j),(k,l)}=\omega_{(i,l),(k,j)}$, and local dimension $1$ is included.