Theorem 3, Eqs. (19)–(20): success-weighted hidden nonlocality

For all positive integers $a,b$ and every positive semidefinite complex matrix $\rho$ of trace $1$ on $\mathbb C^a\otimes\mathbb C^b$, the following bounds hold:

Here $(X^\Gamma)_{(i,j),(k,l)}=X_{(i,l),(k,j)}$, and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$. A separable density matrix means a finite sum $\sum_t w_t A_t\otimes B_t$, where $w_t\ge0$, $\sum_t w_t=1$, and every $A_t,B_t$ is positive semidefinite with trace $1$. The relative entropy of entanglement is $E_{\mathrm{rel}}(\theta)=\inf_{\sigma\text{ separable density}}D_2(\theta\|\sigma)$, where $D_2(\theta\|\sigma)=+\infty$ unless both arguments are density matrices and $\ker\sigma\subseteq\ker\theta$; otherwise it is $\max\{0,\operatorname{Re}\operatorname{tr}[f(\theta)-\theta\log\sigma]/\log2\}$, with $f(x)=x\log x$, matrix functions interpreted by continuous functional calculus, and the real logarithm taking value $0$ at $0$. To define $R_H,T$, a Bell experiment consists of a map $\Phi$ in the closure of finite LOCC maps, with positive finite output dimensions, followed by local POVMs $A_{x,u},B_{y,v}$ in a scenario with positive finite numbers of inputs and outputs; each POVM consists of positive semidefinite matrices summing to the identity. Finite LOCC maps are generated by identity maps and finite complete local Kraus operations, followed by outcome-dependent maps of the same form and summed over outcomes; the closure is taken in the topology on these maps. Its measured box on $\theta$ is $P_\theta(x,y,u,v)=\operatorname{Re}\operatorname{tr}[(A_{x,u}\otimes B_{y,v})\Phi(\theta)]$. For boxes $P,Q$ and a nonnegative input distribution $\pi$ summing to $1$, put $D_\pi(P\|Q)=\sum_{x,y}\pi(x,y)D_{\mathrm{cl},2}(P_{xy}\|Q_{xy})$, where $D_{\mathrm{cl},2}(p\|q)=+\infty$ if some $p_i>0=q_i$, and otherwise equals the nonnegative part of $\sum_{i:p_i\ne0}p_i\log(p_i/q_i)/\log2$. A finite local box is a finite convex combination of deterministic boxes $\mathbf1_{\{u=f(x),\,v=g(y)\}}$; define $\mathcal N(P)=\sup_\pi\inf_{Q\text{ finite local}}D_\pi(P\|Q)$. Write $\rho^{[n]}$ for the grouped tensor power whose entries are $\prod_{t=1}^n\rho_{(i_t,j_t),(k_t,l_t)}$. Let $T_n(\rho)$ be the supremum over Bell experiments on $\rho^{[n]}$ of $\sup_\pi\inf_{\sigma\text{ separable density on }\mathbb C^a\otimes\mathbb C^b}D_\pi(P_{\rho^{[n]}}\|P_{\sigma^{[n]}})$. Let $H_n(\rho)$ be the supremum over such experiments and complete finite local Kraus families, with one selected Kraus operator on each side, of $0$ when the selected output branch $\omega=K\Phi(\rho^{[n]})K^\dagger$ has trace $p=0$, and otherwise of $p\,\mathcal N(P_{\omega/p})$, using the experiment’s local POVMs and $K$ equal to the tensor product of the selected operators. Then $T(\rho)=\limsup_{n\to\infty}T_n(\rho)/n$ and $R_H(\rho)=\limsup_{n\to\infty}H_n(\rho)/n$, with positive block lengths and all these quantities valued in $[0,+\infty]$.