Unnumbered after Eq. (21): dimension-dependent vanishing

Both of the following universally quantified assertions hold. First, let $a_j,b_j$ be arbitrary sequences of positive integers, $\rho_j$ arbitrary complex matrices on $\mathbb C^{a_j}\otimes\mathbb C^{b_j}$, and $\epsilon_j$ arbitrary real numbers, indexed by every $j\in\mathbb N$, including $j=0$. Suppose, for each $j$, that $\rho_j^\Gamma\succeq0$, that $\rho_j$ is positive semidefinite of trace $1$, that there is a separable density matrix $\sigma_j$ with $\|\rho_j^\Gamma-\sigma_j^\Gamma\|_1\le\epsilon_j$, and that $0<\epsilon_j<\tfrac12$. If

then $R(\rho_j)\to0$ and $R_H(\rho_j)\to0$ in $[0,+\infty]$. Second, for every sequence of unitary matrices $U_j$ of size $j+1$ such that every entry has modulus $1/\sqrt{j+1}$, both $R(\widehat\rho_j)\to0$ and $R_H(\widehat\rho_j)\to0$, where $\widehat\rho_j$ is the following explicitly defined matrix, relabelled on each party by the chosen finite-set equivalence with $\{0,\ldots,2(j+1)-1\}$. Set $d=j+1$, define a shield matrix on $\mathbb C^d\otimes\mathbb C^d$ by $X_{(r,s),(t,u)}=(U_j)_{r,s}/(d\sqrt d)$ when $(t,u)=(s,r)$, and $0$ otherwise, and put $Y=\sqrt d\,X^\Gamma$ and $p=(\sqrt d+1)^{-1}$. On two binary key registers and this shield, define $Z$ by its key blocks:

with every other key block zero; regroup $((\alpha,\beta),(r,s))$ as $((\alpha,r),(\beta,s))$ before the finite-set relabelling. This assertion includes $d=1$. In both assertions, $\Gamma$ transposes the second party, $(X^\Gamma)_{(i,j),(k,l)}=X_{(i,l),(k,j)}$, and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$; a separable density matrix is a finite convex combination of tensor products of local positive semidefinite trace-$1$ matrices. The quantities $R,R_H$ have the following literal meaning. A Bell experiment consists of a map in the closure of finite LOCC maps, with positive finite output dimensions, followed by local POVMs in any positive finite Bell scenario; finite LOCC maps are generated by identities and complete finite local Kraus operations with outcome-dependent continuations. For a box $P$, put $\mathcal N(P)=\sup_\pi\inf_Q D_\pi(P\|Q)$, where $\pi$ ranges over input probability distributions, $Q$ ranges over finite convex combinations of deterministic local response boxes, and $D_\pi=\sum_{x,y}\pi(x,y)D_{\mathrm{cl},2}(P_{xy}\|Q_{xy})$. Here $D_{\mathrm{cl},2}(p\|q)$ is $+\infty$ if some $p_i>0=q_i$, and otherwise is the nonnegative part of $\sum_{i:p_i\ne0}p_i\log(p_i/q_i)/\log2$. Measurement probabilities are $\operatorname{Re}\operatorname{tr}[(A_{x,u}\otimes B_{y,v})\theta]$. For each positive block length $n$, $R_n(\rho)$ is the supremum of $\mathcal N$ of the measured outputs of experiments on the grouped tensor power $\rho^{[n]}$. The corresponding $H_n(\rho)$ additionally optimizes over complete finite local Kraus families and one selected operator on each side after the experiment’s map: a branch $\omega$ with trace $q=0$ contributes $0$, and otherwise contributes $q\,\mathcal N$ of the experiment’s measurements on $\omega/q$. Finally, $R(\rho)=\limsup_{n\to\infty}R_n(\rho)/n$ and $R_H(\rho)=\limsup_{n\to\infty}H_n(\rho)/n$.