Proposition 1: positive-coefficient bound and near-unit distillable key

Define $H_m$ as follows: with $d=m^2$, let $F_d$ swap the two $d$-dimensional factors, put $W_\pm=(I\pm F_d)/(d^2\pm d)$, $T_1=((W_++W_-)/2)^{\otimes m}$, and $T_2=W_+^{\otimes m}$, grouping the first coordinates of the tensor factors on one side and the second coordinates on the other. On two parties each indexed by $\{0,1\}\times\{0,\ldots,d-1\}^{m\times m}$, the key blocks of $H_m$ are $c_m((T_1+T_2)/6)^{\otimes m}$ at $00,00$ and $11,11$, $c_m((T_1-T_2)/6)^{\otimes m}$ at $00,11$ and $11,00$, and $c_m(T_2/6)^{\otimes m}$ at $01,01$ and $10,10$, with all other blocks zero and $c_m=(2(1/3)^m+2(1/6)^m)^{-1}$; tensor powers here mean products of the corresponding matrix entries with the same grouping by party. For every natural $m\ge2$, $H_m$ is positive semidefinite of trace $1$, its second-party transpose $H_m^{T_2}$ is positive semidefinite, and, for every finite Bell scenario with positive input and output sizes and every real coefficient family $s_{xyab}\ge0$,

Here $\operatorname{paired}(\alpha,\beta)_{((i,k),(j,l)),((i',k'),(j',l'))}=\alpha_{(i,j),(i',j')}\beta_{(k,l),(k',l')}$; $V_s$ is the supremum of the measured expression $\sum s_{xyab}\operatorname{Re}\operatorname{tr}((M_{a|x}\otimes N_{b|y})\omega)$ over local positive semidefinite measurements summing to the identity, $C_s=\sup_{f,g}\sum_{x,y}s_{xy,f(x),g(y)}$, and $Q_s$ is its supremum over all positive finite local dimensions, density matrices, and local measurements. In addition, after relabeling each party by a chosen bijection with a finite initial segment of the natural numbers, the extended-nonnegative-real distillable key of $\operatorname{paired}(H_{j+2}^{T_2},H_{j+2})$ converges to $1$ as $j\to\infty$. For a state $\omega$ on local dimensions $a,b$, this distillable key is the supremum of the nonnegative real rates $r$, embedded in $[0,\infty]$, with the following property: for every $\epsilon,\delta>0$, all sufficiently large positive $n$ admit positive integers $d,s,t$, a density matrix $\tau$ on $\mathbb C^s\otimes\mathbb C^t$, unitaries $U_0,\ldots,U_{d-1}$ on that shield, and a finite LOCC map taking $\omega^{\otimes n}$ to within trace norm $\epsilon$ of the matrix whose key block $ii,jj$ is $U_i\tau U_j^\dagger/d$ and whose other key blocks vanish, with $(\log_2d)/n\ge r-\delta$. Finite LOCC here consists of the identity and finite recursive Alice or Bob Kraus measurements, each satisfying $\sum_iE_i^\dagger E_i=I$, followed by a branch-dependent finite LOCC map, with the branch outputs summed. The trace norm is $\|Z\|_1=\operatorname{Re}\operatorname{tr}\sqrt{Z^\dagger Z}$, and second-party transpose has entries $Z^{T_2}_{(i,j),(k,l)}=Z_{(i,l),(k,j)}$.