One-ebit cost and asymptotic PPT relative entropy of rho_v

Let $v_j=(|0,j\rangle-|j,0\rangle)/\sqrt2$ for $j=1,2$, where the kets are coordinate basis vectors on $\{0,1,2\}\times\{0,1,2\}$, and set $\rho_V=\tfrac12(v_1v_1^\dagger+v_2v_2^\dagger)$, with $(vv^\dagger)_{u,v}=v_u\overline{v_v}$. The assertion is

as equalities in the extended real line. Here $(\Gamma X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$, and a channel is a finite Kraus sum $X\mapsto\sum_jK_jXK_j^\dagger$ satisfying $\sum_jK_j^\dagger K_j=I$. A PPT channel is a channel whose partial-transpose conjugate $X\mapsto\Gamma_{\rm out}\Lambda(\Gamma_{\rm in}X)$ is also a channel. The quantity $C_{\mathrm{PPT}}(\rho_V)$ is the infimum of real $r\ge0$ such that, for every $\varepsilon,\delta>0$, all sufficiently large positive natural $n$ admit $m\in\mathbb N$, including $m=0$, and a PPT channel with $m\le n(r+\delta)$ and $\|\Lambda(\Phi_{2^m})-\widehat{\rho_V^{\otimes n}}\|_1\le\varepsilon$. The matrix $\Phi_k$ has entry $1/k$ when both row and column have equal party indices, and $0$ otherwise; tensor-power entries are products of the single-copy entries, with party registers grouped and then relabelled by the fixed finite-index bijections indicated by the hat. The other quantity is

where $D_2(X\|S)=\operatorname{Re}\operatorname{Tr}(X\log X-X\log S)/\log2$ when $\ker S\subseteq\ker X$, and $D_2(X\|S)=+\infty$ otherwise, with matrix logarithms interpreted by spectral functional calculus and $\log0=0$.