Exact PPT yield and regularized Rains value of rho_v

Let $v_j=(|0,j\rangle-|j,0\rangle)/\sqrt2$ for $j=1,2$, with coordinate basis kets on $\{0,1,2\}\times\{0,1,2\}$, and let $\rho_V=\tfrac12(v_1v_1^\dagger+v_2v_2^\dagger)$, where a vector’s outer product has entries $v_u\overline{v_v}$. Then, in the extended real line,

Write $(\Gamma X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$ and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. A channel is a finite Kraus sum $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$; a PPT channel is one whose input-output partial-transpose conjugate is also a channel. The quantity $D_{\mathrm{PPT}}(\rho_V)$ is the supremum of real $r\ge0$ such that, for every $\varepsilon>0$, every sufficiently large natural $n$ admits a PPT channel satisfying $\|\Lambda(\widehat{\rho_V^{\otimes n}})-\Phi_{\lceil2^{nr}\rceil}\|_1\le\varepsilon$. Tensor powers have entries given by products, with each party’s registers grouped; the hat means the fixed relabelling by finite initial segments. The matrix $\Phi_k$ has entry $1/k$ if its row and column each have equal party indices, and $0$ otherwise. Finally,

where $D_2(X\|S)=\operatorname{Re}\operatorname{Tr}(X\log X-X\log S)/\log2$ if $\ker S\subseteq\ker X$, and $+\infty$ otherwise. Matrix logarithms use spectral functional calculus with $\log0=0$. The matrices $S$ in this infimum are not required to have trace $1$.