Irreversibility in the complex spectral parametrization

For every complex matrix $\rho$ on $\{0,1,2\}\times\{0,1,2\}$, real $p$, and vectors $a,b\in\mathbb C^3$, assume that $\rho\ge0$, $\operatorname{Tr}\rho=1$, $0<p<1$, and, on defining

one has $\|w\|=1$, $\langle u,w\rangle=0$, and $\rho=p\,uu^\dagger+(1-p)\,ww^\dagger$. Here kets are coordinate basis vectors, norms and inner products are Euclidean, and $vv^\dagger$ has entries $v_x\overline{v_y}$; the vectors $a,b$ themselves are not separately required to be normalized or orthogonal. The conclusion is $D_{\mathrm{PPT}}(\rho)<1$ and $C_{\mathrm{PPT}}(\rho)=1$, with both quantities extended-real valued. Specifically, write $(\Gamma X)_{(i,j),(i',j')}=X_{(i,j'),(i',j)}$, $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$, and let $\Phi_k$ have entry $1/k$ when both row and column have equal party indices and $0$ otherwise. A channel is a finite Kraus sum $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$; it is PPT when its partial-transpose conjugate is also a channel. The quantity $D_{\mathrm{PPT}}(\rho)$ is the supremum of $r\ge0$ such that, for every $\varepsilon>0$, all sufficiently large natural $n$ admit a PPT channel taking $\rho^{\otimes n}$ within trace norm $\varepsilon$ of $\Phi_{\lceil2^{nr}\rceil}$. The quantity $C_{\mathrm{PPT}}(\rho)$ is the infimum of $r\ge0$ such that, for every $\varepsilon,\delta>0$, all sufficiently large positive natural $n$ admit $m\in\mathbb N$ and a PPT channel taking $\Phi_{2^m}$ within trace norm $\varepsilon$ of $\rho^{\otimes n}$, with $m\le n(r+\delta)$. Tensor-power entries are products of single-copy entries, party registers are grouped and relabelled by the fixed finite-index bijections, and $m=0$ is allowed.