Every rank-two two-qutrit antisymmetric state is PPT-irreversible

For every complex matrix $\rho$ indexed by $\{0,1,2\}\times\{0,1,2\}$, assume that $\rho$ is positive semidefinite, $\operatorname{Tr}\rho=1$, its column-space dimension over $\mathbb C$ is exactly $2$, and

Then

These quantities take values in the extended real line and have the following literal definitions. Write $(\Gamma X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$, and $(\Phi_k)_{(i,j),(i',j')}=k^{-1}$ if $i=j$ and $i'=j'$, and $0$ otherwise. A channel is a map $X\mapsto\sum_jK_jXK_j^\dagger$ for a finite family satisfying $\sum_jK_j^\dagger K_j=I$; it is a PPT channel when both it and $X\mapsto\Gamma_{\rm out}\Lambda(\Gamma_{\rm in}X)$ are channels. The matrix $\rho^{\otimes n}$, with the copies of each party grouped together, has entries $\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}$. The quantity $D_{\mathrm{PPT}}(\rho)$ is the supremum of real $r\ge0$ such that, for every tolerance $\varepsilon>0$, 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 real $r\ge0$ such that, for every $\varepsilon,\delta>0$, all sufficiently large positive natural $n$ admit a natural $m\ge0$ and a PPT channel taking $\Phi_{2^m}$ within trace norm $\varepsilon$ of $\rho^{\otimes n}$, with $m\le n(r+\delta)$. In these definitions the grouped input or output bases are relabelled by the fixed bijections with finite initial segments used in the code.