Proposition 3: PPT asymptotic pure-state cost and distillation

Let $d$ be any natural number and let $\lambda=(\lambda_i)_{i=0}^{d-1}$ satisfy $\lambda_i\ge0$ for every $i$ and $\sum_i\lambda_i=1$; these assumptions exclude $d=0$. Write $\log_2 x=\log x/\log2$, let $\rho_\lambda=|\psi_\lambda\rangle\langle\psi_\lambda|$ with $\psi_\lambda=\sum_i\sqrt{\lambda_i}\,|i,i\rangle$, and let $\Omega_K=K^{-1}\sum_{i,j=0}^{K-1}|i,i\rangle\langle j,j|$ for positive $K$. A permitted transformation is a complex-linear map that preserves trace, is completely positive, and remains completely positive after partial transposition on the second subsystem both before and after the map; complete positivity means that every amplification by an identity map of finite dimension, including dimension zero, preserves positive semidefiniteness. Consider all real rates $R$ for which there are sequences of positive integers $n_j,K_j$ with $n_j\to\infty$ and $\log_2K_j/n_j\to R$, together with such transformations taking $\Omega_{K_j}$ to matrices converging in trace norm to the grouped tensor power $\rho_\lambda^{\otimes n_j}$. The infimum of these rates in the extended real numbers equals $-\sum_i\lambda_i\log_2\lambda_i$. Also, the supremum in the extended real numbers of the rates obtained in the reverse direction, using permitted transformations whose outputs on $\rho_\lambda^{\otimes n_j}$ converge in trace norm to $\Omega_{K_j}$, equals the same number:

Here the finite real expression is embedded into the extended reals, zero-probability terms contribute zero under the total real logarithm, and the trace norm of $Z$ is $\operatorname{Re}\operatorname{tr}\sqrt{Z^\dagger Z}$.