Asymptotically Optimal Mixed-State Cloning

We determine asymptotic global fidelities of cloning generic mixed states with simple (non-degenerate) full-rank spectra. In fixed dimension $d$, let $n$ inputs and $m_n$ outputs satisfy $m_n\mathrm{/}n\to \gamma >1$. For a known simple full-rank spectrum $p_1>\cdots >p_d>0$, we prove that the optimal root fidelity between the entire output and the product target state is $$F_{\mathrm{o}\mathrm{r}\mathrm{b}}(\gamma ,p)=\prod _{i<j}\frac{\sqrt{\gamma }+\sqrt{q_{ij}(\gamma -1+q_{ij})}}{\gamma +q_{ij}},q_{ij}=\frac{p_j}{p_i}\mathrm{.}$$ When the spectrum is unknown, one spectrum-independent cloner attains $$F_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}(\gamma ,p)={\left(\frac{2\sqrt{\gamma }}{1+\gamma }\right)}^{(d-1)\mathrm{/}2}{F}_{\mathrm{o}\mathrm{r}\mathrm{b}}(\gamma ,p),$$ uniformly on compact subsets of the simple full-rank chamber and is minimax optimal. The two factors quantify spectral and eigenbasis uncertainty. Our channels combine classical processing of Schur–Weyl labels with a covariant Cartan-sector channel. Local asymptotic normality and a Gaussian fidelity bound against arbitrary channels yield the converses. For flat projector states $\rho =P\mathrm{/}r$, with $P$ a rank-$r$ orthogonal projector, we separately prove the optimum ${\gamma }^{-r(d-r)\mathrm{/}2}$. Furthermore, we estimate the exact asymptotic fidelity of purify–clone–trace (PCT), and show that our cloners strictly outperform PCT at every fixed gain for simple full-rank spectra and rank-$r$ projector states with $1<r<d$.