Asymptotically Optimal Mixed-State Cloning

Abstract

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 $m_n/n\toγ>1$. 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 $$ F_\rm{orb}(γ,p) =\prod_{i<j} \frac{\sqrtγ+\sqrt{q_{ij}(γ-1+q_{ij})}} {γ+q_{ij}}, \qquad q_{ij}=\frac{p_j}{p_i}. $$ When the spectrum is unknown, one spectrum-independent cloner attains $$ F_\rm{univ}(γ,p) =\left(\frac{2\sqrtγ}{1+γ}\right)^{(d-1)/2} F_\rm{orb}(γ,p), $$ 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 $ρ=P/r$, with $P$ a rank-$r$ orthogonal projector, we separately prove the optimum $γ^{-r(d-r)/2}$. 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 $1<r<d$.