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

arXiv:2609.35986 [quant-ph]

(or arXiv:2609.35986v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.35986 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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