发表机构
Leinweber Institute for Theoretical Physics, Stanford University; Zyphra（莱因韦伯理论物理研究所，斯坦福大学; Zyphra）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了通用混合态克隆的渐近最优保真度，给出了已知谱和未知谱下的精确公式，并证明所提克隆器优于纯化-克隆-迹方案。
AI 中文摘要
我们确定了具有简单（非简并）满秩谱的通用混合态克隆的渐近全局保真度。在固定维度 $d$ 中，设 $n$ 个输入和 $m_n$ 个输出满足 $m_n/n\to\gamma>1$。对于已知的简单满秩谱 $p_1>\cdots>p_d>0$，我们证明了整个输出与乘积目标态之间的最优根保真度为 $$ F_{\rm{orb}}(\gamma,p) =\prod_{i<j} \frac{\sqrt{\gamma}+\sqrt{q_{ij}(\gamma-1+q_{ij})}} {\gamma+q_{ij}}, \qquad q_{ij}=\frac{p_j}{p_i}. $$ 当谱未知时，一个与谱无关的克隆器在简单满秩腔的紧致子集上一致地达到 $$ F_{\rm{univ}}(\gamma,p) =\left(\frac{2\sqrt\gamma}{1+\gamma}\right)^{(d-1)/2} F_{\rm{orb}}(\gamma,p), $$ 并且是极小极大最优的。这两个因子分别量化了谱和本征基的不确定性。我们的信道将 Schur--Weyl 标签的经典处理与协变 Cartan 扇区信道相结合。局部渐近正态性和针对任意信道的高斯保真度界给出了逆命题。对于平坦投影态 $\rho=P/r$，其中 $P$ 是秩为 $r$ 的正交投影算子，我们单独证明了最优值为 $\gamma^{-r(d-r)/2}$。此外，我们估计了纯化-克隆-迹（PCT）的精确渐近保真度，并表明对于简单满秩谱和 $1<r<d$ 的秩 $r$ 投影态，我们的克隆器在每个固定增益下都严格优于 PCT。
英文摘要
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 $1<r<d$.
Comments28+24pp; comments are welcome
