发表机构
Perimeter Institute for Theoretical Physics; QuSoft, CWI; Sorbonne Université, CNRS, LIP6; QuSoft, University of Amsterdam（Perimeter理论物理研究所; QuSoft，荷兰数学与计算机科学研究学会; 索邦大学，法国国家科学研究中心，LIP6实验室; QuSoft，阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究结构化纯态的近似克隆复杂度，针对费米子高斯态、玻色子高斯态及相位态等族，构造了最优或显式克隆通道，将复杂度从指数级降至多项式级，并统一了表示论框架。
AI 中文摘要
不可克隆定理是量子力学中的一个基石性结果，它禁止复制一般的量子信息。更定量地说，即使给定未知状态的$N$个副本，任何产生$(N+1)$个副本状态的通道在某些输入上也会产生非零的迹距离误差。反过来，我们可以问近似$N\rightarrow N+1$克隆的样本复杂度：对于期望的小误差$\epsilon$，需要多大的$N$？对于希尔伯特空间$H$上的任意纯态，答案是$N\simeq \dim H/\epsilon$——对于大多数感兴趣的$H$而言，这是天文数字般巨大的。如果输入被承诺位于一个结构化族中呢？我们研究了对多体物理和量子信息至关重要的多种族：$n$模费米子高斯态和Slater态、$n$模玻色子高斯态以及qudit相位态。对于费米子，我们构造了最优克隆通道，将复杂度从指数级降低到$n$的多项式级。对于玻色子高斯态，我们构造了一个显式克隆器，证明了多项式复杂度，且对态的能量没有任何约束；然而，该通道在一般情况下并非最优。费米子和玻色子的结果源于一个统一的表示论框架，该框架推广了Werner以及Chiribella和Yang的先前工作；它也给出了克隆器的酉实现。此外，我们解释了为什么对于许多族，克隆复杂度与族流形的维度成线性关系，这是通过在高$N$下对框架势进行受控鞍点评估得出的。同样的计算也解释了层析成像复杂度的类似缩放。上述框架不涵盖相位态；对于这些态，我们给出了一个具有最优克隆保真度的通道的独立构造。我们的工作补充了最近关于克隆稳定子态的结果。
英文摘要
The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given $N$ copies of an unknown state, any channel producing an $(N+1)$-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate $N\rightarrow N+1$ cloning: for a desired small error $ε$, what $N$ suffices? For arbitrary pure states on a Hilbert space $H$, the answer is $N\simeq \dim H/ε$ - astronomically large for most $H$ of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information: $n$-mode fermionic Gaussian and Slater states, $n$-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in $n$. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high $N$. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.
Comments44 pages main text, 40 pages appendices
