混合态的最优克隆
Optimal cloning of mixed states
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中文总结 AI 辅助
该研究证明混合态近似克隆及近似转置任务中，所需副本数均为Θ(krd/ε)，随机纯化信道的复杂度是最优的，无法进一步降低。
中文摘要 AI 辅助
我们考虑量子态的近似克隆问题：给定n个未知态ρ∈ℂ^(d×d)的副本，制备一个与ρ^(⊗(n+k))具有高保真度的(n+k)副本态。Werner的纯态克隆器是纯态情形下的最优信道，表明要以保真度1-ε克隆k个额外的未知纯态副本，需要且仅需要n=Θ(kd/ε)个副本。随机纯化信道是Werner克隆器向混合态输入的直接扩展：给定n个混合态副本，随机纯化输入，在更大的希尔伯特空间中应用Werner信道，然后求迹消去辅助寄存器，该方法使用n=O(krd/ε)个副本，可克隆秩为r的态。我们研究是否能做得更好，结果表明答案是否定的：必须使用n=Ω(krd/ε)个副本。我们通过研究投影克隆的特殊情况来证明这一下界，其中输入态ρ被限定为P/r的形式，P是秩为r的正交投影算子。作为我们技术的进一步应用，我们考虑密切相关的量子态近似转置问题，即试图将ρ^(⊗n)转换为与(ρ^T)^(⊗k)具有高保真度的k副本态，在此任务中，我们同样证明需要且仅需要n=Θ(krd/ε)个副本。
英文摘要
We consider the problem of approximate cloning of quantum states: given $n$ copies of an unknown state $ρ\in \mathbb{C}^{d \times d}$, prepare an $(n+k)$-copy state with high fidelity to $ρ^{\otimes (n+k)}$. Werner's pure state cloner is the optimal channel for the pure state case, and shows that $n = Θ(kd/\varepsilon)$ copies are necessary and sufficient to clone $k$ additional copies of an unknown pure state to fidelity $1-\varepsilon$. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given $n$ copies of a mixed state, randomly purify your input, apply Werner's channel in the larger Hilbert space, and then trace out the auxiliary registers. This gives a mixed state cloner using $n = O(krd/\varepsilon)$ copies to clone rank-$r$ states. Can one do any better? We show that the answer is no: one must use $n = Ω(krd/\varepsilon)$ copies. We prove our lower bound by studying the special case of projector cloning, in which the input state $ρ$ is promised to be of the form $P/r$, where $P$ is a rank-$r$ orthogonal projector. As a further application of our techniques, we consider the closely related problem of approximate transposition of quantum states, where one seeks to convert $ρ^{\otimes n}$ to a $k$-copy state with high fidelity to $(ρ^T)^{\otimes k}$. Here, we again show $n = Θ(krd/\varepsilon)$ copies are necessary and sufficient for this task.
