发表机构
School of Computer and Communication Sciences - EPFL（洛桑联邦理工学院计算机与通信科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明算子Schmidt秩为三的2×n二分态可分解为rank(ρ)个纯乘积态或至多n+1个混合乘积态，后者紧致，并提供构造性方法，推广了已有结果。
AI 中文摘要
我们证明，一个量子比特与一个n能级系统耦合的每个二分态$\ ho$，当其算子Schmidt秩等于三时，可以写成$\ ext{rank}(\ ho)$个纯乘积态的混合，并且一般可以写成至多$n+1$个混合乘积态的混合。第二个结果在如下意义上是紧的：存在不能用更少数量的项进行分解的态。我们的证明给出了获得这些分解的构造性方法，涉及由态显式确定的一个算子的适当酉膨胀的特征分解。我们的框架恢复了先前关于纯乘积态分解长度的已知结果，并且也能处理混合态分解。对于纯乘积态，我们的分解与现有分解不同。
英文摘要
We prove that every bipartite state $ρ$ of a qubit coupled to an $n$-level system whose operator Schmidt rank equals three can be written as a mixture of $\text{rank}(ρ)$ pure product states, and as a mixture of at most $n+1$ mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.
Comments28 pages, 6 figures
