循环指数多项式与广义量子可分性测试
Cycle Index Polynomials and Generalized Quantum Separability Tests
AI总结：
本文研究量子混合度与可分性测试，证明测试接受概率由有限群的循环指数多项式给出，构建了对称群、循环群对应测试的显式线路，门复杂度分别为$O(k^2)$与$O(k\log(k))$。
摘要：
纯双体态的一个子系统的混合度决定了整体态是否为可分的、非纠缠态。本文研究混合度的量子计算测试，推导了当态的副本数增大时，这类测试接受概率的精确表达式。我们证明该表达式的解析形式由对称群$S_k$的循环指数多项式给出，而该多项式本身与Bell多项式相关。在此基础上，我们推导了一族量子可分性测试，每个测试都由一个有限群生成；我们证明所有这类算法的接受概率都由对应群的循环指数多项式决定。最后，我们构建并分析了这些测试的显式线路构造，表明对应于对称群和循环群的测试可分别用$O(k^2)$和$O(k\log(k))$个受控-SWAP门实现，其中$k$是被测试态的副本数。
英文摘要：
The mixedness of one share of a pure bipartite state determines whether the overall state is a separable, unentangled state. Here we consider quantum computational tests of mixedness, and we derive an exact expression of the acceptance probability of such tests as the number of copies of the state becomes larger. We prove that the analytical form of this expression is given by the cycle index polynomial of the symmetric group $S_k$, which is itself related to the Bell polynomials. After doing so, we derive a family of quantum separability tests, each of which is generated by a finite group; for all such algorithms, we show that the acceptance probability is determined by the cycle index polynomial of the group. Finally, we produce and analyze explicit circuit constructions for these tests, showing that the tests corresponding to the symmetric and cyclic groups can be executed with $O(k^2)$ and $O(k\log(k))$ controlled-SWAP gates, respectively, where $k$ is the number of copies of the state being tested.
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