论文标题
循环指数多项式和广义量子可分离性测试
Cycle Index Polynomials and Generalized Quantum Separability Tests
论文作者
论文摘要
一个纯二分国家的一部分的混合性决定了整个状态是否是可分离的，未进入状态。在这里，我们考虑混合性的量子计算测试，并得出了此类测试的接受概率的精确表达，因为状态的副本数量变大。我们证明，该表达式的分析形式由对称组$ s_k $的循环索引多项式给出，该循环本身与贝尔多项式有关。这样做之后，我们得出了一个量子可分离性测试的家族，每个测试都是由有限组生成的。对于所有此类算法，我们表明接受概率取决于该组的循环指数多项式。最后，我们为这些测试生成和分析明确的电路结构，表明可以使用$ O（k^2）$和$ O（k \ log（k））$ Condroled-Swap Gates进行对称和环状组对应的测试，其中$ k $是该州的副本数量。
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.
