Cycle index polynomials and generalized quantum separability tests
Cycle index polynomials and generalized quantum separability tests
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DOI：
10.1098/rspa.2022.0733
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发表时间：
2022-08
期刊：
影响因子：
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通讯作者：
Zachary P. Bradshaw;Margarite L. LaBorde;M. Wilde
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文献类型：
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作者：
Zachary P. Bradshaw;Margarite L. LaBorde;M. Wilde
纯二部态的一部分的混和性决定了整体状态是否是一个可分离的、不纠缠的状态。在这里，我们考虑混合的量子计算测试，并推导出当状态副本数量变大时这种测试的接受概率的精确表达式。证明了该表达式的解析形式是由对称群Sk的循环指标多项式给出的，该群本身与贝尔多项式有关。在此基础上，我们导出了一组量子可分性检验，每个可分性检验都由一个有限群生成；对于所有这类算法，我们证明了接受概率由群的循环指数多项式决定。最后，我们生成并分析了这些测试的显式电路结构，表明对应于对称群和循环群的测试可以分别使用O（k2）和O（klog (k)）受控交换门执行，其中k是被测试状态的副本数。
The mixedness of one share of a pure bipartite state determines whether the overall state is a separable, unentangled one. 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 Sk, 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 analyse explicit circuit constructions for these tests, showing that the tests corresponding to the symmetric and cyclic groups can be executed with O(k2) and O(klog(k)) controlled-SWAP gates, respectively, where k is the number of copies of the state being tested.