Finite-Function-Encoding Quantum States
Finite-Function-Encoding Quantum States
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DOI：
10.22331/q-2022-05-09-708
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发表时间：
2020-12
期刊：
影响因子：
6.4
通讯作者：
Paul Appel;Alexander J. Heilman;E. W. Wertz;David W. Lyons;M. Huber;Matej Pivoluska;G. Vitagliano
中科院分区：
文献类型：
--
作者：
Paul Appel;Alexander J. Heilman;E. W. Wertz;David W. Lyons;M. Huber;Matej Pivoluska;G. Vitagliano
我们引入了有限函数编码(FFE)态，它编码任意d值逻辑函数，即模d整数环上的多元函数，并研究了它们的一些结构性质。我们还指出了多项式函数编码状态和非多项式函数编码状态之间的一些区别：前者可以与图形对象相关联，我们称之为张量边超图(TEH)，它是超图的推广，每条超图上附加一个张量来编码不同单项式的系数。为了完善这个框架，我们还引入了有限函数编码泡利(FP)算子的概念，它对应于数学上所知的广义对称群的元素。首先，利用这一机制，我们研究了与FFE态相关的稳定子群，并观察了QUDIT超图态是如何在文献[1]中引入的。\cite{2017PhRvA..95e2340S}允许使用特别简单形式的稳定剂。然后，我们研究了FFE态在局域么正(LU)下的分类，在说明了这个问题的复杂性之后，我们重点讨论了二分态的情况，特别是在局域FP操作(LFP)下的分类。我们找到了两个四元组和两个四元组的所有LU类和LFP类，并研究了其他几个特殊类，指出了最大纠缠FFE态与复Butson型Hadamard矩阵之间的关系。我们的研究还展示了FFE态的性质，特别是它们的LU分类与整数上的有限环理论之间的关系。
We introduce finite-function-encoding (FFE) states which encode arbitrary d-valued logic functions, i.e., multivariate functions over the ring of integers modulo d, and investigate some of their structural properties. We also point out some differences between polynomial and non-polynomial function encoding states: The former can be associated to graphical objects, that we dub tensor-edge hypergraphs (TEH), which are a generalization of hypergraphs with a tensor attached to each hyperedge encoding the coefficients of the different monomials. To complete the framework, we also introduce a notion of finite-function-encoding Pauli (FP) operators, which correspond to elements of what is known as the generalized symmetric group in mathematics. First, using this machinery, we study the stabilizer group associated to FFE states and observe how qudit hypergraph states introduced in Ref. \cite{2017PhRvA..95e2340S} admit stabilizers of a particularly simpler form. Afterwards, we investigate the classification of FFE states under local unitaries (LU), and, after showing the complexity of this problem, we focus on the case of bipartite states and especially on the classification under local FP operations (LFP). We find all LU and LFP classes for two qutrits and two ququarts and study several other special classes, pointing out the relation between maximally entangled FFE states and complex Butson-type Hadamard matrices. Our investigation showcases also the relation between the properties of FFE states, especially their LU classification, and the theory of finite rings over the integers.