Unextendible product bases from tile structures in bipartite systems
Unextendible product bases from tile structures in bipartite systems
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DOI：
10.1088/1751-8121/acb099
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发表时间：
2023-01
期刊：
影响因子：
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通讯作者：
Siwen You;Chen Wang;Fei Shi;Sihuang Hu;Yiwei Zhang
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文献类型：
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作者：
Siwen You;Chen Wang;Fei Shi;Sihuang Hu;Yiwei Zhang
不可扩积基是量子信息理论中最基本的概念之一，它的构造备受关注。最近，Shi等人（2020 Phy. Rev. A 101 062329）经由瓦片结构方法构建UPB。他们表明，具有某些属性的m × n网格中的瓦片结构会导致Cm Cn中UPB的显式构造。如果图块结构恰好包含s个图块，则UPB的大小为mn−s+1。在本文中，我们深入分析了瓦片结构方法的范围，表明m × m网格中所需的瓦片结构最多包含m24 +m个瓦片，因此通过该方法构造的Cm Cm中UPB的最小尺寸是m2-m24-m+1。这种瓦片结构和UPB被称为极值瓦片结构和极值UPB。我们提供的算法，用于构建可能的候选人的极值瓦片结构和检查其有效性。在应用上，我们将极值UPB与大秩的局域不可约纠缠态和束缚纠缠态联系起来。
Unextendible product basis (UPB) is one of the most fundamental concepts in the field of quantum information theory, and its constructions attract much attention. Recently, Shi et al (2020 Phy. Rev. A 101 062329) constructed UPBs via a tile structure approach. They showed that a tile structure in an m × n grid with certain properties leads to explicit constructions of UPBs in Cm⊗Cn . The size of the UPB is mn−s+1 if the tile structure contains exactly s tiles. In this paper, we deeply analyze the extent of the tile structure approach, by showing that a desired tile structure in an m × m grid contains at most ⌊m24⌋+m tiles, and thus the smallest size of a UPB in Cm⊗Cm constructed by this approach is m2−⌊m24⌋−m+1 . Such tile structures and UPBs are referred to as extremal tile structures and extremal UPBs. We provide algorithms both for constructing possible candidates of extremal tile structures and checking their validity. For applications, we connect the extremal UPBs to local irreducibility and bound entangled states with large ranks.