In quantum mechanics and quantum information, to establish the orthogonal bases is a useful means. The existence of unextendible product bases impels us to study the `entanglement bases' problems. In this paper, the concepts of entanglement bases and exact-entanglement bases are defined, and a theorem about exact-entanglement bases is given. We discuss the general structures of the orthogonal complete bases. Two examples of applications are given. At last, we discuss the problem of transformation of the general structure forms.In quantum mechanics and quantum information, to establish the orthogonal bases is a useful means. The existence of unextendible product bases impels us to study the `entanglement bases' problems. In this paper, the concepts of entanglement bases and exact-entanglement bases are defined, and a theorem about exact-entanglement bases is given. We discuss the general structures of the orthogonal complete bases. Two examples of applications are given. At last, we discuss the problem of transformation of the general structure forms.

arXiv:quant-ph/0401156Physical Review AOct 14, 20041 citation

Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system

Guang-Bao Xu, Ying-Hui Yang, Qiao-Yan Wen +2 more

As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.

arXiv:1512.06485Scientific ReportsAug 9, 201654 citations

Quantum Entanglement: An Analysis via the Orthogonality Relation

Shengyang Zhong

Foundations of PhysicsAug 1, 20231 citation

Strong nonlocal sets of UPB

Bichen Che, Zhao Dou, Min Lei +1 more

The unextendible product bases (UPBs) are interesting members from the family of orthogonal product states. In this paper, we investigate the construction of 3-qubit UPB with strong nonlocality of different sizes. First, a UPB set in ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{3}}$ of size 12 is presented based on the Shifts UPB, the structure of which is described by mapping the system to a $3\times 3\times 3$ Rubik's Cube. After observing the orthogonal graph of each qubit, we provide a general method of constructing UPB in ${{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}$ of size ${{\left( d-1 \right)}^{3}}+3\left( d-2 \right)+1$ . Second, for the more general case where the dimensions of qubits are different, we extend the tile structure to 3-qubit system and propose a Tri-tile structure for 3-qubit UPB. Then, by means of this structure, a ${{C}^{4}}\otimes {{C}^{4}}\otimes {{C}^{5}}$ system of size 30 is obtained based on a ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{4}}$ system. Similarly, we generalize this approach to ${{C}^{{{d}_{1}}}}\otimes {{C}^{{{d}_{2}}}}\otimes {{C}^{{{d}_{3}}}}$ system which has a similar composition to ${{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}$ . Our research provides a positive answer to the open questions raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist multi-qubit UPBs that can exhibit strong quantum nonlocality without entanglement.The unextendible product bases (UPBs) are interesting members from the family of orthogonal product states. In this paper, we investigate the construction of 3-qubit UPB with strong nonlocality of different sizes. First, a UPB set in ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{3}}$ of size 12 is presented based on the Shifts UPB, the structure of which is described by mapping the system to a $3\times 3\times 3$ Rubik's Cube. After observing the orthogonal graph of each qubit, we provide a general method of constructing UPB in ${{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}$ of size ${{\left( d-1 \right)}^{3}}+3\left( d-2 \right)+1$. Second, for the more general case where the dimensions of qubits are different, we extend the tile structure to 3-qubit system and propose a Tri-tile structure for 3-qubit UPB. Then, by means of this structure, a ${{C}^{4}}\otimes {{C}^{4}}\otimes {{C}^{5}}$ system of size 30 is obtained based on a ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{4}}$ system. Similarly, we generalize this approach to ${{C}^{{{d}_{1}}}}\otimes {{C}^{{{d}_{2}}}}\otimes {{C}^{{{d}_{3}}}}$ system which has a similar composition to ${{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}$. Our research provides a positive answer to the open questions raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist multi-qubit UPBs that can exhibit strong quantum nonlocality without entanglement.

arXiv:2106.08699Jun 16, 20212 citations

Characterization of multiqubit pure-state entanglement (3 pages)

Z. Chen

Physical Review AJan 1, 20060 citations

{{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{3}}

of size 12 is presented based on the Shifts UPB, the structure of which is described by mapping the system to a

3\times 3\times 3

Rubik's Cube. After observing the orthogonal graph of each qubit, we provide a general method of constructing UPB in

{{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}

of size

{{\left( d-1 \right)}^{3}}+3\left( d-2 \right)+1

. Second, for the more general case where the dimensions of qubits are different, we extend the tile structure to 3-qubit system and propose a Tri-tile structure for 3-qubit UPB. Then, by means of this structure, a

{{C}^{4}}\otimes {{C}^{4}}\otimes {{C}^{5}}

system of size 30 is obtained based on a

{{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{4}}

system. Similarly, we generalize this approach to

{{C}^{{{d}_{1}}}}\otimes {{C}^{{{d}_{2}}}}\otimes {{C}^{{{d}_{3}}}}

system which has a similar composition to

{{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}

. Our research provides a positive answer to the open questions raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist multi-qubit UPBs that can exhibit strong quantum nonlocality without entanglement.