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A conjecture on the number of independent vector variables in a multiqubit orthogonal product basis.

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Title: A conjecture on the number of independent vector variables in a multiqubit orthogonal product basis.
Authors: Chen, Yilin1 (AUTHOR) 21091001@buaa.edu.cn, Chen, Lin1 (AUTHOR) 1700398750@qq.com
Source: International Journal of Quantum Information. Jun2026, Vol. 24 Issue 4, p1-25. 25p.
Subjects: Orthonormal basis, Independent variables, Quantum information theory, Quantum computing, Matrices (Mathematics), Quantum states
Abstract: In this paper, we investigate the number of independent vector variables in an n -qubit Orthogonal Product Basis (OPB). Here, the term "independent vector variable" refers to a structural element in the construction of a Unextendible Orthogonal Matrix (UOM). Intuitively, since fixing one state | x 〉 in a qubit basis { | x 〉 , | x ′ 〉 } uniquely determines its orthogonal partner | x ′ 〉 , we treat the pair as a single degree of freedom — one variable — rather than two. Based on this counting measure, we propose the conjecture that the number is upper bounded by 2 n − 1. We show that if the number of orthogonal pairs of variables is minimum, then the matrix corresponding to the OPB contains a column consisting of exactly one variable. In this case the conjecture holds. We also demonstrate more OPBs true for the conjecture, when they have a column containing a small number of independent variables. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Quantum Information is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
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PubType: Academic Journal
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  Data: A conjecture on the number of independent vector variables in a multiqubit orthogonal product basis.
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  Data: <searchLink fieldCode="AR" term="%22Chen%2C+Yilin%22">Chen, Yilin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> 21091001@buaa.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Chen%2C+Lin%22">Chen, Lin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> 1700398750@qq.com</i>
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  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Quantum+Information%22">International Journal of Quantum Information</searchLink>. Jun2026, Vol. 24 Issue 4, p1-25. 25p.
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  Data: <searchLink fieldCode="DE" term="%22Orthonormal+basis%22">Orthonormal basis</searchLink><br /><searchLink fieldCode="DE" term="%22Independent+variables%22">Independent variables</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+information+theory%22">Quantum information theory</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+computing%22">Quantum computing</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+states%22">Quantum states</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: In this paper, we investigate the number of independent vector variables in an n -qubit Orthogonal Product Basis (OPB). Here, the term "independent vector variable" refers to a structural element in the construction of a Unextendible Orthogonal Matrix (UOM). Intuitively, since fixing one state | x 〉 in a qubit basis { | x 〉 , | x ′ 〉 } uniquely determines its orthogonal partner | x ′ 〉 , we treat the pair as a single degree of freedom — one variable — rather than two. Based on this counting measure, we propose the conjecture that the number is upper bounded by 2 n − 1. We show that if the number of orthogonal pairs of variables is minimum, then the matrix corresponding to the OPB contains a column consisting of exactly one variable. In this case the conjecture holds. We also demonstrate more OPBs true for the conjecture, when they have a column containing a small number of independent variables. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of International Journal of Quantum Information is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1142/S0219749926500085
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      – Code: eng
        Text: English
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        PageCount: 25
        StartPage: 1
    Subjects:
      – SubjectFull: Orthonormal basis
        Type: general
      – SubjectFull: Independent variables
        Type: general
      – SubjectFull: Quantum information theory
        Type: general
      – SubjectFull: Quantum computing
        Type: general
      – SubjectFull: Matrices (Mathematics)
        Type: general
      – SubjectFull: Quantum states
        Type: general
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      – TitleFull: A conjecture on the number of independent vector variables in a multiqubit orthogonal product basis.
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              M: 06
              Text: Jun2026
              Type: published
              Y: 2026
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              Value: 24
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