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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]
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Database: Engineering Source
Description
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]
ISSN:02197499
DOI:10.1142/S0219749926500085