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A conjecture on the absolute separability of eight by eight positive semi-definite matrices

Mark Yutong Zhao
pp.9-32 DOI: 10.62051/tdrp2f37 August 13, 2026
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Abstract: The separability problem, a long-standing open challenge in mathematics and quantum information theory. In this work, we propose and investigate a conjecture as a sub-question of the separability problem. Specifically, we investigate whether the form I8-xx*  is separable for any eight-dimensional unit vector x  . This is actually the so-called absolute separability problem. For studying the conjecture, we classify the vectors in a finitely-dimensional bipartite Hilbert space, and construct their equivalence classes in terms of the so-called Schmidt rank, under both product general linear and product linear groups. Then we construct seven equivalence classes of x , using which we partially prove the conjecture. Our results highlight the the complexity of the difference between bipartite and tripartite spaces.
Keywords: Separability, Positive Semidefinite Matrix, Kronecker Product.
APA Citation: Mark Yutong Zhao (2026). A conjecture on the absolute separability of eight by eight positive semi-definite matrices. Transactions on Computer Science and Intelligent Systems Research, 13(1), 9-32. https://doi.org/10.62051/tdrp2f37

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