Equality condition for a matrix inequality by partial transpose

• URL: http://arxiv.org/abs/2508.18644v1
• Date: Tue, 26 Aug 2025 03:42:46 GMT
• Title: Equality condition for a matrix inequality by partial transpose
• Authors: Nalan Wang, Lin Chen,
• Abstract summary: We study the equality condition for a matrix inequality generated by partial transpose.<n>We show that $sumK_j=1 A_j otimes B_j$ is locally equivalent to an elegant block-diagonal form.
• Score: 4.115763274264731
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The partial transpose map is a linear map widely used quantum information theory. We study the equality condition for a matrix inequality generated by partial transpose, namely $\rank(\sum^K_{j=1} A_j^T \otimes B_j)\le K \cdot \rank(\sum^K_{j=1} A_j \otimes B_j)$, where $A_j$'s and $B_j$'s are respectively the matrices of the same size, and $K$ is the Schmidt rank. We explicitly construct the condition when $A_i$'s are column or row vectors, or $2\times 2$ matrices. For the case where the Schmidt rank equals the dimension of $A_j$, we extend the results from $2\times 2$ matrices to square matrices, and further to rectangular matrices. In detail, we show that $\sum^K_{j=1} A_j \otimes B_j$ is locally equivalent to an elegant block-diagonal form consisting solely of identity and zero matrices. We also study the general case for $K=2$, and it turns out that the key is to characterize the expression of matrices $A_j$'s and $B_j$'s.

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