Accretive Partial Transpose Matrices and Their Connections to Matrix Means
Eman Aldabbas, Mohammad Sababheh
TL;DR
This work develops a parallel theory for accretive partial transpose (APT) matrices, extending known PPT results to the accretive setting by leveraging block-matrix positivity, sectorial (accretive) structure, and matrix means. It proves that APT is preserved under weighted geometric means and related mean operations, and it derives eigenvalue, singular-value, and norm inequalities for the blocks involved, including Hiroshima-type bounds adapted to APT. A key contribution is a sufficient condition, using operator-monotone functions, ensuring that fused block expressions like $f(A)\nabla_t f(B)XX^*f(A\nabla_tB)$ are APT under sectorial hypotheses. The results deepen the mathematical toolkit for APT matrices and connect accretive block-positivity with matrix means and functional calculus, with potential implications for quantum-information-inspired block-positivity analyses.
Abstract
Accretive partial transpose (APT) matrices have been recently defined, as a natural extension of positive partial transpose (PPT) matrices. In this paper, we discuss further properties of APT matrices in a way that extends some of those properties known for PPT matrices. Among many results, we show that if \(A,B,X\) are $n\times n$ complex matrices such that \(A,B\) are sectorial with sector angle $α$ for some \(α\in [0,π/2)\), and if \(f:(0,\infty)\to(0,\infty)\) is a certain operator monotone function such that \(\begin{bmatrix} \cos^2(α) f(A) & X X^* & \cos^2(α) f(B) \end{bmatrix}\) is APT, Then \(\begin{bmatrix} f(A)\nabla_t f(B) & X X^* & f(A \nabla_tB ) \end{bmatrix}\) is APT for any \(0\leq t\leq 1\), where $\nabla_t$ is the weighted arithmetic mean.