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Search: All articles in the CMB digital archive with keyword determinant

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1. CMB Online first

Lin, Minghua
A determinantal inequality involving partial traces
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes \mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved an inequality for Schatten $p$-norms: \[ 1+\|\mathbf{A}\|_p\ge \|\tr_1 \mathbf{A}\|_p+\|\tr_2 \mathbf{A}\|_p, \] where $\tr_1, \tr_2$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality \[ 1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n. \]

Keywords:determinantal inequality, partial trace, block matrix
Categories:47B65, 15A45, 15A60

2. CMB 2016 (vol 59 pp. 311)

Ilten, Nathan; Teitler, Zach
Product Ranks of the $3\times 3$ Determinant and Permanent
We show that the product rank of the $3 \times 3$ determinant $\det_3$ is $5$, and the product rank of the $3 \times 3$ permanent $\operatorname{perm}_3$ is $4$. As a corollary, we obtain that the tensor rank of $\det_3$ is $5$ and the tensor rank of $\operatorname{perm}_3$ is $4$. We show moreover that the border product rank of $\operatorname{perm}_n$ is larger than $n$ for any $n\geq 3$.

Keywords:product rank, tensor rank, determinant, permanent, Fano schemes
Categories:15A21, 15A69, 14M12, 14N15

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