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A Determinantal Inequality Involving Partial Traces

Published:2016-03-17
Printed: Sep 2016
• Minghua Lin
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Abstract

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
 MSC Classifications: 47B65 - Positive operators and order-bounded operators 15A45 - Miscellaneous inequalities involving matrices 15A60 - Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]

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