A Determinantal Inequality Involving Partial Traces

Lin, Minghua

Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 585-591

Résumé

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: $$1\,+\,||\mathbf{A}|{{|}_{p}}\,\ge \,{{\left\| \text{T}{{\text{r}}_{1}}\,\mathbf{A} \right\|}_{p}}\,+\,||\text{T}{{\text{r}}_{2}}\,\mathbf{A}|{{|}_{p}},$$ where $\text{T}{{\text{r}}_{1}}$ and $\text{T}{{\text{r}}_{2}}$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality $$1\,+\,\det \,\mathbf{A}\,\ge \,\det {{\left( \text{T}{{\text{r}}_{1}}\mathbf{A} \right)}^{m}}\,+\,\det {{\left( \text{T}{{\text{r}}_{2}}\mathbf{A} \right)}^{2}}.$$

Détail

DOI : 10.4153/CMB-2016-001-5

Mots-clés : 47B65, 15A45, 15A60, determinantal inequality, partial trace, block matrix