  We prove that the sequence space $\ell_{p,q}$ does not embed into
$L_{p,q}(\mathcal{M},\tau)$ for any noncommutative probability space
$(\mathcal{M},\tau)$, $1< p<\infty $, $1\le q<\infty$, $p\ne q$. Several
applications to the isomorphic classification of noncommutative
$L_{p,q}$-spaces are given, which extend and complement several earlier
results.
