|g_i^A g_i^B|), \\
2(1 - \sum_i \lambda_i^2 |g_i^A g_i^B|) &\geq & 2 - \sum_i \lambda_i^2[(g_i^A)^2 + (g_i^B)^2].
\label{eq:col32}\end{aligned}$$ Using inequalities (\[eq:col3\])-(\[eq:col32\]), we get $$\begin{aligned}
2\sum_i |\lambda_i - \lambda_i^2 g_i^A g_i^B| &\leq & 2 - \sum_i \lambda_i^2[(g_i^A)^2 + (g_i^B)^2] \nonumber \\
&\leq & 2(1 - \sum_i \lambda_i^2 |g_i^A g_i^B|) \nonumber \\
&=& (2 - 2\sum_i \lambda_i) + 2\sum_i (\lambda_i - \lambda_i^2 |g_i^A g_i^B|) \nonumber \\
&\leq & (2 - 2\sum_i \lambda_i) + 2\sum_i |\lambda_i - \lambda_i^2 g_i^A g_i^B| \nonumber \\
\Leftrightarrow \sum_i \lambda_i &\leq & 1.\end{aligned}$$ This is the cross-norm or realignment criterion of separability mentioned earlier, which we get as a corollary of the covariance matrix criterion.

There are several other corollaries of the covariance matrix criterion, which enable one to efficiently detect entangled states in several cases. Moreover, the covariance matrix criterion can be improved by using local filtering operation [@local_filter]. See Ref. [@cmc2] for details.

Non-operational entanglement criteria
=====================================

In this section, we will discuss three further entanglement criteria. We will show how the Hahn-Banach theorem can be used to obtain “entanglement witnesses”. We will also introduce the notion of positive maps, and present the entanglement criterion based on it. And finally we will present the range criterion 