are the matrix representations of $\tilde{G}_k^A$ and $\tilde{G}_l^B$ in $\{{\ensuremath{|u_k\rangle}}\}$ and $\{{\ensuremath{|v_k\rangle}}\}$ basis respectively, then using Eqs. (\[eq:ccn1\]) and (\[eq:ccn2\]), we obtain $$\rho_{AB} = \sum_{k = 1}^{d_B^2} \lambda_{k} G_{k}^{A} \otimes G_{k}^{B}.
  \label{eq:ccn3}$$ Eq. (\[eq:ccn3\]) can be interpreted as the Schmidt decomposition of the density matrix $\rho_{AB}$ in operator space, where the singular values $\lambda_k$ are real and non-negative. The cross-norm or realignment criterion of separability is given by the following theorem:

**Theorem 6** [@CCN3]\
*If a shared quantum state $\rho_{AB}$ is separable, then $$\sum_{k} \lambda_{k} \leq 1,$$ where $\lambda_{k}$ are the singular values of $\rho_{AB}$ as given in Eq. (\[eq:ccn3\]). If the inequality is violated, one can conclude that $\rho_{AB}$ must be an entangled state.*

Covariance matrix
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There exist several other operational criteria in the literature to detect whether a quantum state is separable or entangled [@reduction; @Doherty; @other_criteria1; @other_criteria2; @other_criteria3; @other_criteria4]. We will conclude this section by briefly illustrating one such separability criteria, known as the covariance matrix criterion [@cmc1; @cmc2], which provides a general framework to link and understand several existing criteria including the cross-norm or realignment criterion. Like the cross-norm or realignment criterion, this method can identify entangled state for which the partial transposition criterion fails. 