Although the von Neumann entropy is an important notion for quantifying disorder, the theory of majorization is a more stringent quantifier [@majorizationBhatia]: For two probability distributions $x$ and $y$, $x \prec y$ if and only if $x =  Dy$, where $D$ is a doubly stochastic matrix [^4]. Moreover, $x \prec y$ implies that $H(\{x_i\}) \geq H(\{y_i\})$. Quantum mechanics therefore allows the existence of states for which global disorder is greater than local disorder even in the sense of majorization.

A density matrix that satisfies Eq. (\[dibakar\]), automatically satisfies Eq. (\[chandrima\]). In this sense, Theorem 4 is a generalization of Theorem 5.

Cross-norm or matrix realignment
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The cross-norm or matrix realignment criterion [@CCN1; @CCN2; @CCN3] provides another way to delineate separable and entangled states, and more importantly, can successfully detect various PPT entangled states. There are various ways to formulate this criterion. Here we present a formulation given in Ref. [@CCN3] as Corollary 18.

A density matrix $\rho_{AB}$ on a Hilbert space $\mathcal{H}_{A} \otimes \mathcal{H}_{B}$, where $d_{A}$ and $d_{B}$ are the dimensions of the Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ respectively, can be written as $$\rho_{AB} = \sum_{\substack{1 \leq i,j \leq d_A \\ 1 \leq \mu, \nu \leq d_B}} a_{ij}^{\mu \nu} 