The majorization criterion, to be discussed in this subsection, has been shown to be *not* more powerful in detecting entanglement. We choose to discuss it here, mainly because it has independent roots. Moreover, it reveals a very interesting thermodynamical property of entanglement.

Before presenting the criterion, we present a definition of majorization [@majorizationBhatia].\
**Def. 4** *Let $x = (x_1, x_2, \ldots, x_d )$, and $y  = (y_1, y_2, \ldots, y_d )$ be two probability distributions, arranged in decreasing order, i.e. $x_1 \geq x_2 \geq \ldots \geq x_d$ and $y_1 \geq y_2 \geq \ldots \geq y_d$. Then we define “$x$ majorized by $y$”, denoted as $x\prec y$, as* $$\sum_{i=1}^l x_i \leq \sum_{i=1}^l y_i,$$ *where $ l = 1,2, \ldots d-1$, and equality holds when $l = d$*.

**Theorem 4** [@NielsenKempe]\
*If a state $\rho_{AB}$ is separable, then* $$\label{dibakar}
\lambda (\rho_{AB}) \prec \lambda (\rho_A ), \quad  and \quad \lambda (\rho_{AB}) \prec \lambda (\rho_B ),$$ *where $\lambda (\rho_{AB})$ is the set of eigenvalues of $ \rho_{AB} $, and $\lambda ( \rho_A )$ and $ \lambda ( \rho_B ) $ are the sets of eigenvalues of the reduced density matrices of the state $\rho_{AB}$, and where all the sets are arranged in decreasing order.*

