separability problem still eludes us, there has been significant progress in recent years, and we will review some such directions in the following sections.

Operational entanglement criteria
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In this section, we will introduce some operational entanglement criteria for bipartite quantum states. In particular, we will discuss the partial transposition criterion [@PeresPPT; @HorodeckiPPT], the majorization criterion [@NielsenKempe], the cross-norm or realignment criterion [@CCN1; @CCN2; @CCN3], and the covariance matrix criterion [@cmc1; @cmc2]. There exist several other criteria (see e.g. Refs. [@reduction; @Doherty; @other_criteria1; @other_criteria2; @other_criteria3; @other_criteria4]), which will not be discussed here. However note that, up to now, a necessary and sufficient criterion for detecting entanglement of an arbitrary given mixed state is still lacking.

Partial Transposition
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\[sec:PPT\] **Def. 3** *Let $\rho_{AB}$ be a bipartite density matrix, and let us express it as $$\rho_{AB} = \sum_{\substack{1 \leq i,j \leq d_A \\ 1 \leq \mu, \nu \leq d_B}}  a_{ij}^{\mu\nu} (|i\rangle\langle j|)_A \otimes (|\mu\rangle\langle \nu|)_B,
\label{eq:mixed_state}$$ where $\{|i\rangle\}$ ($\{|\mu\rangle\}$) is a set of real orthonormal vectors in ${\cal H}_A$ (${\cal H}_B$), with $d_A = \dim {\cal H}_A$ and $d_B = \dim {\cal H}_B$. The partial transposition, $\rho_{AB}^{T_A}$, of $\rho_{AB}$ with respect to subsystem $A$, is defined as* $$\label{eq_partial_trans}
\rho_{AB}^{T_A} = 
\sum_{\substack{1 