\langle f_i|.
\label{eqn:sepmixed}\end{aligned}$$ **Def. 2** *Separable and entangled mixed states:*\
*A quantum state $\rho_{AB}$ is separable if and only if it can be represented as a convex combination of the product of projectors on local states as stated in Eq. (\[eqn:sepmixed\]). Otherwise, the state is said to be entangled.*

The important point to note here is that the state displayed in Eq. (\[eqn:sepmixed\]) is the most general state that Alice and Bob will be able to prepare by local quantum operations and classical communication (LOCC) [@Werner]. In LOCC protocols, two parties Alice and Bob perform local quantum operations separately in their respective Hilbert spaces and they are allowed to communicate classical information about the results of their local operations. Let us make the definition somewhat more formal.\
**Local operations and classical communication (LOCC):** Suppose Alice and Bob share a quantum state $\rho_{AB}$ defined on the Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Alice performs a quantum operation on her local Hilbert space $\mathcal{H}_A$, using a complete set of complete general quantum operations $\{A_i^{(1)}\}$, satisfying $\sum_i (A_i^{(1)})^{\dagger} A_i^{(1)} = I_A$, and sends her measurement result $i$ to Bob via a classical channel. Depending on the measurement result of Alice, Bob operates a complete 