of separability. All three criteria are “non-operational”, in the sense that they are not state-independent. Nevertheless, they provide important insight into the structure of the set of entangled states. Moreover, the concept of entanglement witnesses can be used to detect entanglement experimentally, by performing only a few *local* measurements, assuming some prior knowledge of the density matrix [@sakkhi12; @sakkhiprl].

### Technical Preface

The following lemma and observation will be useful for later purposes.\
**Lemma 1**\
$\mbox{tr}(\rho_{AB}^{T_{A}}\sigma_{AB}) = \mbox{tr}(\rho_{AB}\sigma_{AB}^{T_{A}})$. **Observation:**\
The space of linear operators acting on ${\cal H}$ (denoted by ${\cal B}({\cal H})$) is itself a Hilbert space, with the (Euclidean) scalar product $$\langle A|B\rangle = \mbox{tr}(A^\dagger B) \qquad A,B \in {\cal B}({\cal H}).$$ This scalar product is equivalent to writing $A$ and $B$ row-wise as vectors, and scalar-multiplying them: $$\mbox{tr}(A^\dagger B)  = \sum_{ij}A^\ast_{ij}B_{ij} = \sum_{k=1}^{(\dim {\cal H})^2} a_{k}^\ast b_k.$$

Entanglement Witnesses
----------------------

### Entanglement Witness from the Hahn-Banach theorem

Central to the concept of entanglement witnesses, is the Hahn-Banach theorem, which we will present here limited to our situation and without proof (see e.g. [@alt:1985] for a proof of the more general theorem):\
**Theorem 8**\
*Let $S$ be a convex compact set in a finite dimensional Banach space. Let 