Comparisons of entanglement witnesses for n-qubit systems

An overlooked problem in witness design is the possibility of phase offset contamination. For example, the singlet EPS 2-qubit state $\frac{1}{\sqrt 2 }\left( {\left| {\uparrow \downarrow } \right\rangle -\left| {\downarrow \uparrow } \right\rangle } \right)$differs from $\frac{1}{\sqrt 2 }\left( {\left| {\uparrow \downarrow } \right\rangle +\left| {\downarrow \uparrow } \right\rangle } \right)$only by a relative phase factor of $e^{i\pi }$, yet both states are fully entangled. We compare in detail several published witnesses on entangled pure and mixed systems with varying degrees of phase offset.