Figure 14: The separable states (green) and the corresponding witnesses,
(Schematic).
Sis indeed an entanglement witness since
〈ψ⊗φ|S|ψ⊗φ〉=∣
∣
∣
∣
∣
∑M
j=1〈ψ|j〉〈j|φ〉∣
∣
∣
∣
∣
2
≥ 0 (7.22)SinceTr S=M, the corresponding normalized witness is
WS=
S
M
=
(
|β〉〈β|)pt
(7.23)and the equality on the right follows from the first line of Eq. (7.8).
Since partial transposition is an isometry, it is clear thatWSlies at the same
distance from the maximally mixed state as the pure state|β〉〈β|. In particular,
the associated vectorwSlies on the unit sphere.
Remark 7.2.In the case that the partitioning is to two isomorphic Hilbert spaces,
N 1 =N 2 ,Sis the swap. In a coordinate free notation
S|ψ〉⊗|ψ〉=|φ〉⊗|φ〉 (7.24)|β〉is then the Bell state.
7.7 Entangled states and witnesses near the Gurvits-Barnum ball
Near the boundary ofBgbone can find entangled states and (non-trivial) entan-
glement witnesses, see Fig. (15).
To see this, consider the bi-partitionningN=M^2. SinceS^2 = 1 andS=S∗,