are orthogonal projections. PS projects on the states that are symmetric under
swap, andPAon the anti-symmetric ones. Hence,
Tr PA=M(M−1)
2
, Tr PS=M(M+ 1)
2
(7.26)
The state
ρ=1 +ε
M(M−1)PA+
1 −ε
M(M+ 1)PS, 0 < ε≤ 1 (7.27)is entangled with the swap as witness. Indeed,
Tr(Sρ) =1 +ε
M(M−1)Tr(SPA) +1 −ε
M(M+ 1)Tr(SPS)=−
1 +ε
M(M−1)Tr(PA) +1 −ε
M(M+ 1)Tr(PS)=−
1 +ε
2+
1 −ε
2=−ε (7.28)Whenεis small,ρis close to the Gurvits-Barnum ball. One way to see this is to
compute its purity
Trρ^2 =(
1 +ε
M(M−1)) 2
TrPA+(
1 −ε
M(M+ 1)) 2
TrPS=
M(1 +ε^2 ) + 2ε
M(M^2 −1)(7.29)
Using Eq. (3.11) to translate purity to the radius one finds, after some algebra,
r(ρ) =r^20(
1 +ε√
N
)
(7.30)
Since partial transposition is an isometry,ρptis also near the Gurvits-Barnum ball.
It is an entanglement witness for the Bell state:
−ε=Tr(ρS) =Tr(ρptSpt) =M〈β|ρpt|β〉 (7.31)and we have used Eq. (7.23) in the last step.
7.8 A Clifford ball of separable states
Here we construct a 2−1 Clifford ball with radiusr 0 , of separable quantum states, in the Hilbert spaceCN=CM⊗CM.is the (maximal) number of anti-commuting
(generalized) Pauli matricesσμ, acting onCM. We call this ball the Clifford ball.