Fig. 5 shows a two dimensional section obtained by picking two pure states
randomly. Since a generic pure state is entangled, the section goes through two
pure entangled states lying on the unit circle.
Figure 5: A numerical computation of a planar cross section through the origin
and through two random pure states (located on the circle). The orange region
shows the separable states and the blue region the entangled states. The section
is quite far from a sphere centered at the origin.
2.2 A 3-D section through Bell states
Consider the 3D cross section given by^7 :
4 ρ(x,y,z) = 1 +√
3
(
xσ 1 ⊗σ 1 +yσ 2 ⊗σ 2 +zσ 3 ⊗σ 3)
(2.2)
The section has the property that both subsystems are maximally mixed
Tr 2 ρ=Tr 1 ρ=1
2
Since the purity is given by
Tr(ρ^2 ) =1
4
+
3
4
(
x^2 +y^2 +z^2)
(2.3)
the pure states lie on the unit sphere and all the states in this section must lie
inside the unit ball.
The matrices on the right commute and satisfy one relation
(σ 1 ⊗σ 1 ) (σ 2 ⊗σ 2 ) =−σ 3 ⊗σ 3(^7) A generic two qubits state is SLOCC equivalent to a point of this section, see [4].