An elementary introduction to the geometry of quantum states with a picture book

|ψ〉〈ψ|

τψ

Figure 7: The figure shows the intersection of the half space in Eq. (3.18) with
the unit ball in the case that xcorresponds to a pure state|ψ〉. The blue dot
represents the stateτψof Eq. (3.20).

clearly lies on ∂DN. Using Eq. (3.17), one verifies that|xψ| = N^1 − 1 saturating
Eq. (3.19).
Eq. (3.19) follows from:

Bgb=

{

x

∣

∣

∣

∣

∣

x·x′≥−

1

N− 1

,|x′|≤ 1

}

⊆

{

x

∣

∣

∣

∣

∣

x·x′≥−

1

N− 1

,x′∈DN

}

⊆DN (3.21)

In the last step we used the fact that the positivity ofρfollows from the positivity
ofρ′by Eq. (3.17).

Remark 3.1.An alternate proof is: By Eq. (3.13) minimizingr(θ)is like minimiz-
ing the smallest eigenvalueλ 1 (θ)ofS(θ) =θ·σunder the constraintsTrS^2 (θ) =N
andTrS(θ) = 0. The minimum occurs when

spec(S) =

(

−

1

r 0

,r 0 ,...,r 0

)

, r 0 =

1

√

N− 1

This, together with Eq. (3.13), givesr 02 for the radius of the inscribed ball of non-
negative matrices.

4 Cross sections

DNhas few sections that are simple to describe, even whenN is large.