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

Figure 11: Dhas, in most direction, a small radius, and extends to the pure states
only in rare direction.

and small variance

E

((

|θ|^2 − 1

) 2 )

=−1 +

∑

αβ

E

(

θ^2 αθβ^2

)

=

2

N^2 − 1

S(θ) =θ·σmay be viewed as an element of the ensemble of theN×Ntraceless
Hermitian random matrices.
By Wigner semi-circle law, whenN is large, the densityμof eigenvalues ap-
proach a semi-circle

dμ=

2

πΛ^2

√

Λ^2 −λ^2 dλ (6.3)

with edges at

Λ^2 =

4

N

E

(

Tr

(

S^2 (θ

))

= 4 (6.4)

WhenN is large, the bottom of the spectrum,λ 1 (θ), a random variable, is close
to the bottom edge at−2 [23]. The radius functionr(θ) is related to the lowest
eigenvalueλ 1 (θ) by Eq. (3.13) and the value forrtin Eq. (6.1) follows^10.

Remark 6.1.To see where the value ofΛcomes from, observe that by Eq. (6.3)

E

(

Tr(θ·σ)^2

)

=

∑

j

E

(

λ^2 j

)

≈N

∫

λ^2 dμ=N

Λ^2

4

(6.5)

(^10) Sinceθis a unit vector only on average, there is a slight relative ambiguity inr. The error
is smaller than theN−^2 /^3 fluctuation in the Tracy-Widom distribution [23].