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

C ̄N− 1

r(θ)

√

1 +(N−^3 1) 2

θ

α

C ̄ 0

C ̄N− 2

1

1

N− 1

Figure 10: The radiusr(θ) along the path from fromC ̄N− 2 toC ̄ 0 has large deriva-
tive,O(N), nearθ= 0.

6 A tiny ball in most directions

A basic principle in probability theory asserts that while anything that might
happen will happen as the system gets large, certain features can become regular,
universal, and non-random [22]. AsNgets large,DN, in most directions, is a ball,
whose radius is
rt≈

r 0

2

(6.1)

Although the radius of the ball is small whenNis large, it is much larger than the
inscribed ball whose radius isr^20. The computation ofrtis a simple application of
random matrix theory [7].

6.1 Application of random matrix theory

Define a random directionθby a vector of iid Gaussian random variables:

θ= (θ 1 ,...,θN (^2) − 1 ), θα∼N

[

0 ,

1

N^2 − 1

]

(6.2)

whereN[μ,σ^2 ] denotes the normal distribution with meanμand varianceσ^2. θ
has mean unit length

E

(

|θ|^2

)

= (N^2 −1)E

(

θ^21

)

= 1