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

6.2 Directions associated with states with substantial purity are rare

Our aim in this section is to show that the probability for finding directions where
r(θ)≥r 0 is exponentially small.^11 More precisely:

Prob(r≥a)≤

(r

0

a

)N^2 +1

(6.6)

In particular, states that lie outside the sphere of radiusr 0 have super-exponentially
small measure in the space of directions.
To see this, letdμbe a (normalized) measure onDN. From Eq. (3.11) we get
a relations between the average purity and the average radius^12 :

∫

DN

Tr(ρ^2 )dμ=

1

N

+

(

1 −

1

N

)∫

DN

r^2 dμ (6.7)

In the special case thatdμis proportional to the Euclidean measure inRN

(^2) − 1
, the
lhs is known exactly [24]:
1
|DN|

∫

DN

Tr(ρ^2 )dx 1 ...dxN (^2) − 1 =

2 N

N^2 + 1

(6.8)

This gives for theradius of inertia

r^2 e=

∫

DN

r^2 dμ=

N+ 1

N^2 + 1

(6.9)

We use this result to estimate the probability of rare direction that accommodate
states with substantial purity. From Eq. (6.9) we have

N+ 1

N^2 + 1

=

∫

DNdΩdr r

(^2) rN^2 − 2
∫
DNdΩdrr

N^2 − 2 =

(N^2 −1)

∫

DNdΩr

N^2 +1(θ)

(N^2 + 1)

∫

DNdΩ r

N^2 − (^1) (θ) (6.10)
Cancelling common terms we find
1
N− 1

=

∫

DNdΩr

N^2 +1(θ)

∫

DNdΩ r

N^2 − (^1) (θ) (6.11)
(^11) Note thatr 0 = 2rtwithrtthe radius of the ball determined by random matrix theory. This
is an artefact of the method we use wherereplays a role. WhenNis larger 0 ≈re. The stronger
result should haver 0 replaced byrtin Eq. (6.6).
(^12) In Appendix A we show how to explicitly compute the average purity for measures obtained
by partial tracing.