70
3 Microscopic Expressions of Nonlinear Polarization
Tr
ρ 2
is then calculated using Eq. (3.10),
Tr
ρ
2
=
m,l
ρ ml ρ lm =
m,l
i
P
i c
i
m c
i
l
∗
⎛
⎝
j
P
j c
j
l c
j ∗
m
⎞
⎠
=
i,j
P
i P
j
m
c
i
m c
j ∗
m
l
c
j
l c
i∗
l =
i,j
P
i P
j
m
c
i
m c
j ∗
m
2
.
(3.53)
We invoke the Schwartz inequality,
|a · b|
2
≤ |a|
2
|b|
2
(3.54)
for two arbitrary non-zero vectors a and b, where the equal relation holds only for
a = b. Suppose that a = c i and b = c j , where c i and c j are the sets of coefficients
c i
m and c
j
m , respectively,
a = c
i
=
⎛
⎜
⎝
c i
1
c i
2
. . .
⎞
⎟
⎠ ,
b = c
j
=
⎛
⎜
⎝
c
j
1
c
j
2
. . .
⎞
⎟
⎠ .
Then Eq. (3.54) is written by
m
c
i
m c
j ∗
m
2
≤
m
|c
i
m |
2
l
|c
j
l |
2
.
(3.55)
Applying Eqs. (3.55) to (3.53), the following relation is derived,
Tr
ρ
2
=
i,j
P
i P
j
m
c
i
m c
j ∗
m
2
≤
i,j
P
i P
j
m
|c
i
m |
2
l
|c
j
l |
2
=
i
P
i
m
|c
i
m |
2
2
= 1.
(3.56)
The equal relation in Eq. (3.56) is realized only in the case that c i
m = c
j
m for all
m, indicating that the states i and j are identical. This condition should be satisfied
between any pair of states in the ensemble. Therefore, the equal relation means the
case that the ensemble of states consist of a single identical state (pure state).