A given density operator could represent a pure or a mixed state. If the
system is pure, then the state is a ray in Hilbert space, and the density operator
can be expressed as
ρ = |ψihψ|, for some |ψi.
Such a density matrix satisfies
ρ
2
= |ψihψ|ψihψ| = ρ, (5.17)
Tr(ρ
2
) = Tr(ρ) = 1. (5.18)
This is not true if ρ represents a mixed state, where
ρ =
X
n
p
n
|ψ
n
ihψ
n
|,
for which ρ
2
=
X
n,m
p
n
p
m
|ψ
n
ihψ
n
|ψ
m
ihψ
m
| 6= ρ. (5.19)
84 Introduction to Quantum Physics and Information Processing
In the orthonormal basis {|ii}, we have
Tr(ρ
2
) =
X
i
X
n,m
p
n
p
m
hi|ψ
n
ihψ
n
|ψ
m
ihψ
m
|ii
=
X
i,n,m
p
n
p
m
hψ
m
|iihi|ψ
n
ihψ
n
|ψ
m
i
=
X
n, mp
n
p
m
|hψ
n
|ψ
m
i|
2
≤
X
n
p
n
!
2
= 1 (5.20)
The equality holds only when hψ
n
|ψ
m
i = a pure phase for all pairs n and m,
which means that the density matrix comprises only one state vector in Hilbert
space: a pure state. In fact, the quantity Tr(ρ
2
) is sometimes called the purity
of the state. A completely pure state has Tr(ρ
2
) = 1 and a completely mixed
state has Tr(ρ
2
) =
1
n
. These ideas will be very useful when we study quantum
information theory. There we will also encounter the notion of entropy as a
measure of information, which can also be used to distinguish pure and mixed
states.
Example 5.1.5. For the state pure state |+i,
ρ
2
p
=
1
4
1 1
1 1
!
2
=
1
4
2 2
2 2
!
= ρ,
Trρ
2
p
=
1
2
+
1
2
= 1.
For the unpolarized electron beam (Example 5.1.1), which is a maximally
mixed state, we have
ρ
2
m
=
1
4
,
Trρ
2
m
=
1
2
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