The density operator on a Hilbert space, defined by Equation 5.4 satisfies
the following properties:
1. ˆρ is Hermitian.
Proof: ˆρ
†
=
X
n
p
∗
n
|ψ
n
i
†
hψ
n
|
†
=
X
n
p
n
|ψ
n
ihψ
n
| = ˆρ. (5.11)
2. ˆρ is non-negative, that is, for any vector |vi, hv|ˆρ|vi ≥ 0. (This translates
to its eigenvalues being non-negative, or det(ρ) ≥ 0.)
Proof: hv|ˆρ|vi =
X
n
hv|p
n
|ψ
n
ihψ
n
|vi
=
X
n
p
n
|hv|ψ
n
i|
2
≥ 0 (5.12)
since the right side is a sum of numbers that are always positive or zero.
3. It satisfies Trˆρ = 1.
Proof: In an orthonormal basis {|ii},
Trρ =
X
i
hi|
X
n
p
n
|ψ
n
ihψ
n
|
!
|ii
=
X
n
p
n
X
i
hi|ψ
n
ihψ
n
|ii
=
X
n
p
n
hψ
n
|
X
i
|iihi|
!
|ψ
n
i
=
X
n
p
n
hψ
n
|ψ
n
i = 1 (5.13)
In general, any operator on a Hilbert space satisfying these properties is
defined as a density operator and can be used to predict the probabilities of
outcomes of measurement on the system, bypassing the state-vector formalism
altogether.
Example 5.1.4. For a system described by continuous variables, for example
position x, the density operator will be expressed as
ρ =
Z
dx dx
0
ω(x, x
0
)|xihx
Mixed States, Open Systems, and the Density Operator 83
For a pure state, we will have
ρ =
Z
dx dx
0
ψ(x)ψ
∗
(x
0
)|xihx
0
|, (5.15)
where ψ(x) = hx|ψi.
Another property of the density operator that will be useful to us is con-
vexity:
Definition 5.3. Convexity: A set of operators {ˆρ
i
} form a convex set if
ρ = λρ
1
+ (1 − λ)ρ
2
, 0 < λ < 1, (5.16)
for every pair ρ
1
, ρ
2
∈ {ρ
i
}.
Convexity has a very simple meaning: any two members of a convex set
can be connected by a straight line without leaving the set. (See Figure 5.1.)
FIGURE 5.1: (a) A convex set, (b) A non-convex set.
Leave a Reply