Introduction to Quantum Physics and Information Processing

Mixed States, Open Systems, and the Density Operator 87

Proof. We can see this by invoking the convexity property (Equation 5.16).

Suppose our pure state density matrix admits such a decomposition

ˆρ

pure

= λ|ψ

1

ihψ

1

| + (1 − λ)|ψ

2

ihψ

2

|

= λρ

1

+ (1 − λ)ρ

2

.

Now since the state is pure, there exists some vector |ui such that

ˆρ

pure

= |uihu|.

Consider an orthogonal vector |vi : hu|vi = 0.

=⇒ hv|ˆρ

pure

|vi = hv|uihu|vi = 0.

=⇒ λhv| ˆρ

1

|vi + (1 − λ)hv| ˆρ

2

|vi = 0.

Since λ and (1−λ) are positive, this equation can only be satisfied if hv| ˆρ

1

|vi =

0 = hv| ˆρ

2

|vi. This means ˆρ

1

and ˆρ

2

are orthogonal to |vi. But |vi can be any

vector orthogonal to |ui. So we must have