Introduction to Quantum Physics and Information Processing

82 Introduction to Quantum Physics and Information Processing

5.1.1 Properties of the density operator

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