Chapter 9

Density Matrices

In this chapter we want to introduce density matrices, also called density operators,which conceptually take the role of the state vectors discussed so far, as they encode allthe (accessible) information about a quantum mechanical system. It turns out that the“pure” states, described by state vectors1 |ψ 〉 on Hilbert space, are idealized descriptionsthat cannot characterize statistical (incoherent) mixtures, which often occur in the ex-periment, i.e. in Nature. These objects are very important for the theory of quantuminformation and quantum communication. More detailed information about the densitymatrix formalism can be found in [17].

9.1 General Properties of Density Matrices

Consider an observable A in the “pure” state |ψ 〉 with the expectation value given by

〈 A 〉ψ = 〈ψ | A |ψ 〉 , (9.1)

then the following definition is obvious:

Definition 9.1 The density matrix ρ for the pure state |ψ 〉is given by

ρ := |ψ 〉 〈ψ |

This density matrix has the following properties:

I) ρ2 = ρ projector (9.2)

II) ρ† = ρ hermiticity (9.3)

III) Tr ρ = 1 normalization (9.4)

IV) ρ ≥ 0 positivity (9.5)

1Remark for experts: It is possible to find a vector representation for every given quantum mechanicalstate, even those represented by a density matrix. This can be done via the so-called GNS (Gelfand-Neumark-Segal) construction. This vector representation need not, however, be of any practical formand the concept of the density matrix is therefore inevitable.

159

160 CHAPTER 9. DENSITY MATRICES

The first two properties follow immediately from Definition 9.1 and property III) canbe verified using the definition of the trace operation for an arbitrary operator D :

Definition 9.2 The trace of an operator D is given by

TrD :=∑n〈n | D |n 〉

where { |n 〉 } is an arbitrary CONS.

Lets e.g. take the operator D = |ψ 〉 〈φ | and calculate its trace

TrD =∑

n

〈n | ψ 〉 〈φ |n 〉 =∑

n

〈φ |n 〉 〈n |︸ ︷︷ ︸1

ψ 〉 = 〈φ |ψ 〉 . (9.6)

Property IV) means that the eigenvalues of ρ are greater or equal to zero, which canalso be expressed as

〈ϕ | ρ |ϕ 〉 = 〈ϕ | ψ 〉 〈ψ |ϕ 〉 = | 〈ϕ | ψ 〉 |2 ≥ 0 , (9.7)

which is an important property because probabilities are always greater or equal to zero.What we still have to ensure is that the expectation value of an observable in the state|ψ 〉 can be reproduced, which we will formulate in the following theorem:

Theorem 9.1 The expectation value of an observable A in a state,represented by a density matrix ρ , is given by

〈 A 〉ρ = Tr (ρA)

Proof:

Tr (ρA) = Tr (|ψ 〉 〈ψ | A) =∑

n

〈n | ψ 〉 〈ψ | A |n 〉 =

=∑

n

〈ψ | A |n 〉 〈n |︸ ︷︷ ︸1

ψ 〉 = 〈ψ | A |ψ 〉 = 〈 A 〉 . q.e.d. (9.8)

9.2 Pure and Mixed States

Now we can introduce a broader class of states represented by density matrices, the so-called mixed states in contrast to the states we have considered until now, the so-calledpure states.

9.2. PURE AND MIXED STATES 161

9.2.1 Pure States

Let’s begin with the pure states. Consider an ensemble of given objects in the states{ |ψi 〉 }. If all the objects are in the same state, the ensemble is represented by a purestate. To make probabilistic statements the whole ensemble of identically prepared sys-tems must be considered.

Let the system be, e.g., in the state |ψ 〉 which we can expand with respect to theeigenstates of an (hermitian) operator A

|ψ 〉 =∑

n

cn |n 〉 , where A |n 〉 = an |n 〉 . (9.9)

The expectation value is then given by

〈A 〉ψ =∑

n

|cn|2 an =∑

n

Nn

Nan , (9.10)

where |cn|2 is the probability to measure the eigenvalue an. It corresponds to the frac-tion Nn/N , the incidence the eigenvalue an occurs, where Nn is the number of times thiseigenvalue has been measured out of an ensemble of N objects.

The state is characterized by a density matrix of the form of Definition 9.1, withthe properties I) - IV) (Eqs. (9.2) - (9.5)), where we can combine property I) and III) toconclude

Tr ρ2 = 1 . (9.11)

9.2.2 Mixed States

Let us next study the situation where not all of theN systems (objects) of the ensemble arein the same state, i.e. Ni systems are in the state |ψi 〉 respectively, such that

∑Ni = N .

The probability pi to find an individual system of the ensemble described by the state|ψi 〉 is then given by

pi =Ni

N, where

∑i

pi = 1 . (9.12)

We can thus write down the mixed state as a convex sum, i.e. a weighted sum with∑i

pi = 1, of pure state density matrices

ρmix =∑

i

pi ρpurei =

∑i

p i |ψ i 〉 〈ψ i | . (9.13)

The expectation value is again given by Theorem 9.1, i.e.

〈 A 〉ρmix= Tr (ρmix A) , (9.14)

162 CHAPTER 9. DENSITY MATRICES

where we can express the expectation value of the mixed state as a convex sum of expec-tation values of its constituent pure states, i.e.

〈 A 〉ρmix=∑

i

p i 〈ψ i | A |ψ i 〉 . (9.15)

Proof:

Tr (ρmix A) = Tr

(∑i

p i |ψ i 〉 〈ψ i | A

)=

=∑

n

∑i

p i 〈n | ψ i 〉 〈ψ i | A |n 〉 =

=∑

i

p i 〈ψ i | A∑

n

|n 〉 〈n |︸ ︷︷ ︸1

ψ i 〉 =

=∑

i

p i 〈ψ i | A |ψ i 〉 . q.e.d. (9.16)

Properties II) - IV) (Eqs. (9.3) - (9.5)) are still valid for mixed states, but property I)does no longer hold

ρ2mix =

∑i

∑j

p i p j |ψ i 〉 〈ψ i |ψ j 〉︸ ︷︷ ︸δij

〈ψ j | =∑

i

p2i |ψ i 〉 〈ψ i | 6= ρmix , (9.17)

where we, w.l.o.g., assumed that |ψ i 〉 and |ψ j 〉 are orthonormal. We can then calculatethe trace of ρ2, which, in contrast to pure states, is no longer equal to 1 but smaller

Tr ρ2mix =

∑n

〈n |∑

i

∑j

p i p j |ψ i 〉 〈ψ i |ψ j 〉 〈ψ j |n 〉 =

=∑

i

∑j

p i p j 〈ψ i |ψ j 〉 〈ψ j |∑

n

|n 〉 〈n |ψ i 〉 =

=∑

i

∑j

p i p j | 〈ψ i |ψ j 〉 |2 =

=∑

i

p2i <

∑i

p i = 1 . (9.18)

The last step in this calculation is obvious, since 0 ≤ p i ≤ 1 and therefore p2i ≤ p i. We

conclude that the trace of ρ2 is a good measure for the mixedness of a density matrix,since it is equal to 1 for pure states and strictly smaller than 1 for mixed states. For amaximally mixed state we have for a given dimension d of the system

Tr ρ2mix = 1

d> 0 . (9.19)

9.3. TIME EVOLUTION OF DENSITY MATRICES 163

9.3 Time Evolution of Density Matrices

We now want to find the equation of motion for the density matrix. We start from thetime dependent Schrodinger equation and its hermitian conjugate

i~∂

∂t|ψ 〉 = H |ψ 〉 †−→ −i~ ∂

∂t〈ψ | = 〈ψ | H . (9.20)

Then we differentiate the density matrix of a mixed state (Eq. (9.13)) with respect totime, we multiply it by i~ and combine this with Eq. (9.20)

i~∂

∂tρ = i~

∑i

pi ( ˙|ψ i 〉︸︷︷︸− i~ H |ψ i 〉

〈ψ i | + |ψ i 〉 ˙〈ψ i |︸︷︷︸i~ 〈ψ i |H

) =

=∑

i

pi (H ρpurei − ρ pure

i H ) =

= [H , ρ ] . (9.21)

Theorem 9.2 Density matrices satisfy the von Neumann Equation

i~ ∂∂t ρ = [H , ρ ]

The von Neumann equation is the quantum mechanical analogue to the classical Liouvilleequation, recall the substitution (2.85).

The time evolution of the density matrix we can also describe by applying an unitary

operator, the time shift operator U(t, t0), also called propagator

U(t, t0) = e−i~ H (t− t0) . (9.22)

It allows us to relate the density matrix at a later time t to the density matrix at someearlier time t0

ρ(t) = U(t, t0) ρ(t0)U †(t, t0) . (9.23)

Furthermore, it helps us to prove, for instance, that the mixedness Tr ρ2 of a densitymatrix is time independent

Tr ρ2(t) = Tr (U ρ(t0) U † U︸︷︷︸1

ρ(t0)U †) = Tr (ρ(t0) ρ(t0) U † U︸︷︷︸1

) = Tr ρ2(t0) , (9.24)

where we used the cyclicity of the trace operation.

164 CHAPTER 9. DENSITY MATRICES

Example: Density matrix for spin 12

Generally, this will be a 2× 2 matrix that can be written as linear combination of theidentity 1 and the Pauli matrices σx, σy and σz , as

ρ = 12

(1 + ~a~σ) . (9.25)

The coefficient ~a is named the Bloch vector and can be calculated as the expectationvalue of the Pauli matrices

~a = Tr (ρ~σ) = 〈~σ 〉 . (9.26)

All spin 12

density matrices lie on or within the so-called Bloch sphere (with radius~a = 1) and are determined by the Bloch vector ~a. The length of the Bloch vector thustells us something about the mixedness, the polarization of an ensemble, i.e. of a beamof spin 1

2particles, e.g. electrons or neutrons. We say the beam is polarized if ai = 1 and

completely unpolarized if ai = 0 , for all i . This means that pure and mixed states canbe characterized via the Bloch vector in the following way

pure state ρ2 = ρ ⇒ |~a| = 1 (9.27)

mixed state ρ2 6= ρ ⇒ |~a| < 1 . (9.28)

A totally mixed state (ai = 0 for all i) can then be written as

ρmix =1

2( | ↑ 〉 〈 ↑ | + | ↓ 〉 〈 ↓ | ) =

1

21 , (9.29)

such that

Tr ρmix = 1 and Tr ρ2mix =

1

2. (9.30)

Remark: Note, the decomposition (9.29) into up | ↑ 〉 〈 ↑ | and down | ↓ 〉 〈 ↓ | states isby no means unique, we can achieve the totally mixed state ρmix = 1

21 in many different

ways.