Contents
4 Density operators 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 4.2 Deﬁnition of a density operator . . . . . . . . . . . . 4.2.1 Traces . . . . . . . . . . . . . . . . . . . . . 4.2.2 Density operators of pure states . . . . . . . 4.2.3 General density operators . . . . . . . . . . 4.2.4 Examples . . . . . . . . . . . . . . . . . . . 4.2.5 Nonuniqueness of ensemble representations 4.3 VonNeumann equation . . . . . . . . . . . . . . . . 4.4 Measurement for density ope
Contents
4 Density operators 3
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2 Deﬁnition of a density operator . . . . . . . . . . . . . . . . . . . . . 44.2.1 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2.2 Density operators of pure states . . . . . . . . . . . . . . . . 54.2.3 General density operators . . . . . . . . . . . . . . . . . . . 64.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.5 Nonuniqueness of ensemble representations . . . . . . . . . 94.3 VonNeumann equation . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Measurement for density operators . . . . . . . . . . . . . . . . . . . 104.5 Mixing and states of composite systems . . . . . . . . . . . . . . . . 114.5.1 Mixing quantum states . . . . . . . . . . . . . . . . . . . . . 114.5.2 Partial traces . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5.3 Entangled states . . . . . . . . . . . . . . . . . . . . . . . . 131
2
CONTENTS
Chapter 4
Density operators
This will be a relatively short chapter: We will merely introduce here the concept of adensity operator, which incorporates the concept of classical probabilities into quantumtheory. Density operators are quantum states and hence generalize the concept of statevectors.
4.1 Motivation
Let us imagine we have a single spin, associated with a Hilbert space
H
C
2
. Wenow throw a coin. In case of heads, we prepare the spin in

0
, in case of tails, weprepare it in

1
. That is to say, with the classical probability
1
/
2
we have

0
, andwith classical probability
1
/
2
we get

1
. How do we capture this situation? Can wedescribe the system by a state vector

+
= (

0
+

1
)
/
√
2?
(4.1)Not quite. This is easy to see: In case of a
σ
x
measurement, we would always get thesame outcome. But this is different from the situation we encounter here. In fact, whenwe make a measurement of
σ
x
, we would get both outcomes with equal probability. Or
−
= (

0
−
1
)
/
√
2?
(4.2)Again, this will not work, for the same reason. In fact, no state vector is associatedwith such a situation, and for that, we need to generalize our concept of a quantumstate slightly: to density operators. This is, however, the most general quantum state instandard quantum mechanics, and we will not have to generalize it any further.In fact, the above situation is an instance of the situation where we prepare withprobability
p
j
,
j
= 1
,...,n
, a system in a state vector

ψ
i
. Since we encounter aprobability distribution, we have
n
j
=1
p
j
= 1
.
(4.3)3
4
CHAPTER4. DENSITYOPERATORS
Such a situation is sometimes referred to as a mixed ensemble. How do we incorporatethat?
4.2 Deﬁnition of a density operator
4.2.1 Traces
A hint we have already available: Since all operations we can apply to state vectorsact linearly (time evolution and measurement), we already know the following: Letas assume that we initially have the situation that with probability
p
j
the state vector

ψ
j
(
t
0
)
is prepared. Then we, say, evolve the system in time, which means that weapply the unitary
U
(
t,t
0
)
to each state vector,

ψ
j
(
t
)
=
U
(
t,t
0
)

ψ
(
t
0
)
.
(4.4)In order to compute the expectation value of some observable
A
at time
t
, we merelyneed to compute the expectation value for each of the initial states, to obtain
n
j
=1
p
j
ψ
j
(
t
)

A

ψ
j
(
t
)
.
(4.5)Before we can deﬁne the density operator, based on this intuition, we quickly remindourselves again of the trace of a matrix that we saw in the second chapter:
Trace of a matrix:
The trace of a
d
×
d
matrix
A
is deﬁned as
tr(
A
) =
d
−
1
j
=0
j

A

j
.
(4.6)The trace has a number of interesting properties: