The representation of a single qubit state on the Bloch sphere can be ex-
tended to the density operator. The Bloch sphere is parametrized by spherical
angles or in terms of the Bloch vector of Equation 4.2, which characterizes
the polarization of the state. Can we use this kind of description for a mixed
state?
Mixed States, Open Systems, and the Density Operator 85
For a single qubit state, ρ is a 2 × 2 matrix and we can represent it as a
linear combination of the Pauli spin matrices and the identity:
ρ = a
0
+ ~a · ~σ, ~a ≡ (a
1
, a
2
, a
3
).
Since ρ is Hermitian, we need a
0
and a
i
to be real. Since Tr(ρ) = 1 and the
Pauli matrices are traceless, we must have p
0
=
1
2
. Thus, if p
i
=
1
2
a
i
, we can
write
ρ =
1
2
( + ~p · ~σ) =
1
2
“
1 + p
3
p
1
− ip
2
p
1
+ ip
2
1 − p
3
#
. (5.21)
Since ρ must be positive, we need det ρ ≥ 0.
det ρ =
1
2
(1 − ~p
2
).
So ρ is non-negative only if
~p
2
≤ 1, (5.22)
with the equality holding for
pure states: |~p| = 1; det ρ = 0. (5.23)
The vector ~p, also referred to as the polarization vector, is a point on or
inside the unit sphere: the Bloch ball. Thus, states of single qubits can be
represented on the Bloch sphere if they are pure and inside the Bloch sphere
if they are mixed.
FIGURE 5.2: Bloch ball: points inside the Bloch sphere represent qubits in
mixed states
86 Introduction to Quantum Physics and Information Processing
Example 5.1.6. For a pure state, the Bloch representation of the density
matrix is of the form
ρ =
1
2
( + ˆp · ~σ),
where ˆp is the unit polarization vector of the state. To see this, use the Bloch
sphere representation of the state vector along the direction ˆp = {θ, φ}:
ρ(ˆp) = |ψ( ˆp)ihψ(ˆp)|
=
“
cos
θ
2
e
iφ
sin
θ
2
#
h
cos
θ
2
e
−iφ
sin
θ
2
i
=
“
cos
2
θ
2
e
−iφ
cos
θ
2
sin
θ
2
e
iφ
cos
θ
2
sin
θ
2
sin
2
θ
2
#
=
1
2
+
1
2
“
cos θ e
−iφ
sin θ
e
iφ
sin θ −cos θ
#
=
1
2
( + ˆp · ~σ).
Exercise 5.6. Calculate the expectation value hˆn · ~σi of the spin along the di-
rection ˆn, in the mixed state characterized by a polarization vector ~p to
validate the interpretation of ~p as the polarization along the direction ˆn.
Exercise 5.7. Locate in the Bloch ball the states given by the following density
matrices: (a)
1
2
“
1 0
0 1
#
(b)
“
1 0
0 0
#
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