=
1
/
2
1
/
2
1
/
2
1
/
2
= ˆ
ρ
1
,
(where
↑
≡ 
z,
↑
are the usual eigenstates of
S
z
).
Now if we were to start in the state

x,
↑
and measure
S
z
, we know that the
system will collapse either to
↑
or
↓
, with probabilities

x,
↑  ↑ 
2
= 1
/
2
and

x,
↑  ↓ 
2
= 1
/
2
.
So if we measured
S
z
but
didn’t look at the outcome of the measurement
, we would
say that our system is in a mixed state with 50% probability of being
↑
and 50%
probability of being
↓
. Alternatively, we could imagine creating a large number of
identical copies of our system, all in the initial state

x,
↑
. Then if we measured
S
z
on all of them, but didn’t look at the results of the measurements, we would know
that half of the copies were in the state
↑
and half were in the state
↓
. In either
case, this is precisely the mixed state that the density matrix ˆ
ρ
2
represents.
4.
Suppose we have a mixed state of made up of pure states

α
1
, . . . ,

α
m
with nonzero
probabilities
W
1
, . . . , W
m
,
m >
1.
We will assume these states are orthogonal, so
α
i

α
j
=
δ
ij
. The density matrix representing this system is
ˆ
ρ
=
m
i
=1
W
i

α
i
α
i

.
Its square is
ˆ
ρ
2
=
m
i
=1
W
i

α
i
α
i

m
j
=1
W
j

α
j
α
j

=
m
i,j
=1
W
i
W
j

α
i
α
i

α
j
α
j

=
m
i
=1
W
2
i

α
i
α
i

.
Since each
W
i
is nonzero and their sum is 1, each
W
i
must be strictly less than 1.
Therefore
W
2
i
=
W
i
, and hence we see that ˆ
ρ
2
= ˆ
ρ
.
2