FOUNDATIONS OF QUANTUM MECHANICS

62 CHAPTER III. THE POSTULATES
system. Consequently, the state of the entire system can, generally, not be derived from measurements
on the separate subsystems. 4
If the partial traces of W = W 1
⊗ W 2 are both pure, W is also pure, as we saw in the exercise
on p. 56, and since the pure partial traces each have only one term W is of the form |u⟩ ⟨u| ⊗ |v⟩ ⟨v|.
On the other hand, a pure state in H is, generally, not factorizable, which we will show in an example.
EXAMPLE
If |u i ⟩ and |v j ⟩ span a basis in H I and H II , respectively, an arbitrary vector |ψ⟩ in H = H I ⊗ H II
is of the form
|ψ⟩ =
∑N I
∑N II
i=1 j=1
c ij |u i ⟩ ⊗ |v j ⟩. (III. 98)
An arbitrary pure state in H is therefore of the form
|ψ⟩ ⟨ψ| =
∑N I
∑N II
∑N I
∑N II
i=1 j=1 k=1 l=1
Consider the following pure entangled state in H,
c ∗ kl c ij
(
|ui ⟩ ⊗ |v j ⟩ )( ⟨u k | ⊗ ⟨v l | ) . (III. 99)
|Φ⟩ = 1 2
√
2
(
|u1 ⟩ ⊗ |v 1 ⟩ + |u 2 ⟩ ⊗ |v 2 ⟩ ) . (III. 100)
The corresponding W is the 1 - dimensional projector
(
W = |Φ⟩ ⟨Φ| = 1 2 |u1 ⟩ ⟨u 1 | ⊗ |v 1 ⟩ ⟨v 1 | + |u 1 ⟩ ⟨u 2 | ⊗ |v 1 ⟩ ⟨v 2 |
+ |u 2 ⟩ ⟨u 1 | ⊗ |v 2 ⟩ ⟨v 1 | + |u 2 ⟩ ⟨u 2 | ⊗ |v 2 ⟩ ⟨v 2 | ) . (III. 101)
This pure state W is not factorizable, and cannot be written in the form (III. 93). But although W
is pure, its partial traces are not pure,
Tr II W =
Tr I W =
∑N II
(
⟨v j | Φ⟩ ⟨Φ | v j ⟩ = 1 2 |u1 ⟩ ⟨u 1 | + |u 2 ⟩ ⟨u 2 | ) , (III. 102)
j=1
∑N I
i=1
⟨u i | Φ⟩ ⟨Φ | u i ⟩ = 1 2
(
|v1 ⟩ ⟨v 1 | + |v 2 ⟩ ⟨v 2 | ) , (III. 103)
and indeed,
W I ⊗ W II = 1 4
(
|u1 ⟩ ⟨u 1 | ⊗ |v 1 ⟩ ⟨v 1 | + |u 1 ⟩ ⟨u 1 | ⊗ |v 2 ⟩ ⟨v 2 | +
|u 2 ⟩ ⟨u 2 | ⊗ |v 1 ⟩ ⟨v 1 | + |u 2 ⟩ ⟨u 2 | ⊗ |v 2 ⟩ ⟨v 2 | ) ≠ W. (III. 104)
4 This aspect of the quantum mechanical state description is, however, analogous to a classical state description with a
probability distribution. The two - particle distribution function ρ(q 1 , p 1 ; q 2 , p 2 ) is not uniquely defined by the marginal
distribution functions
∫
∫
ρ 1 (q 1 , p 1 ) = ρ(q 1 , p 1 ; q 2 , p 2 ) dq 2 dp 2 and ρ 2 (q 2 , p 2 ) = ρ(q 1 , p 1 ; q 2 , p 2 ) dq 1 dp 1 ,
the marginals are, after all, analogous to the partial traces.