FOUNDATIONS OF QUANTUM MECHANICS

60 CHAPTER III. THE POSTULATES
If Tr II W is pure, there is only one term
Tr II W = |u 1 ⟩ ⟨u 1 |, (III. 85)
and substitution in (III. 79) yields
|ψ n ⟩ = |u 1 ⟩ ⊗ |ϕ 1 n ⟩. (III. 86)
Therefore,
W =
N∑
N∑
p n |u 1 ⟩ ⟨u 1 | ⊗ |ϕ n 1 ⟩ ⟨ϕ n 1 | = |u 1 ⟩ ⟨u 1 | ⊗ p n |ϕ n 1 ⟩ ⟨ϕ n 1 |. (III. 87)
n=1
n=1
Analogous to (III. 82) we find for
Tr I W =
N∑
∑N I
∑N I
n=1 i=1 k=1
p n ⟨u k | u i ⟩ |ϕ n i ⟩ ⟨ϕ n k |. (III. 88)
With i = k = 1 and ⟨u 1 | u 1 ⟩ = 1 we have
Tr I W =
N∑
p n |ϕ n 1 ⟩ ⟨ϕ n 1 |. (III. 89)
n=1
Substituting (III. 89) in (III. 87) we see that W = Tr II W ⊗ Tr I W . Indeed, if one of the partial
traces is pure, W is factorizable, and therefore completely determined, by its partial traces.
To show the ‘only if’ - part of the theorem, that Tr II W and Tr I W uniquely define the state W
of the composite system only if at least one of the partial traces is pure, since only in that
case W is factorizable, we decompose them into orthogonal 1 - dimensional eigenprojectors,
where both u i , v j ∈ [0, 1] sum up to 1 as required in (III. 35) for the projectors to be state
operators,
Tr II W =
Tr I W =
It then holds that
∑N I
i=1
∑N II
j=1
Tr II W ⊗ Tr I W =
u i |u i ⟩ ⟨u i | :=
v j |v j ⟩ ⟨v j | :=
∑N I
∑N II
i=1 j=1
∑N I
i=1
∑N II
j=1
u i U i , (III. 90)
v j V j . (III. 91)
u i v j U i ⊗ V j . (III. 92)