FOUNDATIONS OF QUANTUM MECHANICS

III. 5. PROPER AND IMPROPER MIXTURES 63
III. 4. 1
SUMMARY
1. The state operator W ∈ S (H) of a composite system, whether pure or not, is not factorizable
in general.
2. If W is factorizable, the factors are equal to the partial traces of W ,
W = W 1 ⊗ W 2 implies W 1 = Tr II W and W 2 = Tr I W. (III. 105)
3. The partial traces uniquely define W iff, at least, one of the partial traces is pure, in which
case W is directly factorizable, W = W 1 ⊗ W 2 .
4. The partial traces of W are pure iff W is pure and of the form W = ( |u⟩ ⊗ |v⟩ )( ⟨u| ⊗ ⟨v| ) ,
with |u⟩ ∈ H I and |v⟩ ∈ H II .
III. 5
PROPER AND IMPROPER MIXTURES
The states of composite systems shed new insight on the interpretation of mixtures. Suppose that
W I and W II are the partial traces of an arbitrary state operator W , and, with u i , v j ∈ [0, 1], it holds
that
W I =
N I ∑
i=1
u i |u i ⟩ ⟨u i | and W II =
N II ∑
j=1
v j |v j ⟩ ⟨v j |. (III. 106)
W I and W II contain all quantum mechanical information about results of measurements on the subsystems
in H I and H II . The question is whether we can interpret this by assuming that the individual
subsystems are in the pure states |u i ⟩ and |v j ⟩, with probabilities u i and v j , respectively. If this were
the case, the composite system could be divided in subensembles of systems in the states |u i ⟩ ⊗ |v j ⟩
with probabilities depending on possible correlations between the values of i and j. The state would
be of the form
W ′ =
=
∑N I ∑N II
i=1
j=1
∑N I ∑N II
i=1
j=1
p ij
(
|ui ⟩ ⊗ |v j ⟩ )( ⟨u i | ⊗ ⟨v j | )
p ij |u i ⟩ ⟨u i | ⊗ |v j ⟩ ⟨v j |. (III. 107)
The coefficients p ij have to satisfy
p ij ∈ [0, 1],
N II ∑
j=1
p ij = u i ,
N I ∑
i=1
p ij = v j
and
∑N I ∑N II
i=1
j=1
p ij = 1, (III. 108)