FOUNDATIONS OF QUANTUM MECHANICS

Now consider an operator W of the form
III. 4. COMPOSITE SYSTEMS 61
W =
∑N I
∑N II
i=1 j=1
which is, in general, not factorizable.
z ij U i ⊗ V j , (III. 93)
EXERCISE 20. Prove that U i ⊗ V j is a 1 - dimensional projector in H.
The operator W , (III. 93), is a state operator if
z ij ∈ [0, 1] and
∑N I
∑N II
i=1 j=1
z ij = 1, (III. 94)
furthermore, with (III. 69) and (III. 72) we have
and
Tr II W =
Tr I W =
∑N I
∑N II
i=1 j=1
∑N I
∑N II
i=1 j=1
z ij U i (III. 95)
z ij V j . (III. 96)
This system has an infinite number of solutions for the unknown z ij , unless one of the partial
traces is pure, e.g. Tr II W = U 1 . In that case, according to (III. 95) it has to hold for i = 1
that ∑ j z 1j = 1. But then (III. 35) requires that ∑ j z ij = 0 if i ≠ 1, which means that, because
of the non - negativity of the z ij , it has to hold that z ij = 0 if i ≠ 1. Substituting i = 1 in (III. 93)
yields
W =
∑N II
j=1
z 1j U 1 ⊗ V j
= U 1 ⊗
∑N II
j=1
z 1j V j = Tr II W ⊗ Tr I W, (III. 97)
where the last step is in accordance with (III. 96).
We conclude that only if, at least, one of the partial traces is pure, W is factorizable. □
In the foregoing we saw that only if the state operator W of a composite system is factorizable, it
can be uniquely defined. Contrary to classical physics, in quantum mechanics maximal knowledge of
the state of the subsystems is in general not equivalent to maximal knowledge of the state of the entire