FOUNDATIONS OF QUANTUM MECHANICS

58 CHAPTER III. THE POSTULATES
Concerning the expectation values of the quantities of the subsystem S I alone we can replace the
state W by the partial trace, or state operator, Tr II W in H I , analogously for S II . Therefore it is
customary to let the states of the subsystems correspond to the partial traces Tr II W and Tr I W .
For the partial traces it holds that if W is a direct product of state operators W 1 and W 2 in H I
and H II , respectively, W can also be written as a direct product of its partial traces, which we now
show in a lemma.
LEMMA:
If W is a direct product of the form W = W 1 ⊗ W 2 , where W 1 and W 2 are state operators
in H I and H II , respectively, then Tr II W = W 1 and Tr I W = W 2 .
Proof
Tr II W = Tr II (W 1 ⊗ W 2 ) =
∑N II
⟨β j | W 1 ⊗ W 2 | β j ⟩
j=1
∑N II
= W 1 ⟨β j | W 2 | β j ⟩ = W 1 Tr W 2 = W 1 , (III. 74)
j=1
likewise,
Tr I (W 1 ⊗ W 2 ) = W 2 . □ (III. 75)
From this lemma we see that W = W 1 ⊗ W 2 = Tr II W ⊗ Tr I W , and with the first theorem
of this section, p. 56, this leads to the conclusion that if W is a direct product of its partial traces, it
can be uniquely reconstructed from its partial traces. Generally, an arbitrary state operator W of the
composite system can not be defined by its partial traces, which was shown by Von Neumann.
VON NEUMANN’S THEOREM B:
The partial traces Tr II W and Tr I W uniquely define W , iff at least one of the partial
traces is pure, in which case W is factorizable,
W = Tr II W ⊗ Tr I W. (III. 76)
Proof
Let {|u i ⟩} be a basis of eigenstates of W I having non - degenerate eigenvalues. Leaving out the
eigenvalues p n and u i which are equal to 0, expand W and Tr II W in their eigenvectors,
W =
N∑
p n |ψ n ⟩ ⟨ψ n | with |ψ n ⟩ ∈ H (III. 77)
n=1
and
Tr II W =
∑N I
i=1
u i |u i ⟩ ⟨u i | with |u i ⟩ ∈ H I . (III. 78)