FOUNDATIONS OF QUANTUM MECHANICS

56 CHAPTER III. THE POSTULATES
EXERCISE 17. Prove the following statements.
(a) W = W 1 ⊗ W 2 is a state operator if W 1 and W 2 are state operators.
(b) The opposite of (a) is not true; give a counterexample.
(c) W = W 1 ⊗ W 2 is pure iff both W 1 and W 2 are pure.
EXERCISE 18. Prove that for all vectors |ψ⟩, |ψ ′ ⟩ ∈ H I and |ϕ⟩, |ϕ ′ ⟩ ∈ H II we have
(
|ψ⟩ ⊗ |ϕ⟩
)(
⟨ψ
′
| ⊗ ⟨ϕ ′ | ) = |ψ⟩ ⟨ψ ′ | ⊗ |ϕ⟩ ⟨ϕ ′ |. (III. 65)
THEOREM:
If W is a direct product of operators, W = W 1 ⊗ W 2 , the subsystems are mutually
independent, i.e., the probability to find for A⊗11 the value a i and for 11⊗B the value b j
is equal to the product of the separate probabilities. In this case the expectation values
factorize too, such that ⟨A ⊗ B⟩ W 1 ⊗ W 2
= ⟨A⟩ W 1
⟨B⟩ W 2
.
Proof
Let a i and b j be eigenvalues of A and B, respectively. Using (III. 65) we see that the projector on
the eigenstate |a i ⟩ ⊗ |b j ⟩ of A ⊗ B is P |ai⟩ ⊗ P |bj⟩. Therefore, with (II. 102), p. 33,
( )
µ W P|ai⟩ ⊗ P |bj⟩
= Tr ( )( )
P |ai⟩ ⊗ P |bj⟩ W 1 ⊗ W 2
= Tr ( )
P |ai ⟩W 1 ⊗ P |bj ⟩W 2
= Tr P |ai ⟩W 1 Tr P |bj ⟩W 2
=
( ( )
µ W 1 P|ai⟩)
µW
2
P|bj⟩
=
(
µ W P|ai⟩ ⊗ 11 ) (
µ W 11 ⊗ P|bj⟩)
, (III. 66)
which proves the first part of the theorem.
For the factorization of the expectation values, we have, analogously,
⟨A ⊗ B⟩ W 1 ⊗W 2
= Tr (A ⊗ B)(W 1 ⊗ W 2 ) = Tr AW 1 Tr BW 2
= ⟨A⟩ W 1
⟨B⟩ W 2
, (III. 67)
and we see that the expectation values indeed factorize. □
From (III. 67) we also see that, if W = W 1 ⊗ W 2 , then ⟨A ⊗ 11⟩ W = Tr A W 1 = ⟨A⟩ W1
and ⟨11 ⊗ B⟩ W = ⟨B⟩ W2 , but this does not hold for more general statistical operators W .