FOUNDATIONS OF QUANTUM MECHANICS

III. 4. COMPOSITE SYSTEMS 57
With (II. 99), for an arbitrary state operator W , hence in general W ≠ W 1 ⊗ W 2 , the expectation
value of A ⊗ 11 is
⟨A ⊗ 11⟩ W = Tr (A ⊗ 11)W
=
=
=
∑N I ∑N II
i=1
N I ∑
i=1
N I ∑
i=1
j=1
N I ∑
k=1 j=1
N I
(
⟨αi | ⊗ ⟨β j | )( A ⊗ 11 ) W ( |α i ⟩ ⊗ |β j ⟩ )
N II ∑
∑
⟨α i | A | α k ⟩
k=1
(
⟨αi | ⊗ ⟨β j | )( A |α k ⟩ ⟨α k | ⊗ 11 ) W ( |α i ⟩ ⊗ |β j ⟩ )
N II ∑
j=1
(
⟨αk | ⊗ ⟨β j | ) W ( |α i ⟩ ⊗ |β j ⟩ ) . (III. 68)
To find ⟨A ⊗ 11⟩ W , define the operator W I in H I , called the partial trace of W in relation to H II ,
W I = Tr II W :=
N II
∑
⟨β j | W | β j ⟩, W I ∈ S (H I ). (III. 69)
j=1
For this partial trace it holds that
⟨α k | W I | α i ⟩ =
N II ∑
j=1
and substituting (III. 70) in (III. 68) yields
⟨A ⊗ 11⟩ W =
N I ∑
i=1
(
⟨αk | ⊗ ⟨β j | ) W ( |α i ⟩ ⊗ |β j ⟩ ) , ⟨α k | W I | α i ⟩ ∈ R, (III. 70)
N I
∑
⟨α i | A | α k ⟩ ⟨α k | W I | α i ⟩ = Tr AW I = ⟨A⟩ WI . (III. 71)
k=1
Analogously, with W II the partial trace of W in relation to H I ,
W II = Tr I W :=
N I
∑
⟨α i | W | α i ⟩, W II ∈ S (H II ), (III. 72)
i=1
we see that
⟨11 ⊗ B⟩ W = Tr BW II = ⟨B⟩ WII . (III. 73)
EXERCISE 19. Prove that Tr II W and Tr I W are state operators in H I and H II , respectively.