FOUNDATIONS OF QUANTUM MECHANICS

II. 5. DIRECT SUM AND DIRECT PRODUCT 33
We see that (II. 98) is a special case of (II. 97), that is, where c jk = a j b k . The special vectors which
can be written as (II. 98), i.e., in the form |ϕ⟩ 1 ⊗|ψ⟩ 2 , are called direct product vectors, or factorizable.
In a direct sum space H 1 ⊕H 2 all vectors can be written in the form |ϕ⟩ 1 ⊕|ψ⟩ 2 , but in a direct product
space H 1 ⊗ H 2 not all vectors can be written in the form |ϕ⟩ 1 ⊗ |ψ⟩ 2 . Further on we will see that
states for which c jk cannot be written as a j b k give rise to typical quantum mechanical behavior, as
in the thought experiment of EPR where composite systems are considered, corresponding to states
on H 1 ⊗ H 2 which cannot be factorized. Such states are called non - factorizable or entangled states.
If A and B are operators on H 1 and H 2 , respectively, the direct product operator A ⊗ B is the
operator on H 1 ⊗ H 2 , defined by
(A ⊗ B) ( |ϕ⟩ 1 ⊗ |ψ⟩ 2
)
:= A |ϕ⟩1 ⊗ B |ψ⟩ 2 . (II. 99)
It follows that, with operators C ∈ H 1 and D ∈ H 2 ,
(A ⊗ B) (C ⊗ D) = (A C) ⊗ (B D). (II. 100)
Similar to vectors, operators on the direct product space H 1 ⊗ H 2 are not always factorizable. The
total momentum operator P 1 + P 2 and the distance operator Q 1 − Q 2 of EPR, with P as defined in
section I. 2, (I. 1), and Q likewise, are examples of such non - factorizable direct product operators,
P 1 ⊗ 11 2 + 11 1 ⊗ P 2 and Q 1 ⊗ 11 2 − 11 1 ⊗ Q 2 . (II. 101)
EXERCISE 12. Calculate the commutator of these operators, given that [ P i , Q j
]
= −iδij .
The following properties of the direct product of operators will, further on, be used frequently:
A ⊗ 0 = 0 ⊗ B = 0 ,
(A 1 + A 2 ) ⊗ B = (A 1 ⊗ B) + (A 2 ⊗ B),
11 ⊗ 11 = 11,
a A ⊗ b B = a b (A ⊗ B), (II. 102)
(A ⊗ B) − 1 = A − 1 ⊗ B − 1 ,
(A ⊗ B) † = A † ⊗ B † ,
Tr ( bA ⊗ cB ) = b c Tr A · Tr B.
EXERCISE 13. Prove the properties of ⊗ in (II. 102).