We would in general consider not just a single quantum system, represent-
ing one qubit, but a multiple qubit system that will consist of distinct and
non-interacting component single-qubit systems. The quantum states of the
composite system are elements of a larger Hilbert space composed of the single
qubit Hilbert spaces. For this we take what is called a direct product of the
single-qubit basis states, to form basis states of the larger Hilbert space. This
direct product is also called a tensor product, represented by the symbol
⊗. The elements of the tensor product basis consist of ordered sequences of
elements from the bases of each of the component Hilbert spaces.
Postulate 5. The Hilbert space of a composite system S is the direct product
of Hilbert spaces of the components A, B, C…:
H
S
= H
A
⊗ H
B
⊗ H
C
… (3.27)
If the subsystems have basis states {|e
A
i}, {|e
B
i}…, then each basis state of
the full system is a tensor product of the form
|e
i
i = |e
A
i
i
⊗ |e
B
i
i
⊗ …
A general state of the composite system can be expressed as a linear combina-
tion of basis states of the composite Hilbert space.
For example, a 2-qubit system would consist of two non-interacting single
qubits (say the individual z-spins of two isolated electrons), each with a 2-d
Hilbert space H
2
. The Hilbert space of the 2-qubit system is then
H
4
= H
2
⊗ H
2
. (3.28)
If we label the bases of the H
2
s as {|0i
A
, |1i
A
} and {|0i
B
, |1i
B
}, we get the
basis for H
4
as the ordered pairs
|0i
A
, |1i
A
⊗
|0i
B
, |1i
B
=
|0i
A
⊗ |0i
B
, |0i
A
⊗ |1i
B
, |1i
A
⊗ |0i
B
, |1i
A
⊗ |1i
B
. (3.29)
The notation |ai ⊗ |bi is shortened to |abi and we write the basis for H
4
as
{|00i, |01i, |10i, |11i}. (3.30)
We see binary representations of the numbers 0 to 3 emerging in this 2-qubit
system.
In matrix notation, these basis vectors are generated by direct products. To
write the direct product of two matrices, we should realize that every element
of one matrix is associated with every element of the other. This is done in
the following manner:
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