Introduction to Quantum Physics and Information Processing

68 Introduction to Quantum Physics and Information Processing

is possible to construct higher-dimensional states by taking direct products

of lower-dimensional states. However not all higher-dimensional states can be

constructed this way. There will always exist states that cannot be expressed as

a direct product. Such states are called entangled states. This nomenclature

is due to Erwin Schr¨odinger who first discovered the implication of such states

in 1935 [61].

For example, consider two generic qubits

|ψ

1

i = α

1

|0i + β

1

|1i, |ψ

2

i = α

2

|0i + β

2

|1i. (4.6)

If you form the direct product, you get

|ψ

1

i ⊗ |ψ

2

i =

"

α

1

β

1

#

⊗

"

α

2

β

2

#

=











α

1

α

2

α

1

β

2

β

1

α

2

β

1

β

2











. (4.7) ...