Pure qubit state

In quantum information processing, a pure qubit state is a linear superposition of two basis states, conventionally written | 0> and | 1> (ket 0 and ket 1). A pure qubit state is often written as the sum a| 0> + b| 1>, or as a column vector (a, b) where a and b are the complex amplitudes associated with | 0> and | 1>, respectively. Conventionally the coefficients are normalized, i.e., divided by sqrt(|a|²+|b|²). The vector space in which all pure qubit states lie is the two-dimensional Hilbert space.

Unitary transformations are one kind of basic operation which can be performed on qubits. A two-dimensional unitary transformation transforms a pure qubit state into another.

Measurements are another basic operation in which knowledge is gained about the state of the qubit. With probability proportional to |a|², the result of the measurement will be | 0> and with probability proportional to |b|², it will be | 1>. Unless the qubit is in either one of the basis states, there is no way to know in advance what the result will be. Measurement of the state of the qubit alters the values of a and b. If the state | 0> is measured, a is changed to a magnitude of 1 and b is changed to a magnitude of 0 (see magnitude of complex numbers). If the state | 1> is measured, b is changed to a magnitude of 1 and a is changed to a magnitude of 0.

For every pure state there exists a measurement that always yields a predictable, constant result. This is why they are called 'pure'. Pure states are constrasted with 'mixed' states, which do not have this property.