Pure qubit state

In quantum information processing, a pure qubit state is a non-zero superposition of two basis states, conventionally written in bra-ket notation notation as <math>| 0 \rangle <math> and <math>| 1 \rangle <math>. Two pure qubit states are physically indistinguishable iff they are multiples of each other. Accordingly, a pure qubit state ψ can be written as the sum

<math> \psi = a | 0 \rangle + b | 1 \rangle <math>

where a and b are complex numbers such that

<math> 1 = \sqrt{|a|^2 + |b|^2} <math>.

Geometrically, pure qubit states can be represented by elements of the Bloch sphere.

There are various kinds of physical operation that can be performed on pure qubit states.

• Unitary transformation. These correspond to rotations of the Bloch sphere.

• Standard basis measurement is an operation in which information is gained about the state of the qubit. With probability |a|², the result of the measurement will be <math>| 0 \rangle <math> and with probability |b|², it will be <math>| 1 \rangle <math>. Measurement of the state of the qubit alters the values of a and b. For instance, if the state <math>| 0 \rangle <math> is measured, a is changed to 1 (up to phase) and b is changed to 0. Strictly speaking, a measurement cannot be regarded as an operation on pure qubit states, since it transforms a pure state into a mixed state.

For a more general discussion of these concepts see pure state and density matrix. Also see quantum operation.