Chemistry - 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 $| 0 \rangle$ and $| 1 \rangle$ . 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

$\psi = a | 0 \rangle + b | 1 \rangle$

where a and b are complex numbers such that

$1 = \sqrt{|a|^2 + |b|^2}$ .

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 $| 0 \rangle$ and with probability |b|², it will be $| 1 \rangle$ . Measurement of the state of the qubit alters the values of a and b. For instance, if the state $| 0 \rangle$ 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.