# 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 [itex]| 0 \rangle [itex] and [itex]| 1 \rangle [itex]. 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

[itex] \psi = a | 0 \rangle + b | 1 \rangle [itex]

where a and b are complex numbers such that

[itex] 1 = \sqrt{|a|^2 + |b|^2} [itex].

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.

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

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