Pure qubit state

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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.

  • 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.