A quantum state is the mathematical object we use to describe a quantum system.

For a single qubit, choose two basis states:

A general pure state can be written as

where (\alpha) and (\beta) are complex numbers called probability amplitudes.

Amplitudes are not probabilities

If we measure in the computational basis,

Because one outcome must occur,

This is the normalisation condition.

Why complex numbers matter

Two states can produce the same immediate measurement probabilities but behave differently later because their amplitudes have different relative phases.

For example,

and

both give 0 and 1 with equal probability when measured directly. A later Hadamard gate distinguishes them completely. The sign is invisible in the first measurement, but it changes the interference.

State-vector picture

Using vectors,

so

A quantum gate acts as a matrix that transforms this vector.

Recording version

A quantum state is not a hidden classical answer waiting to be revealed. It is a vector of complex amplitudes, and those amplitudes determine both measurement probabilities and later interference.

Show the general qubit state, the normalisation equation, and the plus and minus states.