unitary operator

A unitary operator is a transformation of quantum states that preserves all inner products: if you apply the same unitary to two states, the overlap between them comes out exactly as before. Geometrically it is a rotation of the state space — it can swing state vectors into new directions, but it never stretches or shrinks them, and it keeps every angle between them intact. Lengths and overlaps are sacred under a unitary.

Because overlaps and lengths are preserved, so are probabilities. A normalised state stays normalised, and the chances of all possible outcomes still add to one after the transformation. This is exactly the behaviour you need for any physically legitimate change of a closed quantum system: nothing is lost, nothing is created, the total probability is conserved. A unitary is the quantum embodiment of a reversible, information-preserving process.

Unitary operators are everywhere in quantum theory. The smooth evolution of an isolated system in time is generated by a unitary operator, the time-evolution operator built from the Hamiltonian; the quantum logic gates of a quantum computer are unitaries; and changing from one basis to another is a unitary change of coordinates. Every such operator has an inverse, which is its Hermitian conjugate, so any unitary process can in principle be run backwards exactly.