unitary evolution
Unitary evolution is the precise mathematical name for the smooth, probability-preserving change of a closed quantum system over time. 'Unitary' is a property of the operator that does the updating: it rotates the state vector around without ever stretching or shrinking it. Because the length of the state vector encodes total probability, keeping that length fixed at one means the probabilities of all possible outcomes always add up to exactly one hundred percent.
Think of it geometrically. A unitary transformation is like rigidly rotating an arrow in an abstract, many-dimensional space — the arrow may swing into a completely new direction, but it never gets longer or shorter. This guarantees that nothing leaks out of the description: the system that existed a moment ago is fully accounted for now, just dressed in different proportions of its possibilities.
Unitarity is also reversible. Knowing the rule that took you forward, you could in principle run it exactly backwards and recover the earlier state, with no information lost along the way. This is why the contrast with measurement is so striking: measurement appears to be abrupt and irreversible, whereas the underlying evolution between measurements is reversible and information-conserving. Reconciling those two faces is at the heart of the measurement problem.
A unitary operator preserves inner products, so the total probability stays fixed at one.
Unitary evolution describes a closed, isolated system. An open system coupled to its surroundings appears to evolve non-unitarily — but that is because part of the world has been left out of the description; the larger combined system still evolves unitarily.