Quantum Measurement and Collapse
Measurement projects a quantum state onto a basis outcome with probability given by the squared amplitude, and the state collapses to that outcome.
The measurement postulate
When a qubit in state a|0> + b|1> is measured in the computational basis, the outcome is 0 with probability |a|^2 or 1 with probability |b|^2. This is the Born rule. After the measurement the state is no longer a superposition: it becomes exactly |0> or exactly |1>, matching the observed result. This abrupt change is called collapse.
Irreversibility and information loss
Collapse is not unitary and not reversible. Before measurement the amplitudes a and b held rich information; afterward only a single classical bit remains and the rest is gone. You cannot measure a qubit to learn a and b — a single copy yields one outcome, and the amplitudes are unrecoverable. Estimating amplitudes requires many identically prepared copies.
Measurement in other bases
Measurement is always relative to a chosen basis. Measuring (|0>+|1>)/sqrt(2) in the computational basis gives 0 or 1 with equal probability, but measuring it in the {(|0>+|1>)/sqrt(2), (|0>-|1>)/sqrt(2)} basis gives a definite outcome every time. Choosing the measurement basis is part of algorithm design; a good final basis is one in which the answer is nearly deterministic.
Formal description
A projective measurement is a set of projectors P_i that sum to the identity. Outcome i occurs with probability
Why it matters for computing
Because measurement destroys superposition and returns only classical bits, a quantum algorithm must funnel the answer into measurement statistics through interference before reading out. The whole difficulty of quantum algorithm design lives in this constraint: the exponential state space is real, but the read-out channel is narrow.