Measurement Speed Limits for Quantum State Purification

We derive three matched speed limits on the purification of a monitored system, valid in every finite dimension and for every strategy and efficiency. All three are set by one functional, the square root of the state’s impurity, whose maximal decay rate is a state-independent constant. For a qubit this reduces to the square root of the determinant, whose protocol-independent decay at unit efficiency is a sharp null test. The three rate constants stand in ratio two to four to eight, and attainment freezes at that square root, the half-moment. Numerics confirm the bounds; for the qubit, a conservation law settles the local-versus-global optimality puzzle, and doubling feedback spends exactly twice the minimum.