Editing Quantum Zeno effect (section)

==Periodic measurement of a quantum system==
Consider a system in a state <math>A</math>, which is the [[eigenstate]] of some measurement operator. Say the system under free time evolution will decay with a certain probability into state <math>B</math>. If measurements are made periodically, with some finite interval between each one, at each measurement, the wave function collapses to an eigenstate of the measurement operator. Between the measurements, the system evolves away from this eigenstate into a [[quantum superposition|superposition]] state of the states ''<math>A</math>'' and ''<math>B</math>''. When the superposition state is measured, it will again collapse, either back into state ''<math>A</math>'' as in the first measurement, or away into state ''<math>B</math>''. However, its probability of collapsing into state ''<math>B</math>'' after a very short amount of time <math>t</math> is proportional to <math>t^2</math>, since probabilities are proportional to squared amplitudes, and amplitudes behave linearly. Thus, in the limit of a large number of short intervals, with a measurement at the end of every interval, the probability of making the transition to ''<math>B</math>'' goes to zero.

According to [[quantum decoherence|decoherence theory]], measurement of a system is not a one-way "collapse" but an interaction with its surrounding environment, which in particular includes the measurement apparatus.{{Citation needed|date=March 2025}} A measurement is equivalent to correlating or coupling the quantum state to the apparatus state in such a way as to register the measured information. If this leaves it still able to decohere further to a different state perhaps due to the noisy thermal [[Surroundings (thermodynamics)|environment]], this state may last only for a brief period of time; the probability of decaying increases with time. Then frequent measurement reestablishes or strengthens the coupling, and with it the measured state, if frequent enough for the probability to remain low. The time it expectedly takes to decay is related to the expected decoherence time of the system when coupled to the environment. The stronger the coupling is, and the shorter the decoherence time, the faster it will decay. So in the decoherence picture, an "ideal" quantum Zeno effect corresponds to the mathematical limit where a quantum system is continuously coupled to the environment, and where that coupling is infinitely strong, and where the "environment" is an infinitely large source of thermal randomness.