In a way, the quantum world seems to know when it's being watched. When physicists make measurements on photons and other quantum-scale particles, the measurements always disturb the system in some way. Although an ideal disturbance should still enable physicists to make multiple measurements and get the same result twice, most real measurements cause a greater disturbance than this ideal minimum, and prohibit physicists from making repeated measurements. In a recent study, physicists have demonstrated a new way to make one of the ideal measurements - called quantum non-demolition (QND) measurements - allowing physicists to detect single particles repeatedly without destroying them.
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For mathematical details see the text book "Quantum Paradoxes" by Yakir Aharonov & student.