Coherence and precision in classical systems

The Quick Study “Collaboration and precision in quantum measurement” (PHYSICS TODAY, December 2011, page 72 ) by Rob Sewell and Morgan Mitchell points out that some quantum mechanical coherence, “quantum collaboration” in their language, allows for the magnetization of a gas to be measured with a precision of 1/N, where N is the number of photons. For large N, 1/N is smaller than 1/√N, so improves upon the usual 1/√N measurement limit; the authors comment that for noninteracting particles, the central limit theorem precludes better classical measurements.

However, coherence is not merely a quantum mechanical effect; many classical systems exhibit similar behavior. For example, one can search for ultrahigh-energy neutrino interactions in Antarctic ice by observing the coherent radio pulses emitted by the resulting particle showers. The observed electric field strength of the pulse scales as the square of the number of particles in the shower (reference ; see also the article I wrote with Francis Halzen, PHYSICS TODAY, May 2008, page 29 ), so for a given uncertainty in field-strength measurement, the uncertainty in the number of shower particles scales as 1/N. That is purely classical electromagnetism.

There are also examples of 1/N scaling without coherence. Consider a system consisting of a noninteracting gas in a reservoir at pressure P, and a small valve that controls access to a gas sensor. The best measurement of the valve’s opening time comes from the first gas molecule observed by the sensor. As one increases the pressure (number of probe molecules N), the time delay between the gate opening and the sensor decreases in a 1/N fashion. For large N, that is more accurate than finding the mean arrival time of the molecules, with an accuracy of 1/√N, and trying to correct for the average delay time.

These comments are not to take anything away from the nice study by Sewell and Mitchell. However, measurements that exhibit 1/N scaling are not limited to quantum systems, and are more common than one might imagine.