ScienceWeek

QUANTUM PHYSICS: ON THE LIMITS OF MEASUREMENTS

The following points are made by Samuel L. Braunstein (Nature 2006 440:617):

1) Just how accurate can measurements get? Whereas classical physics places no fundamental limits on how well we can do, in the quantum world it's a different story. New work[1] derives general limits for the precision with which a single variable can be measured quantum mechanically. But is this new? After all, Heisenberg's uncertainty principle -- one of the earliest results in quantum mechanics -- already places a fundamental limitation on the precision with which we can make a measurement. In its simplest form, the uncertainty principle identifies so-called complementary observables, pairs of quantities for which knowing one quantity precisely means that the other can only be poorly known. This fundamental principle makes it impossible to learn everything about a quantum-mechanical system.

2) If we monitor only one quantity, however, there is no such in-principle limitation. In fact, this is exactly the strategy exploited in interferometric measurements, in which light travels down a pair of distinct paths and the difference between the two path lengths leads to an observable change in the output of the device. This path difference can be measured to an arbitrary accuracy. But what if we are given some constraint, such as a total energy budget or total light intensity? We all know that it is easier to see in a well-lit room than in a dim one. Similarly, the higher the energy or light intensity in an interferometer, the higher its resolution. One may therefore ask, for a fixed budget, how small a path difference can be discerned?

3) Our intuition from everyday experience tells us that the most promising strategy for measuring a distance is to choose a measuring stick with marked intervals of length comparable to the distance we wish to measure. We would not, for example, choose a meter stick to measure a molecule. Following similar logic, we might choose the wavelength of light for our interferometer to be comparable to the path difference we want to measure. The result of Giovannetti et al[1] can be used to show that, for optimal quantum strategies, there is no such bias to the size of our measuring stick or the separation of its tick marks.

4) An optimal strategy refers to a measurement procedure that minimizes the effects of noise on a signal. Ultimately, any measurement is limited by the amount of noise in the system: to discern a signal, the signal-to-noise ratio should be around one or larger. This premise underpins all parameter-estimation theory, both classical and quantum. Classically, statistical averaging over N repeated but independent measurements will lead to a N reduction in the noise. This improvement is known to be optimal because it achieves the bound, known as the Cramér-Rao lower bound[2], that expresses the best accuracy that can be accomplished in the statistical estimation of a parameter. When this classical bound is generalized to repeated quantum measurements, the analogous quantum bound provides a tighter form of the uncertainty principle recast in the language of parameter estimation[3]. However, quantum theory allows much more freedom in choosing measurement strategies than is possible in the classical world.[4]

References:

1. Giovannetti, V. , Lloyd, S. & Maccone, L. Phys. Rev. Lett. 96, 010401 (2006)

2. Cramér, H. Mathematical Methods of Statistics 500 504 (Princeton Univ. Press, 1946)

3. Braunstein, S. L. & Caves, C. M. Phys. Rev. Lett. 72, 3439 3442 (1994)

4. Gill, R. & Massar, S. Phys. Rev. A 61, 042312 (2000)

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