In systems with a finite number of degrees of freedom, it is known that entanglement can help increase the precision of measurements. However, the reason behind this effect was never analyzed in depth. Here we give a simple, intuitive construction that shows how entanglement transforms parallel estimation strategies into sequential ones of same precision. We can then employ this argument to obtain a series of new results in quantum metrology and also to reobtain, in a simpler manner, some old ones.

What happens in systems (e.g. electromagnetic radiation) with an infinite number of degrees of freedom? It is clear that infinite resources can lead to infinite precision in the measurement, and it has been suggested that (since infinite energy variance is possible even for systems with finite average energy) infinite phase precision can be achieved using finite energy in interferometric measurements. We prove that this is not true: we give a general bound to the precision of a parameter in terms of the average value of the conjugate observable. Whence, interferometry cannot give infinite precision in phase estimation using finite energy (its conjugate observable). Our inequality is a generalization of the conventional Heisenberg uncertainty relations (and of the Cramer-Rao bounds): there the variance of one parameter is bounded by the variance of a conjugate one. In our bounds, the variance of one is bounded by the average of the other. The same results hold even if one considers the prior information one has on the system.

My talk is based on the results presented in arXiv:1304.7609; Phys. Rev. Lett. 108, 260405 (2012); Phys. Rev. Lett. 108, 210404 (2012).