# Noiseless noisy quantum metrology

As expected, Luiz Davidovich (IF-UFRJ) gave another crystal clear (noiseless) talk, this time about metrology with open (noisy) quantum systems.

The subject of Luiz’s talk was the same as the talk Bruno gave at CBPF, roughly a month ago (see here). Indeed both talks were based on joint work between Bruno, Luiz and some other guys at IF-UFRJ, and stem from a general framework they developed to deal with parameter estimation in noisy situations.

One kind of twist that Luiz emphasized this time, that I hadn’t realized before, is that the metrology framework can also be used to show sort of “uncertainty relations”. Everyone one knows about Heisenberg’s uncertainty relations (okay, maybe not really everybody, but for sure the readership of this blog). These relations come immediately from the non-commutation between observables in quantum mechanics.

But what about the “uncertainty relation” between time and energy, or photon number and phase? Time and phase are not operators, but parameters (there are some attempts to define observables for time and phase, but this seem to be a tricky business). What Luiz&Co showed is that quantum metrology provides the proper framework to understand, and also to formally prove, these relations.

The canonical example here is the phase estimation interferometer. Photons reach a beam splitter (BS), part of the amplitude goes through the upper arm and get a unknown phase $\theta$ in comparison with the lower arm. The beam is then recombined in another beam splitter, and by the photo-counting in the detectors, just after the two ports of the BS, one wants to determine $\theta$. In the case of no losses the Fisher information is maximal, i.e. the error is minimal, when the probes are in the so-called NOON states. There they got Heisenberg scaling limit, and as such, the uncertainty relation between photon number and phase as $\Delta\theta \ge 1/2 \langle n \rangle$. When losses are present, however, they show that the precision scales as $\sqrt{\langle n \rangle}$, and as such the Cramer-Rao inequality gives us a different uncertainty relation. This shows that for parameters like time and phase, that are not observables, the uncertainty relation might depend on your experiment. That’s a very fundamental result coming from the metrology approach.

Just a side remark before closing. Luiz has just turned 67, and there was a small gathering at IF-UFRJ to celebrate his birthday. Here is the link to the event, and you can leave a message to him there. Luiz is one the most important figures behind the quantum information community in Brazil. Luiz, thank you very much and congratulations!!

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