What is the Corrected Heisenberg Limit for Phase Estimation Measurements?

[PeterDonis]

TL;DR

A 2019 paper derives an uncertainty principle limit larger by a factor of $\pi$ than the conventional one.

A 2019 paper by Gorecki et al. derives an uncertainty principle limit that is larger than the conventional Heisenberg limit by a factor of $\pi$:

https://arxiv.org/abs/1907.05428

I'm wondering if any QM experts have seen this and what your thoughts are.

 

[Dr_Nate]

I'm not an expert in this field, but it appears that there is a difference between the uncertainty principle and the Heisenberg limit. The Heisenberg limit seems to involve the uncertainty of a measured parameter in a noisy quantum system.

 

[vanhees71]

It's of course dependent on how you define what the uncertainty is. When it comes to "phases" or "angles" it's not so trivial anyway!

 

[Cthugha]

In order to explain the result of that manuscript in context, one has to discuss the typical applications for which this kind of bound is usually used. The manuscript is about the Heisenberg limit in the case of unitary operations of the kind $\hat{U}|\psi\rangle=e^{2\pi i\theta}|\psi\rangle$, which corresponds to phase shifts. Operations like these are frequently encountered in e.g. quantum computing, where one may want to shift the relative phase of a qubit or where the information of interest is encoded in the phase of a qubit and one needs to know the phase of the qubit to get the result of the quantum computing process. Accordingly, phase estimation is an important process in quantum computing and a standard part of Shor's algorithm. In a similar manner the same problem occurs in atomic frequency measurements, interferometry and some kinds of gyroscopes. The present manuscript deals exactly with this problem: What is the Heisenberg limit for such phase estimation measurements.

This may be visualized quite easily using the example of a phase shifting measurement. Imagine a standard Mach-Zehnder interferometer with a 50/50 beam splitter and insert a thin glass plate of unknown thickness in the upper arm (path $|a\rangle$) of the interferometer and nothing in the lower path (path $|b\rangle$). Now you want to know the phase shift this glass plate causes. Mathematically, the action of the beam splitter is equivalent to a Hadamard gate that creates the state $(| a \rangle +|b\rangle)/\sqrt{2}$. In the upper path, a phase shift will be caused that is proportional to $\hat{a}^\dagger \hat{a}$. The phase shift is directly proportional to the number of photons. This is the important part.

Now the relevant question is, how the error of your phase estimate will depend on the number of measurements you can do. The latter is usually calculated in terms of "resources", which means that using N photons corresponds to N resources, but also tweaking the interferometer such that the photons pass the glass plate N times corresponds to N resources.
Now, the easiest way is to use single photon states N times. This will create the state $|\phi\rangle=(\exp{i\theta}|a\rangle+|b\rangle)/\sqrt{2}$. You may then measure the photon count distribution at the two output ports and will find that it scales as $\cos\theta$ and the uncertainty in $\theta$ will be limited by shot noise, so you get $\Delta\phi=1/\sqrt{N}$ for large enough N.

Along the same lines, if you manage to create entangled N00N states inside the interferometer, so that you have a fixed number of photons inside and assure that all of them take the same path, but yo do not know which one it is, this will create the state $|\phi\rangle=(\exp{i N \theta}|a\rangle+|b\rangle)/\sqrt{2}$. If you measure the photon count distribution in this case, you will find that it scales as $\cos N \theta$. This yields a new uncertainty limit of $\Delta\phi=1/N$. However, one has to be careful about the validity of this limit. In the single photon case, going to large N corresponds to doing many repetitions of the same experiment. For N00N states, this is obviously more complicated, as going to larger N and going to many repetitions is not the same thing anymore. Loosely speaking, the result of the present manuscript is that for N00N states, the hypothetical limit for many repetitions, where the prior information is always taken into account is given by $\Delta\phi=1/N$, while in the more relevant case of a single shot measurement, where you perform only one run of the experiment, it is given by $\Delta\phi=\pi/N$, unless you have prior information, which is not available in a true single shot experiment.