# Measuring uncertainty relations

In the space of two weeks, two works appear in Nature Physics about measuring uncertainty relations. In the first, an experiment is actually performed to test (and, needless to say, verify) the validity of an uncertainty relation which applies to more situations than the one originally devised by Heisenberg. In the second, it is proposed that the techniques of quantum optics may be used to probe modifications of the usual uncertainty relation due to gravity. Now, to have finally a tiny bit of evidence for quantum gravity, this would really be a breakthrough!

Faithful to my principle of not doing “refereeing on demand”, this is not an unrequested referee report: in fact, I have only browsed those papers, certainly not in enough depth to make judgments. The authors are serious so, by default, I trust them on all the technicalities. The question that I want to raise is: what claims can be made from an uncertainty relation?

An uncertainty relation looks like this:

[something related to the statistics of measurements, typically variances or errors] >= [a number that can be computed from the theory]

which has to be read as: if the left hand side is larger than 0, then there MUST be some error, or some variance, or some other form of “uncertainty” or “indeterminacy”. Let me write the equation above D>=C for shorthand.

Now, let’s see what a bad measurement can do for you. **A bad measurement may introduce more uncertainties than are due to quantum physics**. In other words, one may find D(measured)=C+B, where B is the additional contribution of the bad measurement. It may be the case that your devices cannot be improved, and so you can’t remove B. Now, the second paper proposes an experiment whose goal is precisely to show that D(measured)=C+G, where G is a correction due to gravity. Obviously, much more than the mere observation of the uncertainty relation will be needed, if someone has to believe their claim: they will really have to argue that *there is no way to remove G* and not because their devices are performing poorly. The problem is that there is always a way of removing G: a bad measurement can do it for you!

Indeed, **a bad measurement may also violate the uncertainty relation**. Let me give an extreme example: suppose that you forget to turn on the powermeter that makes the measurement. The result of position measurement will be systematically x=0, no error, no variance. Similarly, the result of momentum measurement will be systematically p=0, no error, no variance. In this situation, D(measured)=0. Of course, nobody would call that a “measurement”, but hey, that may well be “what you observe in the lab”. To be less trivial, suppose that the needle of your powermeter has become a bit stiff, rusty or whatever: the scale may be uncalibrated and you may easily observe D(measured)<C.

So, a bad measurement can influence the uncertainty relation both ways, either increasing or decreasing C.

Now, there are reasonable ways of getting around these arguments. For instance, by *checking functional relations*: don’t measure only one value, but several values, in different configurations. If the results match what you expect from quantum theory, a conspiracy becomes highly improbable; and indirectly it hints that your measurement was not bad after all. For instance, this is the case of Fig. 5a of the first paper mentioned above.

Still, I am left wondering if the tool of the uncertainty relation is at all needed, since by itself it constitutes very little evidence. Let me ask it this way: why, having collected enough statistics for a claim, should one process the information into an “uncertainty relation”? The information was already there, and probably much more of it than gets finally squeezed into those variances or errors. OK, maybe it’s just the right buzzword to get your serious science into Nature Physics: after all, “generalized uncertainty relation” will appeal to journalists much more than “a rigorous study of the observed data”.

Posted on March 22, 2012, in Common knowledge, Latest topics. Bookmark the permalink. 1 Comment.

Hi Valerio,

Thanks for raising this important issue.

You have not broken your principle, but I will do so and post a reply to a blog, your blog.

I must first clarify an important issue that our scheme is not based on measuring the uncertainty relation, but rather directly of the commutation relation between position and momenta. The trick is that the position–momentum commutator of the macromechanical oscillator becomes imprinted onto the phase of the radiation field. Nonetheless, your worry still stands and is completely justified. How do I know that I have measured the average of [x,p], and not the commutator between other two observables? I might believe to measure [x,p], but in fact my device measures [x,x] or the commutator [f(x,p),g(x,p)] between two arbitrary functions of x and p. The two cases would lead to completely different conclusions about the aimed measurement of [x,p] even without introducing potential quantum gravity effects.

I think that there is no other way to circumvent this problem than gradually getting confidence in your experimental device. In our case, the contribution to the phase coming from potential quantum gravity effects is proportional to the mass of the oscillator. So, you can choose a light (compared with Planck mass) oscillator first and try to recover the standard quantum mechanical position–momentum commutator, and then increase the mass. I general, as you note, you should check various functional relations until you get sufficient confidence than the symbols you use in your theory, such as “x”, “p”, or “[x,p]” and relation among them correspond to specific operations performed on your device.

An interesting research direction that would overcome problems you have raised, would be to attempt to find “device independent” tests of quantum gravity effects. And I know somebody who works on “device independent” proofs 😉

Best wishes from Vienna,

Caslav