# Are There Experiments with Revised Limits of the non-Heisenberg Uncertainty Principle?

Ref. 1 uses a notion of non-Heisenberg uncertainty relations (UP).

The authors remark that "because spins obey non-Heisenberg uncertainty relations, enables simultaneous precise knowledge of spin angle and spin amplitude."

There are no experiments with revised limits of the standard UP.

The standard UP says that position and momentum can't be measured simultaneously.

One example of the non-Heisenberg uncertainty relations is that the spin angle and spin amplitude can be measured simultaneously. Since spin is not defined in terms of position and momentum, the standard UP doesn't apply.

I tried to find out more about the non-Heisenberg uncertainty relations (and other examples of it) on Google but was unsuccessful.

Revised question:

Does anyone know about the non-Heisenberg uncertainty relations and other examples of it?

References:

1. Simultaneous tracking of spin angle and amplitude beyond classical limits. G. Colangelo et al. Nature 543, 525–528 (2017), arXiv:1702.08888.
• To reopen this question (v1), consider to make clear that you are not asking for actual violation of HUP. Or if that's your question, then make that clear. – Qmechanic Apr 26 '17 at 7:28
• This article is relatively new and I think it's too early to expect useful comment. It's not even clear these experiments have been independently repeated and verified (a cornerstone of the scientific method), so I'd be cautious in taking the meaning you are. I think it's a huge leap from what they actually did to "beating" HUP. – StephenG Apr 26 '17 at 9:33
• To reopen this question (v2), consider to include a definition of the notions of "Heisenberg vs. non-Heisenberg uncertainty relations" in this context. – Qmechanic Apr 26 '17 at 11:04
• This seems like a list-based question – Kyle Kanos Apr 27 '17 at 10:25
• @Mr.Davis: from a quick scan of the article this is an example of squeezing i.e. moving the uncertainty around. The experiment decreases the uncertainty in the azimuthal angle by increasing the uncertainty in the polar angle. This is a well established experimental technique. – John Rennie May 4 '17 at 8:38

Let's take this by parts. When you state

There are no experiments with revised limits of the standard UP.

apart from being gratuitously inflammatory, when you actually look at that statement it means really very little.

The standard UP says that position and momentum can't be measured simultaneously.

That's not correct - it is a suitable summary of the principle but it's too rough for actual use. The uncertainty principle states that the product of the uncertainties in position and momentum, $\Delta x$ and $\Delta p$, on any state of the system, is bounded from below, i.e. that $\Delta x\,\Delta p \gtrsim \hbar/2$.

On a more abstract setting, given any two physical quantities $A$ and $B$, the product of their uncertainties is also bounded from below, but in the most general case the bound reads $$\Delta A \, \Delta B \geq \frac12 \left|\left<[\hat A,\hat B]\right>\right|$$ and it is in terms of the expectation value of the commutator of the operators that correspond to those observables. This is standard fare in any introductory course in quantum mechanics, and it is the standard way to generalize the Heisenberg uncertainty principle to arbitrary observables; its technical name is the Robertson-Schrödinger uncertainty relation. As such, there's plenty of examples around - it's present pretty much any time you're doing quantum mechanics.

In the specific case of the Colangelo et al. paper, these uncertainty relations take the form $$\Delta S_i \, \Delta S_j \geq \frac12 \left|⟨S_k⟩\right|,$$ where $S_i$, $S_j$, $S_k$ are any permutation of the three components of spin; the relation says that you can measure two components simultaneously so long as the third component averages out to zero. This relation is (a trivial consequence of) the cornerstone of the angular momentum algebra in quantum mechanics, and it is taught during any introductory QM course.

This is all to say: the "non-Heisenberg-ness" of the uncertainty relation at play in that paper is nothing new, and nothing much to gawk at. The paper represents important technical and technological achievements, but it does not break any previous conceptual paradigms.

• Are the observables, angular momentum and energy, commutative or non-commutative, and which case would make them measurable simultaneously? – Mr. Davis May 7 '17 at 1:59
• please ignore the confusing question above. Google didn't give me any answers. Here's a rephrase: Are there other quantum observables besides position and momentum that can be measured simultaneously? – Mr. Davis May 7 '17 at 2:38
• Yes, but you cannot measure position and momentum simultaneously. Please accept that these are rather technical aspects of quantum mechanics and the answer won't fall out of the sky after a five-minute shake on google; instead, you should take up a suitable textbook from the beginning. I would recommend the QM section of the Feynman Lectures (freely available online). – Emilio Pisanty May 7 '17 at 11:38