Proving uncertainty: First rigorous formulation supporting Heisenberg's famous 1927 principle

Apr 29, 2014
Nearly 90 years after Werner Heisenberg pioneered his uncertainty principle, a group of researchers from three countries has provided substantial new insight into this fundamental tenet of quantum physics with the first rigorous formulation supporting the uncertainty principle as Heisenberg envisioned it. Credit: P.Busch/York

Nearly 90 years after Werner Heisenberg pioneered his uncertainty principle, a group of researchers from three countries has provided substantial new insight into this fundamental tenet of quantum physics with the first rigorous formulation supporting the uncertainty principle as Heisenberg envisioned it.

In the Journal of Mathematical Physics, the researchers reports a new way of defining measurement errors that is applicable in the quantum domain and enables a precise characterization of the fundamental limits of the information accessible in quantum experiments. Quantum mechanics requires that we devise approximate joint measurements because the theory itself prohibits simultaneous ideal measurements of position and momentum—and this is the content of the uncertainty relation proven by the researchers.

"Curiously, since Werner Heisenberg, one of the founders of quantum mechanics, gave an intuitive formulation of this principle, it was only recently that serious attempts were made to make the statement precise enough so that one could check its validity," said Paul Busch, Professor of Mathematical Physics at the University of York, who collaborated with Pekka Lahti of the University of Turku in Finland and Reinhard F. Werner of Leibniz Universität in Hannover, Germany on the work.

"Our method of defining the error and disturbance in quantum measurements enabled us to prove an error-disturbance trade-off relation just the way Heisenberg envisaged it," Busch said.

Nearly 90 years after Werner Heisenberg pioneered his uncertainty principle, a group of researchers from three countries has provided substantial new insight into this fundamental tenet of quantum physics with the first rigorous formulation supporting the uncertainty principle as Heisenberg envisioned it. Credit: AIP Emilio Segre Visual Archives, Segre Collection

The first step was to prove an uncertainty relation for a special class of approximate joint measurements of the position and momentum; a class with nice symmetry properties. The main difficulty was then to find a way of reducing the most general case to this symmetric case. This involved a fairly complex chain of arguments using some deep ideas of advanced mathematics. Formulating and proving a family of measurement uncertainty relations for canonical pairs of observables resulted in one possible rigorous interpretation of Heisenberg's 1927 statements.

"We were able to define measures of error and disturbance as figures of merit characterizing the performance of any measuring device; thus, our measures describe how well a given device allows one to determine, for instance, the position of an electron, and how much it disturbs the momentum," explained Busch. "We believe that our approach is the first to provide error measures that are not merely mathematically plausible but more importantly, can be estimated from the statistical data provided by the measurement at hand, so that the numbers one identifies as "errors" are in fact indicators of the quality of the experiment."

This work is particularly timely and significant since some recent research calls the Heisenberg principle into question. The quantum mechanical inequality proposed by M. Ozawa in Japan, if its interpretation were correct, would suggest that might be less stringent than had been thought for the last 80 or so years. If these claims were tenable, it would seriously impact our understanding of the workings of the physical world. Busch, Lahti and Werner argue that this approach is flawed as Ozawa's inequality is meaningful as an error-disturbance relation only in a limited set of circumstances.

The results of this research—a proof of a variety of formulations of measurement error (and disturbance) relations—highlights the fundamental limits of measurements in . Since modern technology has been progressing steadily to controlling smaller and smaller objects (e.g., nanotechnology, quantum computation, quantum cryptography), the time is approaching where device performance may confront the ultimate quantum limits. These results may, for example, corroborate the security of quantum cryptographic protocols insofar as these are based on the validity of the and the Heisenberg effect.

What is next for this research? "Surprisingly, we are only just witnessing the beginnings of a systematic conceptualization of measurement error and disturbance," Busch said. "No doubt researchers will find interesting new error relations, for instance, based on entropic measures. It appears that the recent quantum uncertainty controversy has already inspired numerous researchers to start their own investigations in this area."

Explore further: Research team challenges the limits of famous quantum principle

More information: The article, "Measurement uncertainty relations" is authored by Paul Busch, Pekka Lahti and Reinhard F. Werner. It will be published in the Journal of Mathematical Physics on Tuesday April 29, 2014. DOI: 10.1063/1.4871444

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antialias_physorg
5 / 5 (11) Apr 29, 2014
The really insteresting thing about the Uncertainty Principle is that it points to a fundamental limit of information inherent in a system. I.e. it isn't a shortcoming of apparatus that forbids a better measueremnt but that you cannot draw more information out of a system than is in it.

The canonical conjugate entities (of which position and momentum are just one example) point to an underlying nature of the universe we haven't grasped yet. Seeing these two (or any other conjugates) as expressions of one property is hard, because it is so counter-intuitive.

Maybe we should try building a reperesentation of physical laws based on such mutable-yet-constrained information carrying elements and see where that leads?
noel_lapin
5 / 5 (1) Apr 29, 2014
Then where should the Ozawa's formulation fit in this story ?
http://www.nature...221.html
indio007
5 / 5 (1) Apr 29, 2014
Hmmmm... one model based on experiment , the other based upon symbols on paper.
Which one should I take as accurate?

"The data clearly demonstrate that Ozawa's relation always holds, whereas Heisenberg's relation fails for all measurement strengths."
Torbjorn_Larsson_OM
5 / 5 (4) Apr 29, 2014
That's cute. But "the uncertainty principle as Heisenberg envisioned it" is not the physics here.

"The original heuristic argument that such a limit should exist was given by Heisenberg, after whom it is sometimes named the Heisenberg principle. This ascribes the uncertainty in the measurable quantities to the jolt-like disturbance triggered by the act of observation. Though widely repeated in textbooks, this physical argument is now known to be fundamentally misleading.[4][5]

[... notes conflict with experiments (such as "weak measurements")... ]

It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[4] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects."

[ http://en.wikiped...rinciple ]
Torbjorn_Larsson_OM
5 / 5 (2) Apr 29, 2014
@noel lapin: I haven't studied it, but at a guess Ozawa's formulation is an elaborate ad hoc "fix" of Heisenberg's ideas, that tries to better reproduce the results that are derived directly out of the physics of quantum fields (the relativistic physics that maps to naive "classical" wavefunction quantum mechanics).

@india007: Neither, see the Wikipedia link for the actual physics. Or, since the article is complex (because the subject is), to its links to simpler QM physics. The gist (Heisenberg morally wrong, wavefunction/quantum fields correct physics) is covered in basic QM texts.
animah
5 / 5 (3) Apr 29, 2014
@Torbjorn:

Consider first: How would you go about measuring properties of a closed system (where by definition, it is closed and you are outside)?

Consider second: Closed or not, all systems are bound (by definition, otherwise they are not systems i.e. you can't measure what is not finite to start with).

So I think you might be wrong on this one and it seems more likely you must extract information from a system in order to observe it.
Mimath224
5 / 5 (1) Apr 29, 2014
As usual I want take a very simplistic point of view so that those with the knowledge can correct me. If I throw a paper bag at someone it isn't likely to cause a problem but as my projectile becomes more massive (or smaller,more dense & greater v) then the impact will most definitely cause a problem. However, we can take pictures of this and can make accurate m/ments to find the info at requried points. But the picture uses photons to convey the info and being smaller don't disturb the macro system to notice.
On Qm scale this cannot apply because of the energies are similar and therefore any view will disturb the system. Since a photon is the smallest unit that can carry info then to me it seems quite logical that there is a point where we would be unable to ascertain values of all variables at the same instant.
Having said that, I will have to study Ozawa's formulation to see what my reasoning has missed. Would appreciate constructive comments also. Thanks
rah
not rated yet Apr 30, 2014
So they're certain Heisenberg's Uncertainty Principle is certain? I'm not sure.
animah
5 / 5 (1) Apr 30, 2014
Mimath, first you have to get rid of a misleading use of "measurement" in everyday language.

Consider this:

- You see a red ball. You are not measuring its color, but the properties of the light that bounced off it. Proof: if you switch off the light...

So measuring the color of the ball in ambient light is not at all the same as measuring the atomic composition of its paint (which would also allow you to measure its spectral absorption properties i.e. color).

Only the 2nd one is a true measurement of the object - but it requires you to take a sample of the paint, therefore modifying the object.

So the contention I think is that in the end some energy must always be absorbed by your sensors, whatever they are. And if you are measuring an intrinsic property of an object or system, that energy must come directly from it.

Mimath224
5 / 5 (2) Apr 30, 2014
@animah yes I understand and basically that's what I intended to convey but I said 'disturb' instead of 'modifying the object' and I wrote 'system' instead of object. However, I see your post as being 'quite logical' (similar to my own post) being the quantum realm. But this is what bothers me! It is, to me, similar to particle/wave duality in that I find no strangeness here and there are things in ordinary life that display this quality (Eastern philosophies are replete with such ideas). Yes, there are topics in QM that I do find strange but not the two items mentioned...which could mean that I totally misunderstand HUP (and duality). Those that are able to formulate such ideas is what I appreciate because they take that extra (genius) step and make it even more understandable. No matter how many times I read the basic QM math & texts HUP seems acceptable to me.
I wish I could say the same for QM 'dimensional compactification'!
mytwocts
not rated yet Apr 30, 2014
The uncertainty relation is implied by the wave function formalism using the chain rule of differentiation. Can anyone explain to me how anything new can be said about this?
swordsman
1 / 5 (2) Apr 30, 2014
Finally! Quantum Physics is just beginning to catch up with electronic engineering methogology! Electronic engineers have used the method of characterization for nearly a century. We know that every measurement is in error due to the loading of the measurement apparatus. The measurement of every voltage every current and every force is always in error. We have used the method of characterization. It has nothing to do with the "wave function formalism" or the "chain rule of differentiation". This is the real world where you will have to learn about the loading of the measurement tool on the measurement. See my website page for more information:

http://www.scienc...berg.htm
swordsman
1 / 5 (2) Apr 30, 2014
I would like to add an additional comment to my message above. As to the measurements of atoms and molecules, it is possible to determine the exact effect of measurement using a computer program. The SPICE computer program has been used to predict the actions electronic circuits for many years, and it has proven to be quite accurate. Planck's quantum theory was based on an electronic model of the atom, so I modeled the atom to fit the Bohr/Rydberg frequencies, and it was published: "Analyzing Atoms Using the SPICE Computer Program", Computing is Science and Engineering",May/June 2012,Vol. 14, No. 3. The results show the scaled atom energy frequency and the Rydberg frequencies (the frequencies have to be scaled due to the present limitations of SPICE). I have written a followup article which shows the effect of measurement on the forces acting on the atoms. Perhaps some day it may also be published. More info on my website shown above.
richardwenzel987
not rated yet Apr 30, 2014
I have wondered sometimes whether these uncertainty relations might also apply to measurements that would operationally define "space"-- as, for instance, a path that a particle might take in a gravitational field. We assume continuous space, but in any real world measurement if we cannot locate a particle with infinite precision, doesn't that make a "granular" or "quantized" space an emergent property of the uncertainty relations? Or am I comparing apples to oranges?
russell_russell
not rated yet May 01, 2014
Assuming a continuum allows arbitrary precision. And the chance to test uncertainty again and again. If you allow uncertainty universal validity the journey for arbitrary precision stops.

As an aside what does this say about the quantum zeno effect?
Uncle Ira
3 / 5 (4) May 01, 2014
- You see a red ball. You are not measuring its color, but the properties of the light that bounced off it. Proof: if you switch off the light...


@ animah, don't you get mad with me, you are one the smart peoples that I always give the fives to you. Now ol Ira-Skippy is struggling with this and the google ain't helping non. But if I am reading right, even the the light being on and off changes the way the ball is doing when you are dealing with them quantum things. I could be reading what I'm reading wrong but don't the photons on the light change what the "ball" is doing in the small picture? When they is bouncing off the "ball" the ball is different then when there are not the photons bouncing off the "ball"? Is this I'm reading it wrong? But for a "ball" I suppose it wouldn't be much of a difference, because I feel the same when I walk in the dark as I do when the light is on, eh?

Since you are one of the smarter Skippys, you tell to me where I'm getting it wrong, eh?
Pejico
May 01, 2014
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animah
not rated yet May 01, 2014
you must also have a sigma


Something to ponder for the religionists :-)
animah
5 / 5 (2) May 01, 2014
There's a bug in the comments software: If you post, then quote your 1st post in a second post within 3 minutes, the 1st post disappears...

So again:

the photons on the light change what the "ball" is doing


Absolument mon cher :-).

In this instance, you must first illuminate the object in order to observe it. That changes it, and you must extract that effect from your final numbers. There are scenarios where that might be non-trivial.

I was just pointing out to Mimath that "measuring" in this context is somewhat different from general language.

2 quantitatively identical results may qualitatively be fundamentally different. In other words having a result is not enough, you must also have a sigma. This illustrates again that methodology is everything.

Uncle Ira
5 / 5 (2) May 01, 2014
There's a bug in the comments software: If you post, then quote your 1st post in a second post within 3 minutes, the 1st post disappears...

So again:

the photons on the light change what the "ball" is doing


Absolument mon cher :-).

In this instance, you must first illuminate the object in order to observe it. That changes it, and you must extract that effect from your final numbers. There are scenarios where that might be non-trivial.

I was just pointing out to Mimath that "measuring" in this context is somewhat different from general language.

2 quantitatively identical results may qualitatively be fundamentally different. In other words having a result is not enough, you must also have a sigma. This illustrates again that methodology is everything.


I wasn't sure I read it correct. Thank you for that. I wish the Pretend-To-Know-Everything-Skippys would be as patient and understandable as you and the other Probably-Are-Real-Scientist-Skippys are.
Pejico
May 01, 2014
This comment has been removed by a moderator.
jalmy
1 / 5 (1) May 06, 2014

Uncertainty principle should be called the convenience principle. It would be convenient to accept it, but in fact it is wrong.and eventually will be proven so. But for now go on keep believing that the position of anything can be uncertain and "random" is more than just an imaginary concept. I can tell you right now how you can beat uncertainty. For one thing you do not have to be able to measure where something is to KNOW where it is. You just have to be able to measure where it was or where it will be. And then you can predict where it is, was or will be. If you are worried that your measurement will add to the system in a way that changes it, you just have to know how it changed it. And if you cannot allow your measurement to change the measured well physics tells us how to do that also. You simply hit it from opposite sides at the same time. So that the sum of your forces is ZERO. Uncertainty more like UNIMAGINATIVE PRINCIPLE..