(PhysOrg.com) -- Although the uncertainty principle is probably the most well-known example of a fundamental limitation of measurement precision in quantum mechanics, it is not the only one. In fact, every physical system is characterized by a number of variables that do not change their values as the system evolves over time; such variables are called conserved quantities and they are said to obey a conservation law. The fact that some quantities cannot change their values suggests that there might be restrictions on the possible ways in which a measurement device can interact with a quantum object and extract information from it.

This expectation has been confirmed and made precise by the Wigner-Araki-Yanase (WAY) theorem, which was developed in the early 1960s. In a new study, researchers have extended this theorem by showing that the conservation of the total momentum of a quantum object and measuring apparatus places a fundamental limit on how accurately the object’s position can be measured.

Professor Paul Busch and Ph.D. student Leon Loveridge, mathematical physicists at the University of York, UK, have published their study on this previously unknown fundamental limitation in a recent issue of *Physical Review Letters*.

The scope of the original WAY theorem was limited in that its proof only applied to a restricted class of physical variables and conserved quantities. Ever since the discovery of the theorem, researchers have wondered whether it might extend to the important case of the position of a quantum particle. In 1991, Japanese physicist Masanao Ozawa, then at Harvard University, developed a model which seemed to suggest that the position can be measured with arbitrary accuracy and repeatability using an interaction that leaves the total momentum of the quantum object and apparatus conserved.

In the new study, Busch and Loveridge have analyzed Ozawa’s model and found that, contrary to Ozawa’s conclusion, momentum conservation does in fact limit the accuracy and repeatability of position measurements: they have shown that good accuracy and repeatability of a position measurement can only be achieved by using a sufficiently large apparatus.

The researchers also developed an alternative model that identifies a particular condition underlying the WAY theorem: the so-called Yanase condition, which stipulates the compatibility of the indicator variable of the apparatus with the conserved quantity. This alternative model shows that, if it were allowed to disregard and violate the Yanase condition, position measurements could be done with arbitrary accuracy, even with a small apparatus. However, if one tries to exploit this escape route from the WAY theorem, one is only faced by the puzzling prospect of the same limitation reappearing for the apparatus indicator variable.

“This is perhaps surprising for a number of reasons,” Loveridge told *PhysOrg.com*. “Firstly, it is exponentially more accurate than an old model of von Neumann which did not obey the conservation of momentum - one might have thought that by including the conservation law things should get worse. Secondly, in the discrete and bounded case one has to give up repeatability and the Yanase condition for accurate measurements with no size constraint. In this model, we can still have arbitrarily good accuracy and repeatability, without any constraint on the size.”

As Busch and Loveridge explain, understanding these kinds of quantum limitations on measurements is important for developing a more complete description of physical reality. In addition to being of theoretical interest, such limitations must also be taken into account in the engineering of single quantum objects.

“On a fundamental level, it is important to understand any physical theory as thoroughly as possible, and in turn to understand how nature behaves or at least manifests itself through observation,” Busch said. “Some would say that this is the primary goal of scientific investigation. On a more practical level, as discussed in our paper, there are potential ramifications for the processing of (quantum) information in which the information is encoded in a continuous variable.”

**Explore further:**
More accurate than Heisenberg allows? Uncertainty in the presence of a quantum memory

**More information:**
Paul Busch and Leon Loveridge. “Position Measurements Obeying Momentum Conservation.” *Physical Review Letters* 106, 110406 (2011). DOI:10.1103/PhysRevLett.106.110406

## Zed123

Still, its good to hear that scientist are continuing to improve our understanding of the fundamental aspects of the universe.

## ettinone

## d44x

If you ever study quantum mechanics then one of the very first things you are taught is the Heisenberg uncertainty principle - i.e. if you know an objects exact velocity then you don't know its position, or if you know its exact position then you don't know its velocity.

Now am I missing something here, or have they just multiplied the velocity in the Heisenberg uncertainty principle to get his principle, but with momentum instead?

## dirk_bruere

## Zed123

I'm not even sure what you are talking about here. I guess from the smiley faces that your trying to make a joke but you've failed to convey your meaning very well. Did you want to try again or are we best to just ignore you and move on?

## hard2grep

## n0ns3ns0r

## Zed123

LOL this quoted the wrong post. My comments should have been directed @ ettinone.

## Sean_W

## telos

From the paper "For accurate position measurements subject to a WAY-type limitation, a large momentum spreadand thus kinetic energyis required in the apparatus, which conflicts with the low temperatures necessary for the control of a quantum system."

They are not talking about a simultaneous measurement of position and momentum. They are saying that momentum conservation implies a large momentum spread in the measurement apparatus in order to make an accurate position measurement.

A quantum device that measures position accurately would need to have a large momentum spread (higher temperature) ... but this would conflict with the low temperature required to control it.

## johanfprins

Thus, while being accelerated, BOTH the position of an electron's centre-of-mass and the momentum manifests SIMULTANEOUSLY with 100% accuracy. The so-called "uncertainties" (delta)x and (delta)p=(delta)(hbar)k is valid for all harmonic waves and has NOTHING to do with uncertainties in the position and momentum of an electron-"particle" whatsoever. The electron si a wave and its intensity is determined by its mass-energy. Therefore it has a centre-of-mass as Ehrenfest already proved more than 70 years ago.

## Anthony_Milner

(God?). As our four dimensions digress from our personal ratio of resonant constancy within a sliding curve. I believe that this "WAY" is correct. To be solvable, the Heisenberg question inherently demands limitations on degrees of freedom even under the unrealistic assumption of (n) probabilities on a "finite" plane where at some point, relevance & degrees of freedom under the curve appear to trump theoretical fractal strings of events & relativistic observation.

## ennui27

PHEW - thx for saying that, Zed ..... I am not alone. even if I expect there to be some articles that are way over my head - I usually can get somewhere with them. This - nada until telos and johan came along.

## PaulRadcliff

Elitist attitudes can be kept where they belong. Inside one's self...... By the way, the most bewildering article I've read in a long time. Glad I wasn't the only one confused by it.

## Tyzenstein

Someone should write an article about the uncertainty principle in dealing who should be allowed to post comments on physorg. Elitist attitudes should be systematically eliminated from the self and the scientific community as well.

The comments on this article were WAY better than the article itself.

## CHollman82

## RETT

## Gthedon

## hush1

All forms of energy (that we know of) are bounded.

That's why the speed of all light is 'constant'.

Standing wave (room) only. Now why on earth does space treat all forms of energy with such a nose up, thumbs down, limiting attitude? Nothing to get bend out of shape about.

We, as energy forms are at least discrete(discreet). Space, on the other hand, insists anything we do is a dead give-away for what space really is: Our limits.

Harmless principled anthropic humor. :)