Non-Heisenberg Uncertainty Relations

[Larry]

Simultaneous tracking of spin angle and amplitude beyond classical limits

Nature 543, 525–528 (23 March 2017)

www.nature.com/nature/journal/v543/n7646/abs/nature21434.html

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

What are “non-Heisenberg uncertainty relations” and what other quantum observables obey them?

(There were two references about measuring spins. But they were of no help. Ref. 9 was not available on the Web. Ref. 10 was about measuring spins. However, there was no mention of a “simultaneous precise knowledge of spin angle and spin amplitude.”)

Can someone please answer my questions in a way that a layman can understand?

 

[PeroK]

Simultaneous tracking of spin angle and amplitude beyond classical limits

Nature 543, 525–528 (23 March 2017)

www.nature.com/nature/journal/v543/n7646/abs/nature21434.html

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

What are “non-Heisenberg uncertainty relations” and what other quantum observables obey them?

(There were two references about measuring spins. But they were of no help. Ref. 9 was not available on the Web. Ref. 10 was about measuring spins. However, there was no mention of a “simultaneous precise knowledge of spin angle and spin amplitude.”)

Can someone please answer my questions in a way that a layman can understand?

The Heisenberg Uncertainty Principle (HUP) applies to the position and momentum of a particle.

There is, however, a more general uncertainty principle (UP) that applies to any pair of observables. The HUP simply being a special case of this.

A pair of observables is either compatible (in which case the UP doesn't really say anything) or incompatible (in which case the UP does say something). The position and momentum of a particle are incompatible observables. Whereas, the total spin and spin about a given axis are compatible observables.

Whether this is what the above paper is referringh to is another question.

 

[vanhees71]

They refer to the general uncertainty relation (which so far I always considered to be Heisenberg uncertainty relations, although one should call the Heisenberg-Bohr-Robertson uncertainty relations, because Heisenberg had a wront interpretation of his result first and was corrected by the very stuborn Bohr, and it was derived by Robertson mathematically in full generality). For two arbitrary self-adjoint operators, it's easy to show that
$$\Delta A \Delta B \geq \frac{1}{2} \left | \left \langle \frac{1}{\mathrm{i}} [\hat{A},\hat{B}] \right \rangle \right |,$$
where the averaging is with respect to an arbitrary state ##\hat{\rho}## and ##\Delta A## and ##\Delta B## are the standard deviations of the observables in this state.

 

[Larry]

Thank you, PeroK, for your answer. I would like to clarify some things.

The Heisenberg Uncertainty Principle (HUP) applies to the position and momentum of a particle.

There is, however, a more general uncertainty principle (UP) that applies to any pair of observables. The HUP simply being a special case of this.

A pair of observables is either compatible (in which case the UP doesn't really say anything) or incompatible (in which case the UP does say something). The position and momentum of a particle are incompatible observables. Whereas, the total spin and spin about a given axis are compatible observables.

Whether this is what the above paper is referring to is another question.

The authors state that there is "simultaneous precise knowledge of spin angle and spin amplitude." Apparently, both (spin angle, spin amplitude) and (total spin, spin about an axis) obey the more general uncertainty principle and are compatible observables. They can be measured simultaneously and with precise knowledge, correct?

What other compatible observables are there (besides spins) that obey the more general UP?

Are my questions confounding?

 

[PeroK]

Thank you, PeroK, for your answer. I would like to clarify some things.
The authors state that there is "simultaneous precise knowledge of spin angle and spin amplitude." Apparently, both (spin angle, spin amplitude) and (total spin, spin about an axis) obey the more general uncertainty principle and are compatible observables. They can be measured simultaneously and with precise knowledge, correct?

What other compatible observables are there (besides spins) that obey the more general UP?

Position and momentum in orthogonal directions are compatible. For example, momentum in the x-direction and position in the y-direction are compatible observables.

Energy and angular momentum (for example, in the hydrogen atom) are compatible.

 

[Larry]

They refer to the general uncertainty relation (which so far I always considered to be Heisenberg uncertainty relations, although one should call the Heisenberg-Bohr-Robertson uncertainty relations, because Heisenberg had a wront interpretation of his result first and was corrected by the very stuborn Bohr, and it was derived by Robertson mathematically in full generality). For two arbitrary self-adjoint operators, it's easy to show that
$$\Delta A \Delta B \geq \frac{1}{2} \left | \left \langle \frac{1}{\mathrm{i}} [\hat{A},\hat{B}] \right \rangle \right |,$$
where the averaging is with respect to an arbitrary state ##\hat{\rho}## and ##\Delta A## and ##\Delta B## are the standard deviations of the observables in this state.

vanhees71: I liked your reply; thank you for it.

Please allow me two more questions.

Can you post again the formula for the general UP, and the formula for
the HUP below it? I'd like to compare them to see how they are alike and how
they differ. (I won't be able to understand them, but I'd just like to look at their
mathematical forms.)

Also, I replied to PeroK. Will you please answer those two questions also?

 

[PeroK]

Can you post again the formula for the general UP, and the formula for
the HUP below it? I'd like to compare them to see how they are alike and how
they differ. (I won't be able to understand them, but I'd just like to look at their
mathematical forms.)

The general UP is as given above for any observables ##A## and ##B##:

$$\Delta A \Delta B \geq \frac{1}{2} \left | \left \langle \frac{1}{\mathrm{i}} [\hat{A},\hat{B}] \right \rangle \right |,$$

For a particle moving in one dimension, if we let ##A## be position, ##x##, and ##B## be momentum, ##p##, then commutator of these is:

##[\hat{A},\hat{B}] = [\hat{x},\hat{p}] = i\hbar##

Hence:

$$ \Delta x \Delta p \geq \frac{\hbar}{2}$$

And that is the HUP.

 

[Larry]

The general UP is as given above for any observables ##A## and ##B##:

$$\Delta A \Delta B \geq \frac{1}{2} \left | \left \langle \frac{1}{\mathrm{i}} [\hat{A},\hat{B}] \right \rangle \right |,$$

For a particle moving in one dimension, if we let ##A## be position, ##x##, and ##B## be momentum, ##p##, then commutator of these is:

##[\hat{A},\hat{B}] = [\hat{x},\hat{p}] = i\hbar##

Hence:

$$ \Delta x \Delta p \geq \frac{\hbar}{2}$$

And that is the HUP.

Thanks again, PeroK!

 

[Larry]

Thanks again, PeroK!

The general UP is as given above for any observables ##A## and ##B##:

$$\Delta A \Delta B \geq \frac{1}{2} \left | \left \langle \frac{1}{\mathrm{i}} [\hat{A},\hat{B}] \right \rangle \right |,$$

For a particle moving in one dimension, if we let ##A## be position, ##x##, and ##B## be momentum, ##p##, then commutator of these is:

##[\hat{A},\hat{B}] = [\hat{x},\hat{p}] = i\hbar##

Hence:

$$ \Delta x \Delta p \geq \frac{\hbar}{2}$$

And that is the HUP.

One last piece of the jig saw puzzle:

Compatible observables. They can be measured simultaneously and with precise knowledge, correct?

I really appreciate your help, PeroK.

 

[PeroK]

One last piece of the jig saw puzzle:

Compatible observables. They can be measured simultaneously and with precise knowledge, correct?

I really appreciate your help, PeroK.

The UP has nothing to say about the precision of measurements. That's depends on your measuring equipment.

For example, the spin of an electron about a given axis can take only two values. The UP tells you about the probability of getting one or the other. If you know the particle's spin in the z direction, you cannot simultaneously know it in the x-direction.

This is nothing to do with the precision with which you can measure spin.

 

[Larry]

The UP has nothing to say about the precision of measurements. That's depends on your measuring equipment.

For example, the spin of an electron about a given axis can take only two values. The UP tells you about the probability of getting one or the other. If you know the particle's spin in the z direction, you cannot simultaneously know it in the x-direction.

This is nothing to do with the precision with which you can measure spin.

PeroK,

I am extremely grateful for your patience and thoughtfulness.

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

I think the authors are saying that the HUP is not relevant, but only the more general UP.

The general UP then "enables simultaneous precise knowledge" of two compatible observables.

But, doesn't this violate the notion of "quantum fuzziness" which is based on probabilities?

Are the relations fuzzy or are the results of the experiments?

Thanks again, PeroK.

Larry

Could the non-Heisenberg uncertainty relations they describe, be something entirely new and not related to the general UP or the HUP?

 

[PeroK]

But, doesn't this violate the notion of "quantum fuzziness" which is based on probabilities?

Are the relations fuzzy or are the results of the experiments?

Could the non-Heisenberg uncertainty relations they describe, be something entirely new and not related to the general UP or the HUP?

Quantum fuzziness is not a term I've ever come across.

If it's a new uncertainty principle, then again it's not something I know anything about.

 

[Jilang]

It seems to be a bit complicated since any "fuzziness" is a function of uncertainty in that which is being measured (I.e. the preparation) and also the " fuzziness" of what is doing the measuring (which also has quantum uncertainty).

 

[Larry]

Quantum fuzziness is not a term I've ever come across.

If it's a new uncertainty principle, then again it's not something I know anything about.

You are an excellent Science Advisor. You are unassuming and humble.
I am amazed at your breadth of knowledge and appreciate every answer
you provide.

"Quantum fuzziness" is a a not a scientific term. It is used by lay publications to
emphasize that QM is not deterministic.

No new uncertainty principle.

Copyright law permits up to 500 words to be borrowed from a publication, with
appropriate recognition of the source. I would like to provide you with some context
from the article I cited earlier:

"We use high-dynamic-range optical quantum non-demolition measurements to [achieve]
spin tracking with steady-state angular sensitivity 2.9 decibels below the standard quantum
limit, simultaneously with amplitude sensitivity 7.0 decibels below the Poissonian variance.

"The standard quantum limit and Poissonian variance indicate the best possible sensitivity
with independent particles. Our method surpasses these limits in non-commuting observables,
enabling orders-of-magnitude improvements in sensitivity for [many important applications]."

Do not these statements confirm what you said in your previous post?

Larry

 

[PeroK]

I've read the summary of the paper, but it's too specialised and I'm not able to comment further on it.

 

[Larry]

It seems to be a bit complicated since any "fuzziness" is a function of uncertainty in that which is being measured (I.e. the preparation) and also the " fuzziness" of what is doing the measuring (which also has quantum uncertainty).

Jilang, thank you for the clarification.

In my first post, I included a link to an article in Nature.

Here is an excerpt.

Copyright law permits up to 500 words to be borrowed from a publication, with
appropriate recognition of the source.

"We use high-dynamic-range optical quantum non-demolition measurements to [achieve]
spin tracking with steady-state angular sensitivity 2.9 decibels below the standard quantum
limit, simultaneously with amplitude sensitivity 7.0 decibels below the Poissonian variance.

"The standard quantum limit and Poissonian variance indicate the best possible sensitivity
with independent particles. Our method surpasses these limits in non-commuting observables,
enabling orders-of-magnitude improvements in sensitivity for [many important applications]."

Can you summarize, in simple terms, what was done here?

And, do you think the HUP was used in both their preparation and measurement?

Larry