## Thursday, October 04, 2012

A reader named "the guy from Rhinoceros" has brought my attention to yet another article trying to evade quantum mechanics; it has clearly become a fashionable hobby of many people these days. Otherwise sensible semi-technical Ars Technica wrote an article with a deliberately offensive title,
Demolishing Heisenberg with clever math and experiments.
No kidding. It's a title of three lies because the authors aren't demolishing Heisenberg but working tightly within the framework he co-discovered; their maths isn't clever but rather completely trivial; and they have done no experiments.

The subtitle of the Ars Technica article is "Good, general measurement choices eliminate uncertainty." Thank God, we may return to the 19th century again.

The paper on which this hype is based upon is trying to do "pretty much the same general thing" as the guys abusing the misleading concept of a weak measurement. They just have a different name with different misinterpreted maths for the same thing, "quantum non-demolition experiments".

It's a concept that has, much like the "weak measurements", a meaningful definition, but all the actual applications of the concept that attract the media – because they "demolish Heisenberg" – are completely bogus once again. Let's look at the paper.

The paper is called
Evading Quantum Mechanics: Engineering a Classical Subsystem within a Quantum Environment Phys. Rev. X: X stands for excellence (no kidding)

Evading quantum mechanics (arXiv March 2012: a free copy) by Mankei Tsang and Carlton M. Caves.
Note that the paper has a dramatic title, too, although Ars Technica made it even more dramatic, more personal, and more dishonest.

The general dream of these people is to be able to measure systems in the real world so that the measurement doesn't disturb the system. This is, of course, impossible due to the Heisenberg uncertainty relationships. Whenever a physical system carries at least some information – whenever its Hilbert space is at least two-dimensional – there will always be observables that don't commute with a given one (unless it is a multiple of the unit matrix, $$\lambda\cdot {\mathbb 1}$$: and these special matrices don't measure anything because their eigenvalues $$\lambda$$ are constant and independent of the state of the physical system) simply because matrices $$2\times 2$$ and larger don't commute with each other. It's that simple.

The other matrices that don't commute with a given one aren't artificial in any sense. They're pretty much as fundamental as the original matrix. Just think about the three Pauli matrices: they describe three components of the spin and all these three components are clearly equally natural; after all, they are related by the rotational symmetry. The most widespread example – position and momentum – really conveys the general story, too. The position operator and the momentum operator don't commute with each other,$[x,p] = i\hbar$ and it has consequences even for position itself because the time-derivative of the position is the velocity $$v$$ which is nothing else than a multiple of the momentum, $$v=p/m$$. So the position doesn't commute with its own time derivative which implies$[x(t),x(t')]\neq 0.$ The nonzero commutators in quantum mechanics that lead to the uncertainty principle don't relate "good observables" with some "artificial observables" you could dismiss, ignore, overlook, and ban. Even if you pick the most natural and essential observable that all the anti-quantum zealots like, the position of a particle (or anything), it just doesn't commute with itself at later times. Consequently, if you know the value of the position at time $$t$$, the position at time $$t'$$ which is later will only be known probabilistically. They can't have sharp values at the same moment, at least generically.

Now, the "quantum non-demolition measurements" are meant to be measurements of an observable $$O$$ that obeys$[O(t),O(t')]=0.$ This is the "dream" of the anti-quantum zealots because they may worship $$O(t)$$ as one of the "nice observables" that has made the world classical again and that helped to demolish the evil Werner Heisenberg. But do such observables $$O(t)$$ exist?

If you care about interesting, nontrivial, evolving ones in systems that can actually be realized and that are interacting, the answer is a resounding No. The reason is simple: the vanishing of the commutator above also means that $$O(t)$$ commutes with its time derivative. If the spectrum of $$O(t)$$ is discrete and non-degenerate, it means that an initial state which is an eigenstate of $$O(t)$$ isn't allowed to evolve to anything else than a multiple itself.

Indeed, if it evolved into a state containing a admixture of states with different eigenvalues of $$O(t)$$, it would mean that the Hamiltonian wouldn't conserve $$O(t)$$. It would have to contain matrix elements that are off-diagonal in a basis of $$O(t)$$ eigenstates. Their commutator with $$O(t)$$ would therefore inevitably contain off-diagonal elements as well. This commutator would be proportional to $$dO(t)/dt$$ which would therefore refuse to commute with $$O(t)$$ due to these off-diagonal elements.

So you can't do it.

It means that under these assumptions, an operator only commutes with itself at all times if its value is essentially conserved. But in that case, you don't get any information if you make a later measurement. You will get the same value as you did at the beginning.

In the paragraphs above, I assumed that the spectrum of $$O(t)$$ is discrete and non-degenerate. But morally speaking, I didn't have to. If the operator had a degenerate spectrum, there would be a loophole that allows states to evolve into superpositions of states that also contain other states with the same eigenvalue. But that wouldn't change the value of $$O(t)$$, either, so it would still be true that the measurements are giving us no dynamical information.

What about my assumption of the discreteness? It isn't really reducing the strength of the argument in physically interesting situations, either. Momentum-like continuous variables become discrete in a box. Put a system in a large box, physics shouldn't really change, the spectrum will become discrete, and my argument will apply. For a position-like continuous observable, latticize the space to achieve the same thing.

Nevertheless, strictly speaking, the proof that $$O(t)$$ must be time-independent if $$[O(t),O(t')]=0$$ won't be valid anymore and the authors think it's important to look for counterexamples. They find the following pathological two-dimensional harmonic oscillator:$H = \zav{\frac{kx_1^2}{2} + \frac{p_1^2}{2m}} - \zav{\frac{kx_2^2}{2} +\frac{p_2^2}{2m}}$ The first parenthesis is a normal harmonic oscillator. The second one is normal as well except that the energy is counted with a minus sign. Note that the kinetic energy has the inverted sign as well so this is something else than a particle in the inverted potential (which has no discrete energy eigenstates): it is a negative-mass particle. The eigenvalues of the second parenthesis are the same discrete numbers as the eigenvalules of the first term except for the overall minus sign.

Now, one may easily find "quantum non-demolition observables" in this system – in fact, a broader set of "non-demolition observables" that they call "quantum mechanics free subsystem" and they even introduce an acronym QMFS for this manifestly useless concept. It is a set of observables that must obey$\forall j,k:\quad [Q_j(t),Q_k(t')]=0.$ Can you find such observables in the two-dimensional harmonic oscillator above? Yes, you can. Take some positions and momenta in the light-like directions,$x_1+x_2, \quad p_1-p_2,$ and you may check that these two operators commute with each other (at the same moment). The commutator $$[x_1,p_1]=i\hbar$$ simply cancels against $$[x_2,-p_2]=-i\hbar$$. They spend half a page with totally trivial manipulations and field redefinitions to convey this trivial point about the "light-like combination", showing that they're not really friends with basic linear algebra.

Also, you may check that the time derivative of $$x_1+x_2$$, the first operator in the "quantum mechanics free subsystem", is proportional to $$p_1-p_2$$, the second operator. The sign in front of $$p_2$$ is negative because of the negative mass in the second "parenthesis" of the Hamiltonian. Also, the time derivative of $$p_1-p_2$$ is proportional to $$x_1+x_2$$.

So the time derivative of an operator commutes with the operator. It's true for both of them so the operators $$x_1+x_2$$ and $$p_1-p_2$$ will commute with copies of each other at all times. Great.

The only problem is that the Hamiltonian we introduced to realize this dream of a "quantum mechanics free subsystem" (the terminology does betray that those folks would love to be "liberated" from quantum mechanics) is unphysical. It is unbounded from below. No physical system you may construct in Nature may be well described by the Hamiltonian simply because it could make the energy arbitrarily low. A related problem is that all the energy levels of this two-dimensional harmonic oscillators are infinitely degenerate. You may get $$7\hbar\omega$$ as the sum of two integers (or half-integers), a positive one and a negative one, multiplied by $$\hbar\omega$$ in infinitely many ways, e.g. as $$(1007.5-1000.5)\hbar\omega$$. An experimental arrangement trying to emulate the Hamiltonian clearly has to have infinitely many degrees of freedom (atoms).

Equally disturbingly, the two-dimensional harmonic oscillator is too simple and non-interacting. Any extra terms that you introduce to your Hamiltonian – in particular, terms that are necessary for you to be able to actually measure the values of $$x_1+x_2$$ or $$p_1-p_2$$ using an apparatus – will violate the property that $$x_1+x_2$$ commutes with its time derivative.

One must emphasize that there are still the other observables, $$x_1-x_2$$ and $$p_1+p_2$$, and $$x_1-x_2$$ doesn't commute with $$p_1-p_2$$ while $$x_1+x_2$$ doesn't commute with $$p_1+p_2$$. Of course that you're not escaping quantum mechanics in any way. Everything they're doing is done within the framework of quantum mechanics. Everything they do critically depends on the precious insights of Werner Heisenberg. So comments that this is "demolishing Heisenberg" are absolutely preposterous.

Let's stop discussions about the conceptual misinterpretations and invalid motivation of the work. Can the "quantum mechanics free subsystems" be useful for anything? I am less certain about the answer to this question but my guess is No. For example, you may want to use the operators above to measure the time including the phase which could be seen very accurately, something you can't do with a single ordinary photon. Is it possible to extract the phase accurately?

I don't think so. To be able to react to the "current phase" of $$x_1+x_2$$ and $$p_1-p_2$$ (which are rotating along some ellipses, just like in any harmonic oscillator) sufficiently quickly, the apparatus has to have strong enough interactions with the "bizarre two-dimensional harmonic oscillator", and with strong enough interactions, you will modify the commutators and the "quantum mechanics free subsystem" will no longer have sufficiently accurately vanishing commutators, so your attempts will fail, anyway.

Moreover, we know how to measure the phase precisely. If you have a coherent state of many photons, a macroscopic electromagnetic wave (coming from a LASER, for example), the intensity of the electric field may be measured rather accurately. We are redefining the "effective Planck's constant" (I mean a more general parameter measuring how important quantum mechanics is for certain questions) to $$\hbar/N^k$$ where $$N$$ is the number of photons in the same state and $$k$$ is a positive exponent I don't want to calculate now.

These two very authors think that their construction will be useful to eliminate quantum noise. This is just a name for the effects resulting from the intrinsic and unavoidable probabilistic nature of quantum mechanics. I think that this whole reasoning is completely fallacious. First, the term "quantum noise" is misleading because it tries to give a negatively sounding emotional charge to a great and crucial property of quantum mechanics. Second, the "quantum noise" is both unavoidable and dependent on the situation you consider. So the only way how you may "avoid the quantum noise" is to change the situation. But then you're solving a different problem. This is particularly clear in the authors' own example. If you want to reduce the quantum noise in a particular optical device transmitting information to a NASA spaceship, the solution isn't to claim that the optical device should better be an exact two-dimensional harmonic oscillator with energy unbounded from below. It's not! ;-)

Combined with the fact that the required Hamiltonian seems physically impossible, it seems very hard for me to imagine that this construction could be useful for anything. However, I still think that the reason why such papers – despite their having no citations except for self-citations – get to various "excellent" Physical Reviews and are so frequently hyped by the non-expert media is that they seem like they are "demolishing Werner Heisenberg" which they are surely not. I am greatly annoyed by this dishonest activity – to which the researchers contribute mostly (but not only) dishonesty and the journalists contribute mostly (but not only) stupidity.

You won't demolish Heisenberg's insights because they're demonstrably true, idiots!

And that's the memo.

P.S.: If you have one hour of time and nerves for this kind of insanity, you may watch this January 2011 Google talk by Ron Garret, a guy formerly employed at Google and later turned into a professional armchair physicist of a sort (although, the speaker in this talk about quantum physics admits, it doesn't mean he is a physicist).

The video was sent to me by Lazăr Lung. Three sentences from the description summarize what category of a loon this Gentleman is:
The problem is that the vast majority of popular accounts of QM are simply flat-out wrong. They are based on the so-called Copenhagen interpretation of QM, which has been thoroughly discredited for decades. It turns out that if Copenhagen were true then it would be possible to communicate faster than light, and hence send signals backwards in time.
It's a Category 5 loon. Popular sources are full of anti-Copenhagen crackpot pseudoscience and of exactly this kind of bullshit – claims that the proper Copenhagen quantum mechanics implies superluminal or acausal effects (for the latter, see some largely confused fresh text by Nude Socialist about entangled photons in graves, London, and Beijing: this sort of stuff gets produced every minute, it seems) – but you may still find people who think that the presentation should be even more anti-Copenhagen.

The title of his talk was "The Quantum Conspiracy: What Popularizers of QM Don't Want You to Know". LOL. At the beginning of the talk, he says that the title was a joke – but the rest of the talk proceeds just like if he believed the title is serious and accurate. The talk is confused from the beginning to the end – not a surprise given the fact that the first claim is that a measurement must mean the detection of an underlying objective reality. A few minutes later, he says that there's no objective reality, but that's the main insight by the Copenhagen school that there isn't.

At the end, he says lots of correct things – at various points, he says that the measurements don't reflect an underlying objective reality; there are no hidden variables explaining the quantum randomness etc.; there is no collapse; Schrödinger's cat does evolve into linear superpositions, etc. – and a large part of his statements is a complete misunderstanding what the Copenhagen interpretation is – because the Copenhagen folks are the originators of many ideas he uses as a "replacement" of what he calls the Copenhagen interpretation. Also, he identifies the Copenhagen interpretation with the ideas about the real "collapse", something that no one in the proper Copenhagen school even championed. In particular, Heisenberg never used the term collapse, preferring to speak of the wavefunction representing our knowledge of a system, and collapse as the "jumping" of the wavefunction to a new state, representing a "jump" in our knowledge which occurs once a particular phenomenon is registered by the experimenter (i.e. when an observation takes place).

It's just a complete mess – this guy is completely confused about the history which is perhaps less irritating than the folks who are confused about the physics. Well, he's still confused about much of the physics, too. For example, he believes that the polarization relatively to a 45° slanted axis is the same thing as an "unpolarization". It's surely not. It's just another polarization, another pure state. The unpolarized light is a mixed state which is something else.

Around 26:00, he proposes Einstein-Podolsky-Rosen-Garret paradox (my instructor in Prague taught me not to declare myself a co-author with people who don't know about it, in the context of Roland Omnes, and I guess that EPR didn't approve Garret's message). In that paradox, he either proves superluminal action at a distance; or a violation of the complementarity/uncertainty. That's of course impossible to design such an experiment (the error turns out to be as trivial as not realizing that photons entangled with different states of other particles no longer interfere, and he seems to realize that). At that moment, I became totally unable to say which of his statements he made are meant seriously and which of them are ironic or statements he wants to disprove. I suppose he essentially understands those things, he just encapsulates them in a completely confused story.

1. You've had a lot of fun beating up on several anti-QM pieces in the last several months.

There is some interesting sociology / small man syndrome in the desire of many to "demolish heisenberg" or "demolish einstein".

2. Ars Technica even has it wrong. The author writes:

"When we examine these, we find that we can measure the difference in
momentum and position of the two oscillators as accurately as we like,
because the back action affects the sum of the particle positions and
their average momentum."

Not good.

3. I apologize for off-topic, but have you read this new post of Terence Tao? http://terrytao.wordpress.com/2012/10/02/einsteins-derivation-of-emc2-revisited/
Tao's words ("it would be difficult to say anything non-tautological at all about the physical world if one could rely solely on rigorous mathematical reasoning") somehow reminded me of one famous question, asked at the Gelfand's Seminar: "What is the real mathematical meaning of this physical abstraction?"

4. Well, we can't say almost anything about the real world if we only build on rigorous mathematical reasoning. But we can say huge lots of things if we rely on rigorous mathematical reasoning plus a very small number of observational facts.

5. Uh, this strange 2D harmonic oscillator is really scary LOL, do they by chance plan to build an appropriate apparatus to realize it on 21.12.2012 ... ;-P ;-) :-D ?

As the crackpots turn their attention to Heisenberg, poor Einstein can relax a little bit ... ;-)

6. So this guy on the video is not a physicist and he talks to a crowd, for an hour, about the tricky subject of QM, and puts in on the Web ? Obviously his primary goal is not teach the correct things about QM but to reveal his own personal thoughts about it. What a douche !

7. I am watching the talk and except for the history, meaning of interpretations etc., he actually understands QM more correctly than many people employed as physicists...

8. I've just arrived at his double slit experiment explanation : it's boring. I won't continue.

9. 'The authors also say that thus "enables one to beat the standard quantum limit for force detection" That is not really true unless one takes "standard" to mean less correct, is it? I am still curious whether this could lead to improvements in precision for measurements as in a gravitational wave detector. I don't know enough math to answer that but Lubos is where I would put my money.

10. the guy from rhinocerosOct 5, 2012, 6:33:00 AM

Muchos gracias for kicking the anti-QM butts! Always enjoying your logical clarity! Speaking of particles with negative mass, isn't that what Alcubierre requires for his putative warp drive (who says that Hollywood scriptwriters aren't clairvoyant physicists?).

Eagleworks Laboratories Advanced Propulsion Physics Research:

http://ntrs.nasa.gov/search.jsp?R=20110023492. Click the link for the pdf file to the complete paper. Love to read your response.

11. This has so many subtleties. Every system has different limits for force detection. In their system, they may "detect" things more accurately because they know the result in advance, it's an integrable system.

In most of these cases, they are fighting straw men, and not only those people. There is really no universal "standard quantum limit for force detection". It's just some order-of-magnitude estimate that some people may have overgeneralized in one way or another.

It's just not interesting. It only looks interesting to the people who think that they're violating something about quantum mechanics which they're not and they know very well that everything they do is tightly within the framework of QM.

12. On a previous thread which I cannot find, you state that under rigorous treatment of Feynman's formulation, the appropriate measure over the space of paths regards all C^1 paths as having vanishing measure. This is fascinating! Could you please share a reference? I've had Lebesgue theory through Roydin (not Rudin :p )

(This is the only way to contact you easily, I apologize)

13. Dear Arcturus, it's a fun property but it's really elementary. Some good intros to path integrals surely discuss it.

Anyway, the derivation isn't hard. You just look at the behavior of Fourier coefficients. Take functions on an interval, for example. The Fourier coefficients c_n of a smooth C^1 function must asymptotically decrease faster than 1/n^2.

However, the path integral is gives a Gaussian "statistical distribution" for each Fourier coefficient - kind of independently of others - whose width goes like c_n ~ O(1/n) or so, so it's much wider. This comes simply from decomposing the action in the exponent to Fourier coefficients.

The probability that a function (collection of Fourier coefficients) whose Fourier coefficients c_n are distributed with width O(1/n) will have values of Fourier coefficients decreasing as O(1/n^2) is kind of the product of (1/n^2) / (1/n) = product of 1/n over n, and it is zero. So it's infinitely unlikely for a function in the path integral to be in C^1.

14. I think these attempts to show Einstein right all along have completely missed the point about Einstein's work on quantum mechanics. He wasn't so much against the quantum theory as some might lead you to believe. He thought quantum theory was telling us something extraordinarily deep about the world. He once said that he had spent 10 times as much time on quantum mechanics than he had on general relativity. Einstein didn't think quantum mechanics was wrong, but he wanted to have a physical account for what was going on, something which he felt was lacking from standard quantum theory. He believed in hidden variables as a way of explaining quantum correlations. Bell showed that doesn't work -- e.g. you can't simply say the correlation was set up initially, and that determines the future correlation. So, what does account for the correlation? "Telepathy"?

15. Words like "evading" or "demolishing" are better suited to marketing and promotion than math and physics. Following through the errors and your analysis is "interesting" regardless that on merits the authors conclusions are not, Those interested in bashing established theory will forever crack pots just to hear themselves be noticed making noise would be my "theory".

17. They just seem to say that superluminal speeds may be substituted to the Lorentz transformations in some trivial way. That's great but it's still true that special relativity forbids any objects that would have these superluminal frames as rest frames.

I was playing with these simple things at the high school.

18. Thanks Lubos, the light cone of your comments ever shed light on the darkness of aquarium era science.

19. Ars Technica even has it wrong. The author writes:

"When we examine these, we find that we can measure the difference in
momentum and position of the two oscillators as accurately as we like,
because the back action affects the sum of the particle positions and
their average momentum."

Not good.

20. You've had a lot of fun beating up on several anti-QM pieces in the last several months.

There is some interesting sociology / small man syndrome in the desire of many to "demolish heisenberg" or "demolish einstein".

21. This has so many subtleties. Every system has different limits for force detection. In their system, they may "detect" things more accurately because they know the result in advance, it's an integrable system. This is of course totally useless for systems where one cares about the detection, which are not integrable, which are stable with energy bounded from below, and where they can't use any insights from their toy model.

In most of these cases, they are fighting straw men, and not only those people. There is really no universal "standard quantum limit for force detection". It's just some order-of-magnitude estimate that some people may have overgeneralized in one way or another.

It's just not interesting. It only looks interesting to the people who think that they're violating something about quantum mechanics which they're not and they know very well that everything they do is tightly within the framework of QM.

22. On a previous thread which I cannot find, you state that under rigorous treatment of Feynman's formulation, the appropriate measure over the space of paths regards all C^1 paths as having vanishing measure. This is fascinating! Could you please share a reference? I've had Lebesgue theory through Roydin (not Rudin :p )

(This is the only way to contact you easily, I apologize)

23. Dear Arcturus, it's a fun property but it's really elementary. Some good intros to path integrals surely discuss it.

Anyway, the derivation isn't hard. You just look at the behavior of Fourier coefficients. Take functions on an interval, for example. The Fourier coefficients c_n of a smooth C^1 function must asymptotically decrease faster than 1/n^2.

However, the path integral is gives a Gaussian "statistical distribution" for each Fourier coefficient - kind of independently of others - whose width goes like c_n ~ O(1/n) or so, so it's much wider. This comes simply from decomposing the action in the exponent to Fourier coefficients.

The probability that a function (collection of Fourier coefficients) whose Fourier coefficients c_n are distributed with width O(1/n) will have values of Fourier coefficients decreasing as O(1/n^2) is kind of the product of (1/n^2) / (1/n) = product of 1/n over n, and it is zero. So it's infinitely unlikely for a function in the path integral to be in C^1.

Even piecewise linear (or piecewise C^1, with continuous but unsmooth connections like in |x|) functions are of measure zero.

24. I think these attempts to show Einstein right all along have completely missed the point about Einstein's work on quantum mechanics. He wasn't so much against the quantum theory as some might lead you to believe. He thought quantum theory was telling us something extraordinarily deep about the world. He once said that he had spent 10 times as much time on quantum mechanics than he had on general relativity. Einstein didn't think quantum mechanics was wrong, but he wanted to have a physical account for what was going on, something which he felt was lacking from standard quantum theory. He believed in hidden variables as a way of explaining quantum correlations. Bell showed that doesn't work -- e.g. you can't simply say the correlation was set up initially, and that determines the future correlation. So, what does account for the correlation? "Telepathy"?