Friday, February 03, 2017

Lindblad equation can't solve any "problems" of quantum mechanics

"Testing Quantum Foundations With Atomic..." (a blog post elsewhere) has basically praised an October 2016 paper by Weinberg, Lindblad Decoherence in Atomic Clocks, as an example of research that makes the American research of quantum mechanics great again because it brings "a little less talk, a little more action".

Last March, I already discussed Weinberg's efforts to incorporate the Lindblad equation into the discussions about foundations of quantum mechanics. When it comes to the technical content, Weinberg shows how a particular modification of the equations of quantum mechanics, the Lindblad equation (whose extra terms cause some decoherence even in the absence of any "environment"), could be tested by the experimenters' precise gadget that nicely maintains the coherence, the atomic clocks.

Great, it wouldn't be shocking if the most precise "coherent" (i.e. accurately remembering the relative phase between two parts of a wave etc.) apparatuses we have, atomic clocks, could be used to test some hypothesis about new sources of "decoherence" (i.e. a process that makes Nature forget about the relative phase). Weinberg's paper doesn't really end with any constraints for the coefficients of the new terms or inequalities and I would be willing to be 100-to-1 that no such unitarity-violating terms will be found by 2025. But one may discuss the detection techniques for newly proposed modifications of established equations.

What I find more ludicrous is Weinberg's and another physicist's suggestions that such new terms would "solve" something about what they consider mysteries, paradoxes, or problems of quantum mechanics. The first sentence of Weinberg's paper says
In searching for an interpretation of quantum mechanics we seem to be faced with nothing but bad choices.
and the following sentences repeat some of the by now standard Weinberg's critical words about Copenhagen as well as other "interpretations". The message is that this work about the extra "Lindblad terms" solve some mystery of quantum mechanics because they make something like the wave function collapse "more real". Similarly, another physicist's most positive paragraph in favor of these efforts says:
What would really solve the problem, however, is some type of fundamental decoherence, an actual collapse prescription basically. It’s not a particularly popular idea, but at least it is an idea, and it’s one that’s worth testing.
I don't think that the right word is "unpopular" to describe the statement that such "fundamental decoherence" would "really solve the problem". Instead, this statement is self-evidently wrong.

Even if the extra Lindblad parameters \(\lambda_{mn}\) were nonzero and discovered, and it won't happen, we would't find any "more enlightening" version of quantum mechanics. We would still have similar equations with the same objects and with some new terms that used to be zero but now they are nonzero. If a conceptual change appeared at all, the situation would clearly get more mysterious, not less so. If someone finds neutrinos mysterious, the discovery of the nonzero neutrino masses hardly makes things easier for him. Or consider the same sentence with the QCD theta-angle, CP-violating phases, cosmological constant, or any other parameter that could have been zero but wasn't. If you couldn't understand the theory with a vanishing value of these parameters, the more complex or generalized theory with the new nonzero parameters will be even harder for you, won't it?

OK, the Lindblad equation is the following equation for a density matrix:\[

\dot \rho(t) &= -i[H,\rho(t)]+\\
&+\sum_\alpha \left[ L_\alpha \rho(t) L^\dagger_\alpha-\frac 12\left\{ L_\alpha^\dagger L_\alpha,\rho(t) \right\} \right]

\] This equation is the most general linear equation for the density matrix \(\rho(t)\) that preserves its trace (total probability) and the Hermiticity. The sum over \(\alpha\) runs over at most \(N^2-1\) new terms. Aside from the Hamiltonian matrix \(H\), one must pick many new operators \(L_\alpha\) and their conjugates to define the laws of physics.

I've divided the equation to two lines. The first line is the normal equation for the density matrix, one easily derived from the Schrödinger equation for \(\ket\psi\). The second line contains all the new terms that are zero according to contemporary physics but proposed to be nonzero by Weinberg (and others) and that should be tested by atomic clocks.

Note that \(\rho(t)\) is Hermitian, and so is therefore the left hand side. The first, normal term of the right hand side is a commutator with \(H\) which is Hermitian. For the commutator to be Hermitian as well, the coefficient has to be pure imaginary. On the contrary, the new Lindblad terms have a real coefficient.

To see what these terms are doing or "should do", it's better to look at an Ansatz for a solution – which is Weinberg's equation (3):\[

\rho_{mn}(t) = \rho_{mn}(0) \times \exp\left[ -i(E_m-E_n)t -\lambda_{mn}t \right].

\] The Ansatz was written in an energy eigenstate basis. The oscillating part of the exponent looks just like in Heisenberg's papers and the frequency is \(E_m-E_n\). The diagonal elements of \(\rho(t)\) don't change at all while the off-diagonal elements have a phase that changes with time with this frequency. What's new is the extra, exponentially decreasing factor of \(\exp(-\lambda_{mn}t)\). The off-diagonal elements don't have a constant absolute value, as they should have in unitary quantum mechanics, but they're exponentially damped with some rate \(\lambda_{mn}\) which are parameters bilinear in the matrix elements of the \(L_\alpha\) matrices in the Lindblad equation.

These off-diagonal elements of the density matrix contain the information about the relative phases of the wave function. Decoherence makes them go to zero. Here they are going to zero exponentially so it's "some kind of decoherence". Except that this is proposed to be decoherence due to new terms in the fundamental laws of physics, not due to the interaction with a subsystem labeled the "environment".

The Lindblad equation may appear as an effective equation for an open system that interacts with some environment that we can't trace so instead, we trace over it. But does it make any sense to consider it as a fundamental equation? I don't think so.

First, the modification back to \(\lambda_{mn}=0\) is just prettier and better.

I decided to place this objection at the top. The point is that the addition of all these \(\lambda_{mn}\neq 0\) damped factors is extremely artificial and it makes sense to cut this whole line of generalization by Occam's razor. If the Lindblad equation for some \(H\) and some \(L_\alpha\) has some nice properties, you may be pretty sure that the equation where you simply set \(L_\alpha=0\) is at least equally pretty. You can't lose any virtue by that. On the contrary, you lose virtues when you consider nonzero \(L_\alpha\).

Second, lots of new operators have to be defined on top of the Hamiltonian.

This is an addition to the first complaint but it may be viewed as an independent one. In normal quantum mechanics, we only determine one matrix on the Hilbert space, the Hamiltonian (or directly the S-matrix etc.). Here we must choose the Hamiltonian and about \(N^2-1\) additional operators on the Hilbert space \(L_\alpha\). Who are they? What deeper principle could possibly determine or at least constrain them?

Third, the Lindblad equation doesn't allow any Heisenberg picture at all.

The normal equation has \(L_\alpha=0\) and only contains the commutator with \(H\) in the evolution. Consequently, the evolution in time is a unitary transformation. You may pick a time-dependent basis of the Hilbert space in which the coordinates of \(\ket\psi\) or \(\rho\) will look constant and the operators such as \(x(t),p(t)\) will be time-dependent instead. This is the Heisenberg picture. With the Lindblad equation, you can't do that. There's no basis in the Hilbert space in which \(\rho(t)\) could be constant – after all, its eigenvalues are changing with time. Consequently, you won't be able to write this theory in any Heisenberg picture.

This is a far deeper problem than people like Weinberg may realize. One reason is that the equations for the operators in the Heisenberg picture basically emulate the classical evolution equations for \(x(t),p(t)\) etc. The Heisenberg picture is an elegant way to see that quantum mechanics reduces to classical physics. Now, because you can't write the Weinberg-Lindblad theory in the Heisenberg picture, you won't be able to show the right classical limit. So in fact, by adding the new Weinberg-Lindblad terms, you have made the theory less compatible with classical physics that Weinberg loves so much, not more so!

For this reason, I also suspect that you wouldn't need any atomic clocks to falsify this theory. This theory almost certainly predicts some completely wrong unobserved things for physical systems that are highly classical.

Fourth, the new terms are pretty much by definition proofs that "you are missing something"

I've mentioned that the Lindblad equation may be obtained as an effective equation if you eliminate some environment you can't track. I would argue that the converse is true, too. If you have the Lindblad equation, it shows that it's some effective equation, you have eliminated some degrees of freedom, you should return to the blackboard and see what this deeper physics that you have ignored is and where it is hiding! Weinberg is acting as he believes that the opposite is true: If he found the ugly new terms that normally emerge in effective theories only, he would be led to believe that he has found a more fundamental theory. This thinking clearly seems upside down.

OK, what are you missing when you see these new effective terms?

Bonus: the Lindblad equation is a quantum counterpart of "classical physics with Brownian random forces"

In classical deterministic physics, if you know the point \(x_i(t),p_i(t)\) in the phase space at one moment, you may calculate it at later moments \(t\), too. To explain the Brownian motion, Einstein (and the Polish guy) considered a generalization of deterministic classical physics in which the particle is also affected by classical but random forces (from the surrounding atoms) which are described by some distributions.

So even if the precise position and momentum were known at one moment, they would be unknown after some time of the Brownian motion. The peaked distribution on the phase space would get "dissolved".

This is exactly how you should think about the effect of the new Lindblad terms. They're like some random forces described in terms of the density matrix. Is something getting dissolved as well? Is the exponential decrease of the off-diagonal elements equivalent to the classical spreading of the distribution on the phase space?

You bet. It's not obvious in the basis that Weinberg chose – if the diagonal entries of \(\rho\) don't change. But if you pick any different basis, even the diagonal entries will change – they will be evolving towards values that are closer to each other and that's equivalent to the dissolution of the peaked distribution in the phase space. So there should be some molecules etc. that are causing this randomization of the pollen particle etc.!

Fifth, the new terms violate the conservation laws and/or locality

In a 1983 paper that Weinberg is aware of, Banks, Susskind, and Peskin argued that the equation violates either locality or energy-momentum conservation. Weinberg mentions this paper as well as a 1995 paper by Unruh and Wald which claims to have found some counterexamples to Banks et al. I don't quite understand what those guys have done but I am pretty sure that the counterexamples would have to be extremely artificial.

Look at the formula for \(\rho_{mn}(t)\) above. You see that if you want to preserve the energy conservation law, you really want the exponential decrease to affect the off-diagonal elements in an energy basis only. It means that the matrices \(L_\alpha\) in the extra terms must be able to determine or "calculate" what the energy eigenvectors are. If you just place some generic matrices, the conservation laws will be violated.

Sixth, CPT theorem trouble

Also, the solution to the Lindblad equation has entries that are exponentially decreasing in time. That's an intrinsic time-reversal asymmetry. Well, the legality of these solutions and the elimination of the opposite ones contradicts the existence of any CPT-symmetry. So the CPT-theorem just couldn't hold in any generalized Weinberg-Lindblad theory of this kind. You could ask whether it should hold at all.

Well, I think it should. The CPT transformation is just a continuation of the Lorentz group, the rotation of the \(t_Ez\)-plane by 180 degrees which just happens to make sense even in the Minkowski signature. So the CPT symmetry is closely linked to the Lorentz symmetry. None of this reasoning may be quite applied to the Weinberg-Lindblad theory because operations (in particular, the evolution operations) are not identified with unitary transformations in that theory etc. But I think it must lead to inconsistencies – either non-locality or a violation of the conservation laws.

I am convinced that under reasonable assumptions, it leads to problems with both – conservation laws as well as locality and/or Lorentz symmetry. One "morally non-relativistic" aspect of the Lindblad laws is that the evolution in time isn't represented just by a unitary operator while the translation i.e. evolution in space is still just a unitary transformation. So the temporal and spatial components of a four-vector (energy-momentum) seem to be qualitatively different. I would be surprised if the Lorentz invariance could be preserved by laws like that – at least if these laws are determined by some principles, instead of just by an artificial construction designed to prove me wrong.

Seventh, it just doesn't help you with any "mysteries of quantum mechanics"

But as I said, the most important problem isn't any particular technical flaw in the equations even though I do believe that the troubling observations above are flaws of the theory. The main problem is that these analyses have nothing to say about the "broader problem" that Weinberg talks about, namely his problems with the foundations of quantum mechanics.

Imagine that the new terms exist and are nonzero. So there exists an experiment, e.g. one with an atomic clock, that may show that some \(\lambda_{mn}\neq 0\). This experiment must be accurate enough – so far, similar experiments couldn't see any violation of normal quantum mechanics i.e. they couldn't have proven any \(\lambda_{mn}\neq 0\). The evidence that the new parameters are nonzero is increasing with some time – because these terms cause some intrinsic decoherence that deepens with time.

OK, so even if you said that the experiment for times \(t\gt t_C\) that are enough to see the new Weinberg-Lindblad effects proves that "things are less mysterious" because the relative phases have dropped almost to zero, it would still be true that for \(t\lt t_C\), the damping is small or negligible and the system basically follows the good old unitary rules of quantum mechanics. So the "trouble with quantum mechanics" when applied to your experiment at \(t\lt t_C\) would be exactly the same as it was before you introduced the new terms! The effect of all the new terms would be small or negligible, just like in all experiments that have been confirming unitary quantum mechanics so far.

The idea that the damping of some elements of the density matrix reduces the mystery of quantum mechanics is utterly irrational. At most, the Lindblad-Weinberg equation – if a natural version of it could exist, and I feel certain that it can't – could pick a preferred basis of the Hilbert space e.g. of your brain that would tell you which things you may feel and which you can't. Except that even in normal quantum mechanics, it's not needed. Even without decoherence, any density matrix may be diagonalized in some basis. So you may always view it as the basis that may be would-be classically perceived, if you adopt the viewpoint is that the non-vanishing off-diagonal elements clash with the perception.

And like ordinary decoherence, this Lindblad-induced decoherence doesn't actually pick one of the outcomes. Decoherence makes a density matrix diagonal but it doesn't bring it to the form \({\rm diag}(0,0,1,0,0,0)\) or a similar one.

To summarize, even if pieces of the analyses of atomic clocks are correct, the broader talk about all these things is completely wrong. None of these hypothesized new terms can "solve" any of the "problems" that Weinberg talks about. Weinberg has confined these wrong comments about the interpretation to the first paragraph of his paper. But the other physicist didn't confine them. Let me mention his or her sentences that aren't right:
Each time a quantum state interacts with an environment – air, light, neutrinos, what have you – it becomes a little less quantum.

So how come on large scales our world is distinctly un-quantum?
Our world is never un-quantum. Our world – and both small and large objects in it – obey the laws of quantum mechanics. If you think that any observation of large objects we know disagrees with quantum mechanics, and it's the only meaning of "un-quantum" I can imagine, then you misunderstand what quantum mechanics actually does and predicts.
It seems that besides this usual decoherence, quantum mechanics must do something else, that is explaining the measurement process.
Decoherence is not "needed" for anything. It's just an effective re-organization of the dynamics in situations where a part of the physical system may be viewed as an environment, a re-organization that explains why the relative phases are being forgotten – and therefore one of the first steps needed to explain why a classical theory is sufficient to approximately describe everything (decoherence is needed for that because the main thing that classical physics refuses to remember are the relative quantum phases). But the forgetting still obeys the laws of quantum mechanics, it in no way contradicts it.

If "someone" is doing something else, it's just not quantum mechanics. The dynamical laws of quantum mechanics are performing the evolution of the probability amplitudes – either in the state vector, density matrix, or operators. The rest is to connect these probability amplitudes with the observations. But this isn't done by Nature. Instead, it's done by the physicist. It's the physicist who must understand what a probability amplitude or a probability means and that's what allows him to apply the calculations of the unitary evolution on objects around him. But the application of the laws isn't something that "Nature does". Instead, it is what a "physicist does". And if she doesn't know how to do it right, or if she has some religious or psychological obstacles that prevent him or her from doing it at all, it's his or her defect, not Nature's. (Note that I have used "he or she" and "him or her" in order to be politically correct.)

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