A Effective Dynamics of Open Quantum Systems: Stochastic vs Unitary Models

[Mentz114]

Right. Expectation values by themselves are not sufficient for something to have an approximately definite position. To give an extreme example: If there is a 50% probability of my being in Seattle and a 50% chance of being in New York City, then it is not very meaningful to say that, approximately, my location is somewhere in South Dakota. Or to use another example: If my left foot is in boiling water and my right foot is in ice, it's not really meaningful to say that my feet are in water that is approximately 122 degrees F.

Coarse-graining is only going to give you approximately classical objects (with approximately definite positions) if the probability distribution is strongly peaked around the expectation value. That's what I don't understand about environmentally induced collapse. Why should the distribution become strongly peaked? Is there really an argument that it should be? I don't see how there could be such an argument, using just the minimal interpretation of quantum mechanics (just unitary evolution). My feeling is that the mathematics that shows such an effect must, in some nonobvious way, be incorporating a collapse assumption.

I broadly agree with what you say but it is even more complicated because "Coarse-graining is only going to give you approximately classical objects (with approximately definite positions) if the probability distribution is strongly peaked around the expectation value" is not always the case. For instance a transition from microscopic to macroscopic can be caused by diffusion in which a probability peak is smoothed and spread. Sometimes it takes coherence not decoherence to get to the classical regime.

 

[A. Neumaier]

Expectation values by themselves are not sufficient for something to have an approximately definite position. To give an extreme example: If there is a 50% probability of my being in Seattle and a 50% chance of being in New York City, then it is not very meaningful to say that, approximately, my location is somewhere in South Dakota. Or to use another example: If my left foot is in boiling water and my right foot is in ice, it's not really meaningful to say that my feet are in water that is approximately 122 degrees F.

Note that expectations never come alone. Expectations together with the standard deviations are sufficient for something to have an approximately definite position. Even in your extreme examples, both together (with the usual $3\sigma$ rule in statistics) give an under the given circumstances quite appropriate description of your uncertain location (namely ''somewhere in North America'') in the first case and the assumed description by a single temperature in the second case.

the mathematics that shows such an effect must, in some nonobvious way, be incorporating a collapse assumption.

The mathematics incorporates, in some nonobvious way (through the Markov approximation), an assumption of chaotic behavior, which together with the multimodality coming from the nonlinear evolution of the variables kept in the reduced description of a metastable system, produce dissipation and settlement in one of the local minimizers corresponding to definite measurements. It is not so different from what you get when you bend a classical, rotationally symmetric rod using a force in the direction of the axis of the rod: if the force exceedss the threshold where the straight rod becomes metastable only, the rod will bend into a random, but definite direction. Randomness from Hamiltonian dynamics plus the tiniest amount of uncertainty about the deviations from perfect symmetry.

 

[vanhees71]

In the Weinberg's paper we have exponentially decreasing terms which are negligible at large but finite times, so the infinite time is not essential. But note that "collapse" in Weinberg's paper is not the same thing as "collapse" in most of the literature. For Weinberg, the "collapse" is merely a transition to a density matrix without non-diagonal terms.

This is the ONLY meaning "collapse" can have in physical terms. That's what I fight against all the time in our discussions: There is no instantaneous collapse, but it's all quantum dynamics of relevant (coarse-grained) observables of the macroscopic system (and measurement devices are macroscopic systems). I'll have a careful look at Weinberg's paper this evening. He's THE no-nonsense physicist, and I hope the usual clear statement against any "esoterics" will be given (as in his marvelous textbook on quantum mechanics).

 

[vanhees71]

I broadly agree with what you say but it is even more complicated because "Coarse-graining is only going to give you approximately classical objects (with approximately definite positions) if the probability distribution is strongly peaked around the expectation value" is not always the case. For instance a transition from microscopic to macroscopic can be caused by diffusion in which a probability peak is smoothed and spread. Sometimes it takes coherence not decoherence to get to the classical regime.

Classical behavior is, according to quantum theory, always only "approximate", but it's a so damn good approximation for very many macroscopic observables that it took at least 300 years (from Newton to Heisenberg) to figure out that classical physics is not the full story!

 

[Demystifier]

This is the ONLY meaning "collapse" can have in physical terms.

Perhaps, but we have another, more common expression for that: decoherence.

 

[atyy]

This is the ONLY meaning "collapse" can have in physical terms. That's what I fight against all the time in our discussions: There is no instantaneous collapse, but it's all quantum dynamics of relevant (coarse-grained) observables of the macroscopic system (and measurement devices are macroscopic systems). I'll have a careful look at Weinberg's paper this evening. He's THE no-nonsense physicist, and I hope the usual clear statement against any "esoterics" will be given (as in his marvelous textbook on quantum mechanics).

Unfortunately you are quite wrong (probably due to being mislead by Ballentine and Peres). Weinberg abandoned you already in his textbook, stating collapse explicitly with an equation. Rubi has also abandoned you by using consistent histories, which does not have deterministic unitary evolution as fundamental.

 

[vanhees71]

Yes, I never use the workd "collapse" in context of quantum theory. The only collapse in physics that's really interesting is the gravitational collapse of a star at the end of its life, and that's really happening, while the collapse of the quantum state is just fiction!

 

[vanhees71]

Unfortunately you are quite wrong (probably due to being mislead by Ballentine and Peres). Weinberg abandoned you already in his textbook, stating collapse explicitly with an equation. Rubi has also abandoned you by using consistent histories, which does not have deterministic unitary evolution as fundamental.

Weinberg leaves the answer about which interpretation is correct (or even whether there is a completely satisfactory interpretation) open in his book. He spends quite some time in his book to show that the Born rule cannot be satisfactorily derived from the other postulates (concerning kinematics and dynamics of standard quantum theory):

[Weinberg: Lectures on Quantum mechanics, p. 96]
There is nothing absurd or inconsistent about the decoherent histories
approach in particular, or about the general idea that the state vector serves only
as a predictor of probabilities, not as a complete description of a physical system.
Nevertheless, it would be disappointing if we had to give up the “realist” goal of
finding complete descriptions of physical systems, and of using this description
to derive the Born rule, rather than just assuming it. We can live with the idea
that the state of a physical system is described by a vector in Hilbert space rather
than by numerical values of the positions and momenta of all the particles in the
system, but it is hard to live with no description of physical states at all, only
an algorithm for calculating probabilities. My own conclusion (not universally
shared) is that today there is no interpretation of quantum mechanics that does
not have serious flaws, and that we ought to take seriously the possibility of find-
ing some more satisfactory other theory, to which quantum mechanics is merely
a good approximation.

I would have stopped after the 1st sentence, but of course you can have the view that physics should be more than a mere accurate quantitative description of the world. I don't think so. That's the only purpose of physics (or more generally the natural sciences). Everything else (call it metaphysics or perhaps even religion) is another level of human experience and does not belong to science but to the subjective side of our worldview.

 

[atyy]

Weinberg leaves the answer about which interpretation is correct (or even whether there is a completely satisfactory interpretation) open in his book. He spends quite some time in his book to show that the Born rule cannot be satisfactorily derived from the other postulates (concerning kinematics and dynamics of standard quantum theory):

Sure, but the interpretation you are advocating is not among those he considers having a chance to be correct. Weinberg allows minimal interpretation with collapse, MWI, BM, consistent histories etc. He does not allow Ballentine's and Peres's erroneous handwaving.

 

[vanhees71]

Where does he say so? If I interpret Weinberg correctly, he does not advocate the collapse flavor of Copenhagen Interpretations as satisfactory, because it claims that there's something outside of the validity of quantum dynamics. If I understand him right, there's no solution to this problem yet, i.e., either one should be able to derive the collapse within quantum dynamics, and then it's superfluous or it may be that quantum theory is incomplete and thus must be substituted by some more comprehensive theory that makes the collapse superfluous in some other way.

I think Ballentine's and Peres's view are far from erroneous. They just take QT in the minimal interpretation literally and consider it as a formalism to predict prbabilities. They don't claim "completeness" in the one or the other sense for quantum theory either. I think that's a very valid and pragmatic view, reflecting how QT is used in the labs around the world to invent new experiments, analyze them and describe their outcome. So far the result is that minimally interpreted QT without any collapse explains the outcome of the experiments. The experiments are silent on what may be behind the probabilities predicted by QT, and even whether there is something else behind it. So what?

 

[atyy]

Where does he say so? If I interpret Weinberg correctly, he does not advocate the collapse flavor of Copenhagen Interpretations as satisfactory, because it claims that there's something outside of the validity of quantum dynamics. If I understand him right, there's no solution to this problem yet, i.e., either one should be able to derive the collapse within quantum dynamics, and then it's superfluous or it may be that quantum theory is incomplete and thus must be substituted by some more comprehensive theory that makes the collapse superfluous in some other way.

I think Ballentine's and Peres's view are far from erroneous. They just take QT in the minimal interpretation literally and consider it as a formalism to predict prbabilities. They don't claim "completeness" in the one or the other sense for quantum theory either. I think that's a very valid and pragmatic view, reflecting how QT is used in the labs around the world to invent new experiments, analyze them and describe their outcome. So far the result is that minimally interpreted QT without any collapse explains the outcome of the experiments. The experiments are silent on what may be behind the probabilities predicted by QT, and even whether there is something else behind it. So what?

The collapse in standard terminology is just a way of calculating probabilities, and not necessarily physical. You make the error of asserting that it cannot be physical. Ballentine and Peres mislead people into believing that the minimal interpretation can do without a classical/quantum cut, without state reduction, and has no measurement problem.

BTW, you have understood Weinberg correctly. What he says counters the claim you, Ballentine and Peres make that collapse can be derived from deterministic unitary quantum dynamics in a minimal interpretation.

Also, the idea that if collapse cannot be derived from deterministic unitary quantum dynamics, then a more comprehensive theory is needed is one way of stating that the minimal interpretation does have a measurement problem.

 

[vanhees71]

But there is no measurement problem! We use QT to analyze our experiments and find it to be correct to high accuracy. So where is a problem? It's for sure not in physics. Maybe you (and obviously also Weinberg) are dissatisfied from a philosophical point of view. If you then can come up with a more comprehensive theory, it's great, but it is very unlikely to find a new theory just pondering philosophical quibbles, at least I'm not aware of any great theory being found without a sound and solid foundation in true observations in nature.

 

[atyy]

But there is no measurement problem! We use QT to analyze our experiments and find it to be correct to high accuracy. So where is a problem? It's for sure not in physics. Maybe you (and obviously also Weinberg) are dissatisfied from a philosophical point of view. If you then can come up with a more comprehensive theory, it's great, but it is very unlikely to find a new theory just pondering philosophical quibbles, at least I'm not aware of any great theory being found without a sound and solid foundation in true observations in nature.

It is fine to assert that there is no measurement problem, in the sense that the classical/quantum cut is a feature. However, in any minimal interpretation there is a classical/quantum cut. One can say this is not a problem, as Bohr and Heisenberg did, and I respect that view (actually I quite like it). One can also say there is a problem, as Dirac did. Either way, no one is disagreeing with the success of the theory.

However, what is not acceptable is to say there is no classical/quantum cut and no collapse in a minimal interpretation.

 

[vanhees71]

Where is a classical/quantum cut by applying standard quantum mechanics to predict probabilities and then measure them by looking at large ensembles of preparations?

 

[A. Neumaier]

no collapse in a minimal interpretation.

There is no collapse in a minimal interpretation since nothing at all is asserted about a single system. The statistical assertions about multiple measurements are equivalent to the assertion that the diagonal entries of the density matrices are the observed probabilities. No cut is needed either since the observations are made by quantum systems in the environment, reading macroscopic pointer variables, i.e., expectation values.

Weinberg's view that the collapse in an open system is equivalent to the decay of the off-diagonal entries of the density matrix in the pointer basis is also in agreement with the minimal interpretation - where only the density matrix matters, though not with the Copenhagen interpretation (which makes assertions about the state of single systems).

 

[A. Neumaier]

A paper by Weinberg that appeared today may be relevant here:
http://lanl.arxiv.org/abs/1603.06008

I am quite disappointed with his paper. He just assumes the reduced dynamics of an open system (rather than deriving it from unitarity, with the associated approximate nature this entails), and simply investigates conditions on the coefficients of the Lindblad equation that make it an acceptable description of a measurement process.

 

[atyy]

Where is a classical/quantum cut by applying standard quantum mechanics to predict probabilities and then measure them by looking at large ensembles of preparations?

If the wave function and collapse are not taken to be necessarily real, but the experimental apparatus and results are real, then we have to have a cut somewhere.

 

[Demystifier]

I am quite disappointed with his paper. He just assumes the reduced dynamics of an open system (rather than deriving it from unitarity, with the associated approximate nature this entails), and simply investigates conditions on the coefficients of the Lindblad euation that make it an acceptable description of a measurement process.

If you are disappointed that he didn't derive the Lindblad equation from first principles, well, he cited some references were it was done.
You might also want to see a derivation of the Lindblad equation in the Appendix of my
http://arxiv.org/abs/1502.04324 [JCAP 04 (2015) 002]

 

[A. Neumaier]

If you are disappointed that he didn't derive the Lindblad equation from first principles, well, he cited some references were it was done.
You might also want to see a derivation of the Lindblad equation in the Appendix of my
http://arxiv.org/abs/1502.04324 [JCAP 04 (2015) 002]

I was disappointed that Weinberg didn't really do anything new in the paper. I know many derivations of Lindblad equations under many different assumptions.

 

[Demystifier]

I was disappointed that Weinberg didn't really do anything new in the paper.

That was my impression too.

 

[A. Neumaier]

If the wave function and collapse are not taken to be necessarily real, but the experimental apparatus and results are real, then we have to have a cut somewhere.

It is enough to know that the density matrix of the experimental apparatus makes very accurate predictions of the macroscopic variables. This doesn't require a cut.

 

[atyy]

It is enough to know that the density matrix of the experimental apparatus makes very accurate predictions of the macroscopic variables. This doesn't require a cut.

Or to put it the way Weinberg does - there are two rules for time evolution - deciding which one to use when requires outside input.

 

[A. Neumaier]

Or to put it the way Weinberg does - there are two rules for time evolution - deciding which one to use when requires outside input.

No. It is decided by the way the system is coupled to the detector - which is a property of the combined quantum system. Before and after the system passes the detector, the system is approximately isolated and hence the unitary dynamics is an appropriate approximation. During the interaction with the detector, the system is obviously coupled and the appropriate approximation is a reduced dynamics in Lindblad form.

No outside input is needed - except of course for deciding about which system to consider. But the latter kind of input is also needed when describing classical subsystems of a classical universe. In this sense you should probably claim that classical systems also need an observer to make sense, and hence a cut.

 

[stevendaryl]

Where is a classical/quantum cut by applying standard quantum mechanics to predict probabilities and then measure them by looking at large ensembles of preparations?

I would say that "measure them" means to make a persistent, macroscopic record of some property. So the act of measurement necessarily involves macro/micro distinction. Probabilities don't apply to microscopic properties; if an electron is in the state "spin-up in the z-direction", it doesn't really make sense to say that it has a probability of 50% of being spin-up in the x-direction. It only makes sense to say that a measurement of the spin in the x-direction will result in spin-up. So probabilities only apply to measurements, which are necessarily macroscopic events.

That to me is the measurement problem: How do probabilities arise for a macroscopic event, when the macroscopic event is (presumably) just made up of many, many microscopic events for which probability is not meaningful? You can say that it arises, as it does in classical statistical mechanics, from coarse-graining, but I don't think that's true. Coarse graining introduces probabilities through ignorance of the details of the actual state, but Bell's theorem shows that quantum probabilities cannot be interpreted as being due to ignorance (at least not without nonlocal or retrocausal influences).

 

[A. Neumaier]

the act of measurement necessarily involves macro/micro distinction.

But this is not a cut. ''macro = 10^10 or more atoms'' is fully sufficient for typical accuracies.

Bell's theorem shows that quantum probabilities cannot be interpreted as being due to ignorance

But Bell's theorem is about a microscopic system. The detectors are macroscopic and produce a unique result. That it looks random can be understood in the same way as random buckling in a classical symmetry breaking situation. The Bell state effectively acts like an external force breaking the metastability of the detector state.

 

[stevendaryl]

But this is not a cut. ''macro = 10^10 or more atoms'' is fully sufficient for typical accuracies.

But Bell's theorem is about a microscopic system. The detectors are macroscopic and produce a unique result. That it looks random can be understood in the same way as random buckling in a classical symmetry breaking situation. The Bell state effectively acts like an external force breaking the metastability of the detector state.

You've said things along those lines before, but I don't understand how that can be true. Or to put it more strongly: I do not think that it is true. I do not believe that the appearance of definite outcomes is explained by analogy to classical symmetry breaking.

It certainly is the case that measurement requires a metastable system. That's the way that microscopic events get magnified to macroscopic events. But I think you're mixing up two different notions of "definite outcome" when you propose that the "collapse" of a metastable system results in a definite state.

If I have a coin balanced on its edge, that is one kind of "indefinite state" between "heads" and "tails". It's metastable, in that the slightest perturbation will result in either in "heads" or "tails". That's a classical type of "collapse". But quantum mechanics introduces another kind of indefinite state: superpositions. If it made sense to talk about the wave function of something as large as a coin, then presumably you could have a state of a coin that is a superposition of the coin being heads and the coin being tails. That is an indefinite state that is completely unrelated to the classical metastable state of a coin balanced on its edge.

To me, it seems that by invoking metastability to explain why there are definite outcomes, you are mixing up two completely different notions of "indefinite state".

 

[Demystifier]

But Bell's theorem is about a microscopic system.

I disagree. It is about directly measurable phenomena explained in terms of variables which are not directly measurable. The former (directly measurable phenomena) are macroscopic almost by definition, while the latter (variables which are not directly measurable), in principle, may be either microscopic or macroscopic.

 

[A. Neumaier]

you are mixing up to completely different notions of "indefinite state".

There is only a single notion of state, and it is very definite. Both in classical and in quantum statistical mechanics, it is a density operator, if one uses for classical mechanics the Koopman representation, where the operators are multiplication operators in some Hilbert space of functions on phase space. The only difference between the classical and the quantum case is that in the former case, all operators are diagonal. Decoherence shows that in a reduced description, the density matrices soon get very close to diagonal, recovering a Koopman picture of classical mechanics after a very short decoherence time. While the Koopman dynamics is strictly linear in terms of the density matrix (comparable to a quantum Lindblad equation), the resulting dynamics is highly nonlinear when rewritten as a classical stochastic process (comparable to a quantum jump or quantum diffusion process). Thus decoherence in principle provides (though only very few people think of it in these terms) a reduction of the quantum mechanics of open systems to a highly nonlinear classical stochastic process.

This stochastic process is no different in character from the stochastic process governing the dynamics of a metastable inverted pendulum, and hence has the same kind of multistable effective behavior that leads to definite classical outcomes within the accuracy due to the approximations involved. I recommend reading papers on optical bistability, e.g. this one or this one, where it is shown how semiclassical bistability arises from a quantum model by projecting out irrelevant degrees of freedom.

 

[A. Neumaier]

I disagree. It is about directly measurable phenomena explained in terms of variables which are not directly measurable. The former (directly measurable phenomena) are macroscopic almost by definition, while the latter (variables which are not directly measurable), in principle, may be either microscopic or macroscopic.

Of course the theorem itself is about classical hidden variables, but the experiments violating the resulting inequality are about a microscopic 2-photon system interacting with two macroscopic devices.

 

[stevendaryl]

There is only a single notion of state, and it is very definite. Both in classical and in quantum statistical mechanics, it is a density operator

I think that's very misleading. If you are invoking "metastability" as an explanation for why pointers or cats, or whatever, have (apparently) definite positions, or approximately definite positions. Saying that the density operator is definite is using a different notion of "definite".

The issue, which metastability does not address, and neither does the use of density matrices, or whatever, is this: Why do classical objects such as cats or pointers have approximately definite positions (and approximately definite momenta, etc.)? Metastability has nothing to do with it.

I think that Many Worlds gives real insight into this question, even if you don't believe in it. If you consider the wave function of the entire universe, then there is no reason to think that macroscopic objects have approximately definite positions. There can perfectly well be a state for the whole universe that is a superposition of a state where I am in Seattle, and another state where I am in New York City. Metastability does not at all imply that such superpositions can't exist, or are unstable. But what we can argue is that there are negligible interference effects between these two elements of the superposition. So these two elements of the superposition will evolve independently as if they were completely separate worlds. The "me" in each branch can consistently believe that his is the ONLY branch, and he will never face a contradiction. If there is no way to observe the other branches, then you can apply Occam's razor and get rid of them from your ontology. But there is no physical event corresponding to "getting rid of the other branches". There is no physical collapse, whether mediated by metastability or not.

Even if you don't buy the Many Worlds interpretation, one aspect of it is true for any interpretation: If macroscopic objects failed to have definite positions, it would be unobservable.