Does chaos amplify inherent quantum level randomness?

[Physicist248]

Does chaos amplify inherent quantum level randomness/uncertainty to macroscopic level?

 

[Filip Larsen]

In principle yes. If you have a chaotic system of, say, interacting quantum particles then in principle you cannot know the exact initial condition and hence, at some point the quantum uncertainty has grown to the level of the macroscopic scale to which the motion is bounded.

 

[PeterDonis]

If you have a chaotic system of, say, interacting quantum particles

You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.

 

[Filip Larsen]

You can't, because the quantum evolution equation, Schrodinger's Equation, is linear

I understand that the coupling between chaos and quantum effects are not as straight forward as my intuition from the two fields alone tries to tell me, and that good examples of experiments should a clear coupling are indeed difficult to find.

However, I imagine that chaos enabling non-linearity should be possible to "arranged" by suitable choice of geometry and/or driving fields, e.g. to enable topological mixing of quantum particles. I mean, if we can have chaos in a classical Hamiltonian system of particles (e.g. the general three-body problem in gravity) then it is easy to imagine that replacing the classical observables with their quantum counterparts would allow quantum uncertainty to influence the non-linear interactions.

Perhaps I am too naive thinking it should be easy. The Phys Letter B paper Can chaos be observed in quantum gravity, while being a bit over my engineering head, do seem to conclude there are plenty of challenges pointing to a general link between quantum effects and chaos.

 

[Filip Larsen]

Perhaps a more simple example of coupling between quantum effects and chaotic system would be Chau's circuit where the (statistical) effects of thermal noise has been included. In a classic model of the circuit the state is fully deterministic, but if one includes the relevant effects of thermal noises in the evolution of the state then one should expect any uncertainty in state to grow exponentially to macroscopic scale.

It could be interesting to know if it is possible to make Chua's circuit at nano-scale involving only a discreet number of electrons put into a "non-linear" coupling with each other.

 

[Jarvis323]

You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.

What about collapse though?

 

[PeterDonis]

What about collapse though?

Collapse doesn't have any dynamics in basic QM; it's just a mathematical operation you perform after you know the result of a measurement. Discussion of particular interpretations that give more meaning to collapse than that belongs in the interpretations subforum.

 

[PeterDonis]

I imagine that chaos enabling non-linearity should be possible to "arranged" by suitable choice of geometry and/or driving fields, e.g. to enable topological mixing of quantum particles.

Nothing like this occurs in standard QM. Whether quantum gravity can allow something like this is an open question.

The Phys Letter B paper Can chaos be observed in quantum gravity, while being a bit over my engineering head, do seem to conclude there are plenty of challenges pointing to a general link between quantum effects and chaos.

This paper is proposing a speculative quantum gravity framework that goes beyond standard QM and standard GR. Even if such a framework turns out to be right, however, it is still not clear that "chaos amplifying quantum level randomness" would be a valid description of what it says. The chaos this paper is talking about comes from the fact that the Einstein Field Equation is nonlinear.

 

[gentzen]

You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.

I don't think that this argument is valid. The Schrödinger equation is linear on configuration space, but chaos happens in normal 3D space. And since it is part of a probabilistic description, it is not even a big surprise that it is linear. The Liouville equations provide an analogous probabilistic description of classical systems (where the probabilistic description just described the uncertainty in the initial conditions), and these are also linear.

 

[Fra]

You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.

A problem here is that in QM the Hamiltonian is supposed to exist and be precise; it is not subject to inference in the foundations(which we need not discuss further here wether it ought to or not). So the unitary evolution is essentially postulated. But check this with experiment one could ask: if the system appears chaotic, does it make sense to presume that the process tomography to infer the hamiltonian (from a chaos) makes sense? If not, does it make sense to use the argument against chaos in the first place?Revisiting the role of chaos in quantum tomography
"Though quantum systems show no sensitivity to initial conditions, due to unitarity of evolution, they do show sensitivity to parameters in the Hamiltonian [2]. This leads to an interesting question for quantum tomography and, more generally, quantum simulations. Under what conditions are the system dynamics sensitive to perturbations, and how does this aect our ability to perform quantum tomography?"
-- https://arxiv.org/abs/2203.07692

/Fredrik

 

[S.G. Janssens]

You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.

One must be careful with such overly general statements. See, for example, Linear Chaos by Erdmann and Manguillot.

I don't think that this argument is valid. The Schrödinger equation is linear on configuration space, but chaos happens in normal 3D space. And since it is part of a probabilistic description, it is not even a big surprise that it is linear. The Liouville equations provide an analogous probabilistic description of classical systems (where the probabilistic description just described the uncertainty in the initial conditions), and these are also linear.

Your remark fits well into the context of ergodic theory. (Also see the book mentioned above.)

 

[Jarvis323]

Collapse doesn't have any dynamics in basic QM; it's just a mathematical operation you perform after you know the result of a measurement. Discussion of particular interpretations that give more meaning to collapse than that belongs in the interpretations subforum.

But if the temporal behavior of a quantum system depends also on collapse, then shouldn't we say the quantum system includes more than just unitary evolution?

 

[Nugatory]

But if the temporal behavior of a quantum system depends also on collapse…

There are interpretations (for example MWI) that don’t include collapse, yet successfully predict that temporal behavior. So it can’t be that the behavior depends on collapse.

 

[PeterDonis]

The Schrödinger equation is linear on configuration space, but chaos happens in normal 3D space.

No, chaos happens in whatever space the dynamics happens in. For classical mechanics, that is normal 3D space. But for QM, or at least for non-relativistic QM, it's configuration space. The point is that chaos can only happen if the dynamics is nonlinear.

 

[PeterDonis]

if the system appears chaotic, does it make sense to presume that the process tomography to infer the hamiltonian (from a chaos) makes sense?

If you want to substitute a different theory for standard QM, and it makes correct predictions, go ahead and publish it and then we can discuss it here. But until then, this forum is discussing standard QM, not some speculative replacement for it.

 

[Fra]

If you want to substitute a different theory for standard QM, and it makes correct predictions, go ahead and publish it and then we can discuss it here. But until then, this forum is discussing standard QM, not some speculative replacement for it.

QPT is just a practical inference scheme formalized to infer the dynamical laws presuming QM. So it's not a new theory, its just an inverse problem in QM.

I just note that in this scheme, in addition to the impurity of the state, in practical applications additional uncertainty in the hamiltonian may add to the chaos?

Yes a new theory may involve taking this practical problem and elevate it to something that has fundamental implications, but without doing that which i didnt, I think this is an intetesting point relevant to the topic.

/Fredrik

 

[PeterDonis]

QPT

What is QPT?

 

[Filip Larsen]

The point is that chaos can only happen if the dynamics is nonlinear.

I don't think anyone here is contending that.

The OP question, as I understand it, is to ask if quantum uncertainty can be expected to be amplified to macroscopic level given the system is exhibiting (classical) chaotic dynamics. Chaos in such a system is, as far as I know, considered an emergent behavior that does not have to be brought about by the underlying interactions themselves, but more due to the macroscopic motion of the system with both local amplification along the flow (i.e. positive Lyapunov exponent) and topological mixing of the flows in state space.

So, given we already have a system that exhibit chaotic motion on macroscopic scale, is it the to be expected or even possible for the inherent underlying quantum uncertainty to "leak" into the dynamics to be amplified to macroscopic scale? Or to take a specific experiment, will thermal noise in Chua's circuit be amplified to macroscopic level over time?

 

[Fra]

What is QPT?

Quantum Process Tomography - trying to infer the hamiltonian of a system from a large set of interactions.

/Fredrik

 

[gentzen]

No, chaos happens in whatever space the dynamics happens in. For classical mechanics, that is normal 3D space. But for QM, or at least for non-relativistic QM, it's configuration space. The point is that chaos can only happen if the dynamics is nonlinear.

For a moment I feared that our disagreement is over the meaning of 3D space and configuration space. Luckily, on closer reading of your response I see that this is not the case.

Our disagreement seems easiest to discuss in Bohmian mechanics: The particle trajectories live in a 3N dimensional space, their guiding equation is nonlinear, and we (probably) both agree that they exhibit chaotic behavior. But you say that this is not the relevant dynamic, and I even agree with this. The relevant dynamic is that of the wavefunction, which lives in some (rigged) Hilbert space of square integrable functions over $\mathbb R^{3N}$.

You want to conclude from the linearity of that dynamics (of the wavefunction) that it cannot exhibit chaos, and "I don't think that this argument is valid". My reason for this doubt is that you are completely ignoring that 3N dimensional space of the particle trajectories. This space gives us a notion of closeness of initial conditions (like a Wasserstein metric), and even the unitary evolution of the wavefunction given by the linear Schrödinger equation exhibits (exponential) sensitive dependence on initial conditions with respect to that notion of closeness. And this sensitive dependence on initial conditions is a defining characteristic of chaotic behavior.

 

[Jarvis323]

I don't think anyone here is contending that.

It is true if the system is finite dimensional. But chaos is possible for an infinite dimensional linear system.

 

[PeterDonis]

Quantum Process Tomography - trying to infer the hamiltonian of a system from a large set of interactions.

Do you have a good reference for this?

 

[PeterDonis]

Chaos in such a system is, as far as I know, considered an emergent behavior that does not have to be brought about by the underlying interactions themselves, but more due to the macroscopic motion of the system with both local amplification along the flow (i.e. positive Lyapunov exponent) and topological mixing of the flows in state space.

Do you have a reference for this?

 

[PeterDonis]

Our disagreement seems easiest to discuss in Bohmian mechanics

Only if by "Bohmian mechanics" you mean a different theory from standard QM, not just an interpretation of standard QM. As an interpretation of standard QM, the nonlinearity of the guiding equation is irrelevant since individual particle positions are unknown and are not used to make predictions about the results of experiments; the only role the particle positions play in actual predictions of measurement results (as opposed to the story that is told about measurement results after they are known) is in the assumption that their initial distribution matches the probabilities given by the wave function.

If Bohmian mechanics is treated as a different theory from standard QM, capable of making different predictions for some experiments, then the chaotic properties of the guiding equation can become relevant--but in order to make use of this at all, you would have to find some experiment where the predictions of standard QM were falsified and the predictions of this Bohmian-mechanics-as-a-different-theory were confirmed. Which has not happened and nobody seems to be trying to make it happen.

 

[PeterDonis]

chaos is possible for an infinite dimensional linear system.

Do you have a reference for this?

 

[Filip Larsen]

Do you have a reference for this?

Which part? That chaos is considered an emergent phenomenon? Or chaos theory in general?

 

[Jarvis323]

Do you have a reference for this?

Infinite-Dimensional Linear Dynamical Systems with Chaoticity​

The authors present two results on infinite-dimensional linear dynamical systems with chaoticity. One is about the chaoticity of the backward shift map in the space of infinite sequences on a general Fréchet space. The other is about the chaoticity of a translation map in the space of real continuous functions. The chaos is shown in the senses of both Li-Yorke and Wiggins. Treating dimensions as freedoms, the two results imply that in the case of an infinite number of freedoms, a system may exhibit complexity even when the action is linear. Finally, the authors discuss physical applications of infinite-dimensional linear chaotic dynamical systems.

https://link.springer.com/article/10.1007/s003329900069Linear Chaos

It is commonly believed that chaos is linked to non-linearity, however many (even quite natural) linear dynamical systems exhibit chaotic behavior. The study of these systems is a young and remarkably active field of research, which has seen many landmark results over the past two decades. Linear dynamics lies at the crossroads of several areas of mathematics including operator theory, complex analysis, ergodic theory and partial differential equations. At the same time its basic ideas can be easily understood by a wide audience.

Written by two renowned specialists, Linear Chaos provides a welcome introduction to this theory. Split into two parts, part I presents a self-contained introduction to the dynamics of linear operators, while part II covers selected, largely independent topics from linear dynamics. More than 350 exercises and many illustrations are included, and each chapter contains a further ‘Sources and Comments’ section.

The only prerequisites are a familiarity with metric spaces, the basic theory of Hilbert and Banach spaces and fundamentals of complex analysis. More advanced tools, only needed occasionally, are provided in two appendices.

A self-contained exposition, this book will be suitable for self-study and will appeal to advanced undergraduate or beginning graduate students. It will also be of use to researchers in other areas of mathematics such as partial differential equations, dynamical systems and ergodic theory.

https://link.springer.com/book/10.1007/978-1-4471-2170-1

 

[PeterDonis]

Which part? That chaos is considered an emergent phenomenon?

Yes.

 

[PeterDonis]

@Jarvis323 thanks for the references, I'll take a look at them.

 

[Filip Larsen]

Do you have a reference for this?

It is almost by definition of what emergence means in general, i.e. in this case the structures in chaotic systems, like strange attractors, which appear only when you have the full system, that is, the emergent structure cannot be identified as to originate from a single part of the system. It is very common to denote structures or patterns formed by complex systems as emergent structures.

I am a bit unclear if I have to take your request for a reference to indicate that you are skeptical about chaotic motion being an emergent property of the system, or if you are just engaged in some linguistic nitpicking so to speak. If you really take the position that chaos is not an emergent phenomenon then I guess I could equally well ask you to provide a reference for that claim.

Anyways, a couple of papers that mentions both chaos, quantum and emergence in their title or abstract:

Observing the emergence of chaos in a many-particle quantum system​

Probing quantum chaos in multipartite systems​

 

[Filip Larsen]

Another interesting paper that seems relevant to the discussion in this thread:

Exploring quantum chaos with a single nuclear spin​

Most classical dynamical systems are chaotic. The trajectories of two identical systems prepared in infinitesimally different initial conditions diverge exponentially with time. Quantum systems, instead, exhibit quasi-periodicity due to their discrete spectrum. Nonetheless, the dynamics of quantum systems whose classical counterparts are chaotic are expected to show some features that resemble chaotic motion. Among the many controversial aspects of the quantum-classical boundary, the emergence of chaos remains among the least experimentally verified. Time-resolved observations of quantum chaotic dynamics are particularly rare, and as yet unachieved in a single particle, where the subtle interplay between chaos and quantum measurement could be explored at its deepest levels. We present here a realistic proposal to construct a chaotic driven top from the nuclear spin of a single donor atom in silicon, in the presence of a nuclear quadrupole interaction. This system is exquisitely measurable and controllable, and possesses extremely long intrinsic quantum coherence times, allowing for the observation of subtle dynamical behavior over extended periods. We show that signatures of chaos are expected to arise for experimentally realizable parameters of the system, allowing the study of the relation between quantum decoherence and classical chaos, and the observation of dynamical tunneling.

 

[gentzen]

Our disagreement seems easiest to discuss in Bohmian mechanics:

Only if by "Bohmian mechanics" you mean a different theory from standard QM, not just an interpretation of standard QM. As an interpretation of standard QM, the nonlinearity of the guiding equation is irrelevant since ...

I don't get to your objection. I even explicitly started by acknowledging that the nonlinearity of the guiding equation is irrelevant:

their guiding equation is nonlinear, and ... But you say that this is not the relevant dynamic, and I even agree with this.

And I also explicitly stated where I see our disagreement:

You want to conclude from the linearity of that dynamics (of the wavefunction) that it cannot exhibit chaos, and "I don't think that this argument is valid".

My point of using Bohmian mechanics to discuss our disagreement was that it is closer to QM than the Liouville equations I mentioned in my initial comment, while still allowing essentially the same explanation where the chaos is hiding, and even with essentially the same omissions. (But I see that Jarvis323 now provided nice references where the chaos hides in infinite dimensional linear dynamical systems. I fully understand that those are preferable to my "hints" at explanations with their "intentional omissions".)

 

[PeterDonis]

It is almost by definition of what emergence means in general

General statements about emergence are not the same as a specific derivation of chaos as emergent from the particular dynamics of QM. The latter is what is relevant to this thread.

 

[PeterDonis]

I even explicitly started by acknowledging that the nonlinearity of the guiding equation is irrelevant

And then you contradicted yourself by arguing that it is. If it's irrelevant, then it's irrelevant and can't be used as the basis of any argument that is relevant. If it can be used as the basis of a valid argument for this topic, then it's not irrelevant.

In any case, Bohmian mechanics is an interpretation of QM, and interpretation discussions are off topic in this forum. They belong in the interpretations subforum.

 

[PeterDonis]

@Filip Larsen thanks for the references. I'll take a look at them.