Ah, that smell of hype!

Quantum dynamics is time symmetric. What breaks a symmetry usually are the unobservable degrees of freedom (environment). If you have access to them, there is no surprise you can reverse the dynamics (they are part of the system now). So quantum mechanics works.

But sure, let's call it a time machine. As if we don't have enough misunderstanding and confusion in public about quantum technology.

Classic dynamics is also time symmetric :)

Well, that's the thing — not really. In the classical world there's the second law of thermodynamics and there's no way to go around it without going quantum.

How do you know that? Why there is no second law in quantum systems? You can not derive any of that from the dynamics, it is just some statistics laws that seems to be working for relatively big environments which are in equilibrium.

I didn't read the paper carefully but what's the difference between a pendulum with an external force that compensates friction, and the stuff they have? It looks like a quantum pendulum with an external force. Is there something new to it, except more advanced algebra?

How do you know that?
Because these are the classical laws.
Why there is no second law in quantum systems?
Roughly speaking, because in quantum systems entropy has a different meaning: it's not the inherent property of the system, as in a classical case, but rather a statement about our knowledge of the system. Correspondingly, the second law in quantum case tells us that some information gets irreversibly lost to the environment if the system is coupled to it. The system is then said to be in a mixed state. If the system is closed, its state is pure, and in principle we are able to retrieve the full information about it, so its entropy is zero.
I didn't read the paper carefully but what's the difference between a pendulum with an external force that compensates friction, and the stuff they have?
It's not about that. They have several qubits, which interact to each other. They say that some subset of these qubits is «the system» and the other is «the environment». If they just let the system interact with the environment, it quickly decoheres — following the standard thermodynamics. But unlike standard thermodynamics, the information is not lost, since they have control over the environment. They can reverse the process, and restore the original state of the system by directing the flow of information from the environment to the system.

It's a neat concept for studying quantum thermodynamics, but not a time machine in any sense.

I can construct a classical system that will reverse the entropy. Given, that there is no friction or it is negligible small. Just inverse all the velocities at some point and it will evolve back in time. I can easily do it with billiard balls. Yeah there will be problem with friction, that I can not reverse it, but this is just technical problem. If I could it would work in the same way. With the quantum mechanics, strictly speaking you also never can do it, there would be always some interaction with the environment, which is out of control. So, it seems that all is about control. If you go to few mK and use two level system based on a single atom, then sure there is more control, than with billiard balls, which consists of 10^23 atoms. But, apart from that, what is the difference?

I can construct a classical system that will reverse the entropy.
That's the point: you cannot, without doing work. That's the point of the second law: the increase in entropy is irreversible. You cannot make the heat flow from a cool body to a hot body. That's not the case in quantum systems (under certain very specific conditions). You could return the system to initial state without doing work, by simply allowing the evolution to go with a different sign (if you have control over the states).

Mind, that it's not the same as going back in time. It's merely returning to the same quantum state.

But, apart from that, what is the difference?
That's the whole difference. That's exactly the difference between classical and quantum. If you had control over all 1023 atoms, you'd get a quantum system on your hands, not a classical one. So, by definition, if you have a classical state, you cannot violate the second law, because for that you need a fine control over the full state of your classical system, which would require you to go quantum.

This is very awkward inverse definition. But don't they apply work?

Sorry, I didn't mean «by definition» literally. The second law hold for classical systems, and you cannot violate it. Because in order to violate it, you need access to quantum degrees of freedom. Another way of putting it: entropy is an emergent property, which holds for statistical ensembles (like large classical systems). For quantum systems, it doens't have the same meaning, and thus the second law is not directly applicable. See e.g. on the wiki.
But don't they apply work?
If it's a unitary evolution, you don't apply work. They don't have a perfectly unitary evolution (imperfections of the experiment), but they apply much much less work than they would need in the classical case.

Consider a different perspective on your example with billiards: you have a real system with friction, which dissipates energy into heat. In classical physics, there's no way you can convert this heat back to motion of the billiard balls, no matter what you do with your table (how you manage friction). In quantum physics, you can have access to the degrees of freedom, to which the energy goes due to friction, and return this energy back to the billiard balls. That's because you didn't really loose the information about the system (energy), it got preserved in the environment.

Unless there is a big instability in the motion, like sort of stochastic dynamics, the billiard balls will go back in almost same trajectories. Especially if there are only few of them (like two :) ). But this is true: I need to apply work to reverse velocities. So in the end, no contradiction to the second law — usual Maxwell demon story. I this paper, however they claim (if I understood it right) that there is no energy exchange. This I don't understand because in order to have some useful information in q-bits one has to switch between 0 and 1 states, that costs energy. There is a whole different story about highly degenerate non-abelian states, but this not the case here.

So in the end, no contradiction to the second law — usual Maxwell demon story.
True.
This I don't understand because in order to have some useful information in q-bits one has to switch between 0 and 1 states, that costs energy.
That's the point: you don't do anything, you just let the system evolve freely. In the first setting, this free evolution entangles the main qubit with auxiliary (environmental) qubits. If you just look at the main qubit, it looks as if it's in a mixed (thermal) state. Of course, that's because it's entangled with the environment. In the second setting, this evolution is set to have an opposite sign — and the system disentagnles, leaving the main qubit in a pure initial state.

Consider a simple case: you shine a photon on a beam-splitter (which is a unitary evolution operation). It goes either left, or right. If you now put a detector in one of the arms, you'd see a thermal statistics (i.e. a photon half of the times). Now, apply another unitary evolution, another beam-splitter, where you recombine the two beams. (you end up with a Mach-Zehnder interferometer). You can tune the path length such that the photon after the recombination appears always only on the right ouput. Here's your pure state again.

Alright, you optics example clarifies a lot. It looks, like there is only one thing left. How do they put those interferometers dynamycally (in time)? Because from this step 1 — step 4 description it seems like parameters are changed in time. I guess it should be some adiabatic tuning.

Well, one thing that's important here: they simulate the behaviour of an electron using a semiconductor quantum computer. That is, they have a very simple interaction Hamiltonian (electron scatters off an impurity), which they can easily model using a few quantum gates on a quantum computer. Time reversal in this case is simply applying the same gates in the reversed order. (They play a few tricks in order to implement it efficiently on a small quantum computer, so in the paper it's a bit less direct, but it's irrelevant). So the whole process goes as follows: initialise a qubit in a simple ground state, let it evolve through the model of a Hamiltonian, let it evolve through a reversed Hamiltonian, measure, compare the final state with the ininital state. Say, your evolution is H=U1U2U3. Time reversal is simply applying U3U2U1 (conjugated). No surpise the result is a unity, since U1,2,3U*1,2,3 = 1, by definition of a unitary evolution.

Sure, they have superconducting qbits. Unitary does not mean, that there is no energy transfer. Like NOT gate, that flips a qbit does not conserve energy. If I am not mistaking only when [U,H] = 0 the energy is conserved.

Unitary does not mean, that there is no energy transfer.
Generally not, true.

Like NOT gate, that flips a qbit does not conserve energy.
Yes, but applying two NOT gates conserves energy (it must, since UU*=1). Applying two classical NOT gates always spends energy.

By the way, let's for the sake of this discussion separate the simulation part from the thing they're simulating. Because quantum computer, of course, isn't very efficient, depending on the particular architecture. However, the dynamics they simulate is reversible in the same way as quantum NOT gate is reversible, and there is no energy spent on returning the system to its initial state — this return is a free evolution of a system, not a driven evolution.

What do you mean by applying a gate? I think changing the system evolution operator from U also U* costs energy? At least there is an experimental apparatus that changes the magnetic field orientation, or whatever is there. Unfortunately I don't know how it works exactly in the classical case, but as I remember there is a lower bound on work which has to be done to flip the classical bit. Is there something like this for qbits, or is it better? If I stay withing a Hamiltonian framework of classical mechanics, I can not change the H just by my wish, and say I apply now another "gate" or I have another H (e.g. with opposite mass so that I go backwards). Usually if some parameter changes in time, it means that external force is applied to the system.

What do you mean by applying a gate?
You «apply a gate» = let system evolve under the gate operator.
I think changing the system evolution operator from U also U* costs energy?
If you change the system — then yes. But you don't have to: as in the example of two beam splitters above, you just put a second beam splitter, and let the system evolve through it. With a qubit, you send a microwave pulse to excite a system |0>-> |1>, spending one quantum of energy, and on transition |1>-> |0> you get it back.
I remember there is a lower bound on work which has to be done to flip the classical bit.
Landauer principle. It holds for irreversible dynamics. It holds for quantum systems as well — whenever there's loss. However, when there's no loss, the system is perfectly reversible, and no energy needs to be spent on the reverse evolution, since all the information remains in the system and is accessible for observation.
Usually if some parameter changes in time, it means that external force is applied to the system.
Yes, but you cannot arrange a classical system to undo the decoherence.

Let me put it differently: in classical world, once you have dissipated energy from the system into the heat, there is absolutely no way to retrieve it classically. Even if you're a Laplace demon, still. In quantum world, if you lost some information, it means that you're entangled with other degrees of freedom that you cannot observe. You could in principle retrieve it, if you had access to those degreees of freedom.
But you don't have to: as in the example of two beam splitters above, you just put a second beam splitter, and let the system evolve through it.

Exactly, this brings me back to my previous question:
How do they put those interferometers dynamycally (in time)? If I do have control on when to split and when to combine, then it seams that I am interfering with a system. Like trying to stimulate the emission of that microwave photon. With a beam spliter it is just some internal property of the system, that tells how photons will travel along. So is there just similar internal frequency between steps 1-4 or is there a control?

Landauer principle.

Right! thanks.

How do they put those interferometers dynamycally (in time)?
I don't really understand the question. First, there're no interferometers in there, at least in the superconducting QC. In a photonic QC you literally have semi-transparent mirrors. Second, why would you need to change anything dynamically? You set your interferometer, and let the photons fly through. Or send a sequence of microwave pulses.
So is there just similar internal frequency between steps 1-4 or is there a control?
There's no control, you just send a sequence of microwave pulses to different qubits. This sequence is pre-programmed. You don't decide to change anything based on what you've measured (at least in this protocol). The sequence of gates is deterministic, as the beam splitters are.

Yes, technically you can say I set up mirrors, I set up pulses. Both do some gates — problem solved. But well, how is it differs from: I set up some heaters, I set up some fridges, I heat sample, I cool sample. Starting energy equals final energy, no quantum mechanics at all — problem solved. Is there a mistake that I have in my though experiment, or how the results of two differ?

Difference is: if you set up fridges, you need to put work to cool a sample. If you set up beam splitters, no work needed, it's just free evolution of a system, that allows the energy to flow from hot to cold (figuratively speaking). In classical case, for the whole system (including you) you have a net loss of energy. In quantum case you could in principle have a set up where there's no loss.

I agree, this clarifies a bit. But on the other hand this is where my confusion comes from. It comes firstly, from the difference between beam splitters and qubits. The elastic reflection from the beam splitter only transfers some momentum to it, so for our entropy discussion this shouldn't matter. However, with the superconducting qbit and microwave photons it doesn't look like this, photons are absorbed and then emitted back. OK, you may say in the end the energy is conserved, so it is also in this sense elastic.

I am still trying to understand where is the boundary between system and environment? In my classical example you said that I've applied some work to my sample. I agree. This work I get by burning some coal or by using an electric battery. In the end the total entropy in my lab increased, but a part of the system did undergo similar transitions of heating and cooling.

Ok, In quantum world to increase and later decrease the entropy locally it is not necessary to heat and cool (and thus perform Carnot cycle and loose some work) you can entangle and disentagle a part of the system with another part. The total entropy of two qbits does not change in this case (unlike in my classical example where some coal is burnt in the end).

I see the latter case, like an interference effect. And the interference does not need to be quantum. So, let me try with another example. Two detuened coupled oscillators that create a beating effect. The energy is pumped back and forth between two oscillators. The visible loss you will see after many oscillations, which you will also see if you put many beam splitter.

However, with the superconducting qbit and microwave photons it doesn't look like this, photons are absorbed and then emitted back. OK, you may say in the end the energy is conserved, so it is also in this sense elastic.
Well, yes, you have a resonant system, if you pump it on resonance, it kind of elastic, in the sense you write: the energy is conserved in the end. However, I'm a bit hesitant to discuss this for a quantum computer, since there it's the operations are really not reversible in practice currently. They might become such in a distant future, may be in photonic quantum computers, but not yet. So I would focus on a conceptual discussion of classical vs quantum.

I am still trying to understand where is the boundary between system and environment?
Yeah, that's a good question. That's why I think these expriments are fun, but not really useful. We don't know, and realistically we'd never be able to control the coupling to the environment. Only in a very isolated settings (like in QC) it is possible to maintain some coherence for a decent amount of time.

Two detuened coupled oscillators that create a beating effect. The energy is pumped back and forth between two oscillators. The visible loss you will see after many oscillations, which you will also see if you put many beam splitter.
I don't think it's that: here we talk about a particular kind of loss. You have just a coherent energy exchange. Strictly speaking, you have zero entropy of your osillators (there's no temperature or statistics involved). The statistical properties of these oscillators don't change, only their amplitude does. And they still remain coupled. That's not the same as entanglement: there the systems are correlated, but their dynamics is not changed by entanglement. So, you can hardly talk about thermodynamical principles based on coupled oscillators model.
So, you can hardly talk about thermodynamical principles based on coupled oscillators model.

I agree, the system is too small and actually it is out of equilibrium. The entropy of a single oscillator is however not necessary always zero. If it is a subject to a random forces (kind of Brownian motion story), it is possible to assign some entropy to it with respect to oscillator's configuration space. I also can easily add random forces to my model, then statistics comes into play, however the beating behavior will still be there (for some time).

That's why, I conclude from our discussion the following. This entanglement entropy is a different animal, it does not directly links to the thermodynamics. It is a property of a wave function at a given moment of time. While the Bolzman entropy is about system statistics, and about what will a big system do (e.g. follow the second law). And we can not reason that easy about small system, especially out of equilibrium. What do you think?

The entropy of a single oscillator is however not necessary always zero.
Well, sure, but you didn't mention that in your example:)
The entropy in this case would be associated not with beating, but with random forces, and that would we the dissipation channel. The coupling of the two oscillators is irrelevant for the discussion, only the random forces are.
What do you think?
Entropy in quantum world does have a different meaning, connected to the properies of the state (or rather our ignorance about it). However, it is directly connected to the classical entropy (and the way of deriving the second law from statistical mechanics follows this route).

I would say it's the quesiton of the level of description of the system: you could describe a large system in terms of its quantum state. If you select to do that, you have access to all the information stored in microscopic degrees of freedom. You could describe the same system macroscopically, in statistical terms. Then you end up with the second law and classical dynamics.
May be a slightly different perspective: in quantum world, any system could in principle be reversible, if you had access to all degrees of freedom of the environment. In classical world, the systems are not reversible, since dissipation is governed by quantum laws, and you cannot describe it in classical terms.

If the quantum mechanics is reversible, then classical should also be, because underline dissipation is described by quantum laws which was shown to be reversible!

As far as I know the dissipation argument in quantum mechanics is also statistical. Because decoherence is quite random, it is more likely that everything would decohere and entangle with more and more stuff (rather then disentagle) Again, correct me if I am wrong about it.

If this is true, then it is similar to what you say in classical system, because motion of molecules is quite random they will more likely go opposite to gradient of concentration (either literally particle concentration or just empty energy states concentration). Then this statement can be generalized accurately to all corresponding thermodynamic potentials.

If the quantum mechanics is reversible, then classical should also be, because underline dissipation is described by quantum laws which was shown to be reversible!
Well, yes, as I wrote in the beginning: you cannot describe classical reversibility without going quantum. But «classical» is an emergent concept anyway, our world is fully quantum.
As far as I know the dissipation argument in quantum mechanics is also statistical.
Yes, of course: on any practical level quantum systems are not reversible. The whole point is that they could be in principle, unlike classical systems.

Of course, any quantum system will decohere, and we would never have control over that.
The whole point is that they could be in principle

My point, is that classical also could be in principle. At least I am not aware of any proof of the second law based on the Hamilton mechanics. What happens in the quantum system is the same thing — reversible Schrodinger equation with hermitian Hamiltonian.

The only irreversible process in quantum mechanics is the wave function collapse. Another interpretation which is the decoherence interpretation, does not require such fundamentally irreversible process, but rather uses a statistical argument. Then (to my honestly poor and hand-waving understanding) it is kind of the generalization of the second law.

The entropy in this case would be associated not with beating, but with random forces, and that would we the dissipation channel

Why you are saying that randomness (some notion of statistical ensembles) is required for classical system, but not for quantum? In the two beams examples and the beam splitter there is no randomness, so this makes me very confused. Maybe there is a good reference to learn about connection between entanglement entropy and more conventional entropy, do you know?

Maybe I can put another way. Imagine the internal oscillator's frequency is faster then our time resolution, but the beating frequency has large time scale. The I don't know much about how exactly oscillators are oscillating, I just see very fast wiggling and very slow wiggling, but have no idea about positions. Can I count such situation already as statistical physics problem? Otherwise there is another shady boundary between the air in the room and small sized systems like oscillators.

At least I am not aware of any proof of the second law based on the Hamilton mechanics.
Sure, because Hamilton mechanics doesn't consider lossess. But that's the idealization. In quantum case any non-ideal system in principle could be reversed. In classical world it's not the case.

Another interpretation which is the decoherence interpretation, does not require such fundamentally irreversible process, but rather uses a statistical argument.
Well, okay, that's a different story, but decoherence is not related to collapse: there is decoherence in Copenhagen interpretation too. Moreover, measurement cannot be explained through decoherence. You can have a many-worlds interpretation (I've got a post about it), where indeed this all is treated through statistics.
Why you are saying that randomness (some notion of statistical ensembles) is required for classical system, but not for quantum?
Well, we're off a solid ground here, but yeah — randomness in quantum world only appears during measurement.
In the two beams examples and the beam splitter there is no randomness, so this makes me very confused.
Why do you say there's no randomness? If you measure after the first beam splitter, you get your random distribution.
Maybe there is a good reference to learn about connection between entanglement entropy and more conventional entropy, do you know?
Hmm, it's a direct extension of classical entropy, and is defined in the same way as Gibbs entropy. I'm not sure how deep you want to dive in that, there's tons of literature on the meaning of quantum entropy.
Can I count such situation already as statistical physics problem?
I don't think so, at least in this context. You need random dynamics to talk about entropy — since it's a measure of the system itself, not your measurement device.

The way to add losses to Hamilton mechanics is to treat it statistically. Firstly, use micro-canonical ensemble, from which, by separating it into the system and environment and making some assumption one can get canonical ensemble. This requires that the environment is very big and postulating that only the most probable micro-configurations are relevant. Then one can use fluctuation dissipation theorem to argue about friction.

The similar ideas applies to a quantum mechanics. If I think, for example, about Lindblad equation.

but decoherence is not related to collapse

Sorry, I was thinking more about entanglement, anyway this is related.

If you measure after the first beam splitter...

If we like to stay with many-worlds (or something like that, because there are two many kinds of options) then you add your randomness form the environment. It is like I suddenly add a very large friction to my oscillator (i.e. couple it to the environment) and it stops immediately. Then, if I do it many times I will get a random distribution which will depend on a way I set up the beating experiment.

Sure, in quantum mechanics there is more to it, because there is a violation of Bell inequality, for example. Coming back to the density matrix. I think we treat it as an entropy, because we assume wave-function collapse to be a random process by definition (Born's rule). Basically even though we know everything about the wave-function, we don't know much about the system itself (I mean we have only probabilities of the measurement outcome). This is a huge difference to the classical case, there if I know everything about x and p, I really know everything about the system.

This brings me to yet another conclusion. Your example with the beam splitter, is the following sort of game: The outcome of a measurement after the first split you don't know. This is not because your actual knowledge of the system changed, rather you prepared it in a way that you don't know how it will interact with the detector. If it were just basis rotation, that this would be not so surprising, because you can always rotate the basis of the detector. For the entangled states it is not possible to fix the detector, though. In the end it sounds more in the lines with classical entropy feature.

I am not sure if I am able to continue to play this game. I.e. by introducing a random factor into the measurement devices used for my coupled oscillators model.
I think there is already one QM interpretation like this on the market :)

The way to add losses to Hamilton mechanics is to treat it statistically.
Well, yes, that's how you derive thermodynamics:)

Point is, thermodynamics is an addition to Hamilton dynamics, and you cannot have a lossy Hamiltonian of the system. Of course, it's straightforward to implement it, just sayin'.

I think we treat it as an entropy, because we assume wave-function collapse to be a random process by definition (Born's rule).
Soo, that really depends on the interpretation you consider. But generally, it's not a measure of randomness as in a classical system, but a measure of the amount of information you could extract from the system. It's a subtle difference, but sometimes becomes important in interpretational questions.
Basically even though we know everything about the wave-function, we don't know much about the system itself (I mean we have only probabilities of the measurement outcome).
I'm not sure I follow here: the wave function is everything there is. There's nothing else (at least in ontological interpretations).
In the end it sounds more in the lines with classical entropy feature.

To be honest, I got a bit lost in our conversation :)
Well, yes, that's how you derive thermodynamics:)

Just mentioned that, to make sure that we are talking about the same thing.

I'm not sure I follow here: the wave function is everything there is. There's nothing else (at least in ontological interpretations).

I mean hypothetically. As a measure of information you may have about the system, from statistics point of view. I was trying to formulate, why people see it as entropy, and what is the difference to classical picture.

To be honest, I got a bit lost in our conversation :)

The meta-physics behind QM is out of scope of a rational discussion. If this made you confused, I did not appealed to this much.

P.S. Thanks for the discussion. I learned something from it, and it was helpful to think through QM and statstical physics concepts one more time.

I was trying to formulate, why people see it as entropy, and what is the difference to classical picture.
I think people are trying to find the emergence of classical concepts from quantum. Defining it as von Neumann did, is a very natural thing and carries onto classical concepts in a relatively straightforward way (as you described).

By the way, I started googling after the last comment, and found a curious paper. They discuss the differences in Gibbs and Boltzmann entropies when applied to quantum world. Some interesting points made there, although I haven't finished reading yet.

The meta-physics behind QM is out of scope of a rational discussion. If this made you confused, I did not appealed to this much.
I guess I just wasn't sure which point exactly we were discussing, since it quite far away from the starting comments:)

I actually have some trouble seeing the real physics behind entropy in quantum world — especially when it's so dependent on the interpretation. Like, yeah, I get that from a standard frequentist (aka classical) point of view. But when I start from quantum (like in many worlds or Bohmian), I start loosing the grip on it. Have to learn more about that…

Thanks for the discussion.
Thank you too! I've quite forgotten some things since the university times, it was nice to get thinking about that:)