My question is going deeper than that. It is asking if perhaps even the classical component is to some extent non-deterministic, and if so by how much? Is it purely by an amount dictated by the Planck correction, or is it more complicated than that?

This is a great question, and I don't think that I know the answer, so instead I'm going to do my usual trick of trying to restate the question in a way that I think I can outline an answer to ;)

The point behind Maxwell's daemon (as I'm sure you know) is not about his magical powers, but more concretely it is about the amount of physical resources that he needs to break thermodynamics. This distinction is important because it enables us to pose questions about what the daemon can and can't do, and even allows us to make daemons (which is also done).

So in the classical case, conceptually it is relatively straightforward.

So let's slightly rephrase your question. The daemon is going to be provided a bit stream, and his task is to convert the register into some fiducial state, let's say all zeros.

He has only three operations that he can apply to each bit in the register -

1) measure a bit.

2) do nothing

3) flip the state of the bit

This tasks is, of course, formerly equivalent to the conventional task of Maxwell's daemon.

We are also going to simulate the 'magical power' of the daemon by allowing him access to a memory and predictive algorithms - but we are going to require these algorithms and memory to be bounded by physical laws.

He also knows that the zeros and ones are the result of the sensing of a physical system interacting with its environment, and the distinctions here are between a classical and quantum environment.

Here of course we should clarify. If the daemon can predict the next bit more cheaply than he can measure it, then he should use his algorithms to do the sorting. However, if the measurement is the cheapest way to proceed prior to sorting, then he has functionally a random input.

I suppose that we are also going to allow our daemon some error rate, so that we start with a register at some finite temperature (measure of the relative fraction of zeros and ones) and the action of the daemon is to cool the register to some finite (ideally 0) temperature.

But of course, work is being done by the daemon in doing that cooling. Measurement to finite precision takes energy, as does flipping the state of the bit.

The classical environment is at some level completely deterministic, and so the daemon can invest resources into probing the environment and predicting (with an appropriate model) the next state of the bit stream. In general, the minimal energy for the sorting should occur when the daemon can completely predict the next bit without having to measure any of the preceeding bits, so that the bit stream is entirely deterministic.

Conversely, the maximum cost to initialise the register must come from having to measure every bit and (let's say) flip 50% of the bits.

Now, we are in a position to perform a mathematical solution to this (of course under assumptions). But the key will be how long it takes for the predictability of the next bit to go to 0% as a function of the memory and the algorithm. In the quantum case, we can assume true randomness, and hence memory will be (probably) fairly useless, or at least restricted at some level of usefulness. Conversely, in the classical case, increasing the size of the memory will always asymptotically add additional information, and therefore improve the predicatibility of the protocol. Hence you could generate some appropriate scalings for some toy models and compare the results.

Finally, returning to the real world - my suspicion is that the thermal noise is ultimately quantum, based on stochastic measurements caused by the environment - but of course there are finite correlations.

NB the finite correlations are one reason for me trying to pose your question in the fashion that I did, namely because you really want to interpolate between the two limits, and I think that the framework above allows that.

@Andy. First, my question doesn't rely on the nature of the demon....other than the fact the demon is not magical enough to predict a quantum outcome. His powers do have that limitation. He can do just about anything else. He is the unholy spawn of Maxwell's demon; and not necessarily exactly the same as Maxwell's.

Second, the part of your answer that says: "my suspicion is that the thermal noise is ultimately quantum, based on stochastic measurements caused by the environment" gets to the heart of my question. The real question is to ask to what extent we can say the classical noise is non-deterministic itself. In my thinking, it must have a non-deterministic component--simply because the reality is that it isn't purely classical (re-read paragraph 4 of my question, and let me know if you agree with that).

To what extent are classical fluctuations driven by underlying quantum ones? I don't think it suffices to say "well, let's cool to ~0 Kelvin and whatever is there is the quantum part"; because my paragraph 4 implies that at room temperature the classical fluctuations may be "amplifying" the quantum fluctuations to some extent. Can we really say that all quantum non-determinism is totally washed out at room temperature, or can we say that the classical fluctuations have in fact piggybacked on them and thus are party non-deterministic?

I wonder if anything has been written on this?

Hi Derek,

Actually, I don't think that I did misunderstand your question/comments ;). Equally, I'm happy to give the daemon magical powers, but I him to properly account for them based on physical principles. I think that this helps when trying to frame the question because my intuition tells me that the difference in the cases that you are looking at hinges on the divergence/convergence of the predictability, or equivalently, the amount of work needed to be done to predict the next bit.

So again, we return to our model for the noise.

If we assume that the system is classically chaotic, then we have a very nice model for how to understand the classical determinism. Depending on where we are in the phase space of the chaotic environment, we can determine the predictability of our system.

Let us add an additional constraint here - let us assume that the daemon can measure the bit to a finite precision, and that it costs him less to measure to poor precision than to high precision.

Now, if the evolution of the environment is such that it follows a classically stable trajectory, then low precision measurements are probably a pretty good idea. You are pretty sure you know which state your bit is in, but every so often you just want to check (Alternatively, assume you only have high-fidelity measurements, but a random choice of when to perform a measurement).

But this approach is predicated on some intrinsic determinism in the evolution. Increasing the knowledge of the system (or preserving more memory) must update our predictability, asymptotically reaching complete predictability with infinite memory. But of course the rate of approach is going to vary with precisely where in the phase space of our system we are at.

We can contrast that with the quantum version which will have intrinsic, irreducible noise at some finite level due to how important the quantum fluctuations are to the evolution. For example, predictability could be zero if the bit stream came about from an initially prepared equal superposition of |0> and |1>.

So, the testable consequence of this hypothesis would be to start with a system that we can solve, and determine the relationship between the number of memory bits that we use in our prediction and the fidelity of that prediction for a deterministic model (pseudo random), vs one with a true random distribution.

I think that you will find that the two cases will have similar behaviour, up to the limit set by the true random nature.

Now, I suppose I am implicitly assuming here that everything is going on at 0K because I'm assuming that a deterministic model for the classical behaviour can be used, but I think that this is a fair way to the view the problem based on how you've posed it - otherwise, at finite temperature you are really washing out the effects that you are trying to explore by introducing true random noise that is presumably classical. What I'm proposing still has the possibility to identify classical signals which functionally act as noise, and hence could be used to derive an effective temperature, but for which thermal effects are not the origin of the disorder.

And finally, I think that you will find that classical thermal noise is not in fact classical, but is quantum!

There has been quite a bit of work in this space, although I don't think that I've seen anything quite like what we've discussed: but then I haven't looked hard enough, either!

The bit where you say: "I think that you will find that classical thermal noise is not in fact classical, but is quantum!" is in fact my suspicion too, and that is what I am trying to get to the bottom of. If anyone knows any references on this, I'd be grateful.

Le us consider a quantum system exposed to thermal fluctuations of the background. The thermal noise is a result of classical random fluctuations, while the quantum fluctuations of the system compete the thermal noise so that during a sufficiently long time the quntum fluctuations are fully suppressed.

Yes, but the question is then to what extent have the resulting classical fluctuations piggybacked on some of the quantum non-determinism? That is what my question is really getting at. Surely the resulting classical noise is not 100% deterministic?

Thermal noise results from classical fluctuations of the reservoir. If it (weakly) interacts with a quantum object, then, because of infinite number of degrees of freedom of the reservoir, the quantum fluctuations are necessarily destroyed . This is well known and rigorously proved result in statistical physics. Hence, as a result, the thermal noise competes and finally fully destroys the quantum fluctuations.

Yes, but see the 4th paragraph of my initial question. Can you explain why those things would not be happening?

This is not a question about which type of noise is destroyed. This is question about determinism vs. non-determinism. Surely if you have the superposition of a deterministic process plus a non-deterministic process, the result will have a non-deterministic component. If not, I want to know why. Please explain.

In this case the term nondeterminstic seems not to be appropriate. A quantum object is described as a wave packet and a contact with a macrospic object (measurement) projects one quantum state of the packet. This quantum state fulfills the Heisenberg uncertainty principle. It seems that this is You call indeterminism, do You?

In fact, in many particle systems with complex spectra (e.g. highly excited spectra of nuclei) there can happen, that the level distances are very small when compared to the quantum width resulting from the Heisenberg principle. Then the levels strongly repulse each other and localized states appear. In some conditions the levels which mix within the quantum width of the level can not be distinguished from each other -they are not separable (the wave function can not be written as a product of both) . These mixed states can coexist with the states of the spectra which are separable. Then, we have both type of states coexisting in the phase space. Such a situation seems to exist at the boarder between classical and quantum world. These problems are intensively studied in a current literature.

Dear Eva,

I think that you've missed exactly what Derek's question is getting at. We all know the canonical explanation for thermal noise and tracing over infinite states, what he's trying to get to is a fundamental knowledge of whether at any level the classical picture can be considered correct, and indeed whether classical physics is self-consistent in a noise picture.

So let's start by going back to what I believe to be Newton's idea of the Universe as a machine.

The idea here is that the Universe evolves by deterministic and known laws. Given that all of those laws are known, all that is required is to determine the initial condition (or indeed _any_ state) to enough precision so as to determine any future or past state. Now there are treatments that involve predictability and classical chaos to understand whether future predictions are in some sense efficient or divergent, based on complexity theory.

This was certainly in my mind when I proposed my thought experiment above. The interesting question is how the daemon's prediction diverges, but Derek is more interested in the limits, i.e. can he ever predict the future to arbitrary accuracy (albeit with infinite resources).

Now a naive application of quantum mechanics would also seem to support the concept of infinite predictability. After all, the Schroedinger equation is deterministic, and with some contrivance (and infinite resources) one might assume that the environment could be measured into some fiducial state and then left to evolve. With enough knowledge of the state and its evolution one might therefore be led to believe that at some particular times (but certainly not all times) it would be possible to find a state that could be deterministically predicted to arbitrary accuracy.

This is, of course, not the question that Derek is asking.

Instead he's asking if noise always has to be result of quantum fluctuations, and we should be quite clear what we mean by this. What we really mean is that the noise is due to the environment effectively measuring out states where those states are not eigenstates of the measurement Hamiltonian. If such processes are going on, then they constitute irreducible noise and therefore provide a hard limit on any predictions that can be made.

Note that I'm implicitly using the language of decoherence theory here, but equally the same argument can be cast in many-worlds language.

But Derek is actually asking an even stronger question. Not only do quantum states provide a noise floor, but do they in fact provide the ONLY noise floor that makes any physical sense.

My suspicion is that quantum does provide the only logical noise floor, and I've used the above thinking to come to what I believe to be a testable gedanken experiment, but I don't pretend to know the answer to it rigorously.

Hope I got it right, Derek ;)

Dear Andrew,

thank You for Your explanations. Hovewer, I am feeling a bit uneasy due to several misundestandings. The most serious, which I mean is the most important, is the term "eigenstates of the measurement Hamiltonian". I really do not understand why do You suppose the measuring device to be quantum? Then we would be left with a sequence of measurements ... The measuring system is always classical, macroscopic system and this brings with it destroying quantum fluctuations.

Remember, the temperature in the Fermi-Dirac or Bose Einstien distributions i.e. the temperature of respective fermions or bosons is the temperature of the macroscopic reservoir .

Maybe let me put my question more succinctly: is classical thermal noise 100% deterministic? Yes or no.

If no, I can accept that. If yes, I want to hear your explanation as to why non-deterministic quantum effects don't have an underlying impact on the classical noise.

The second part of the question is how many thermal noise sources (from resistors) do we need to XOR together to make the output as good as non-deterministic?

First of all let us make us clear about semantics:

1.Your understanding of the word "deterministic." As used in statistical mechanics,

the thermal Gaussian noise of a time dependent random force is defined by its correlation function (second momentum), which in the case of white noise is Dirac delta function of the time difference, in the case of colour noise a function of difference of respective times. This noise is CLASSICAL noise.

From Your concept it seems to me, that this kind of noise You call deterministic.

Do You feel the contradiction -randomness versus determinism?

2. Schroedinger equation is deterministic, but interpretation of its solution is probabilistic. Heisenberg principle is the principle of uncertainty. Here the correct term is UNCERTAINTY. Why to call it non-deterministic? For this case it is unusual.

3. For completeness, deterministic chaos as a kind of dynamics which results from deterministic nonlinear equations, but is complex , i.e. unpredictable from the dynamic equations because of independence of its behaviour on initial conditions.

All,

Derek is asking about the Johnson noise: current/voltage fluctuations in a resistor originating from the thermal motion of electrons. The motion of a single electron is a purely quantum effect. The scattering of the electron on the lattice and with each other are elementary quantum phenomena. For example, the acoustical phonon scattering is totally inelastic and has no phase/momentum memory. Thus the elementary stochastic motion is a purely quantum process. The measurable current/voltage is the sum of these processes. Its distribution and energy balance satisfies classical thermodynamics. This is the same question as, if the thermal radiation is quantum or classical. People often say that it is classical. However, it consists of photons and, to get the correct spectrum in the short wavelength limit, in 1900, Max Planck had to introduce the E = hf energy quantum in the radiation. That was the the first quantum formula. Five years later, Einstein identified this as the photon. Similarly, macroscopic Johnson noise can be explained by classical thermodynamics, but when we go to very small sizes or very high frequencies, we will need quantum physics to describe the observations. Concerning randomness, Johnson noise is "more random" than elementary quantum processes because, to the quantum processes here classical high-degree-of-freedom effects (including chaos, which works even in low degrees-of-freedom) contribute, too.