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Hi there! I'm Dc987 at Wikipedia.

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Completion status: this resource is ~25% complete.

Subject classification: this is a physics resource.

RD Experiment

Abstract. Sort of.

This experiment illustrates the idea of the entropic principle for the single quantum event - or possible influence of the entropic considerations on the outcome of a single quantum event. Two alternative versions of a real experiment setup are proposed. An attempt is made to realize one of the experiment versions.

Introduction

The idea is to make a {laser/symmetric beam splitter/photo-multipliers/counters} setup, make one branch 'more entropic' and see if that would make any change to the counter values distribution.

A photon just came out of a laser. There are two potential paths. Which path would the photon 'chose'? With what probability? Would the asymmetry (heater/'entropy generator' on the path A) of the setup affect that 'choice'?

Would $N_{A}/N_{B}\!$ show a statistically significant deviation from $50/50\!$ ?
Would switching on the largish electric heater change the rate of the counter $N_{B}\!$ ?

From the classical perspective the answer would be definite no (and would result in all kinds of paradoxes). From the QM perspective, my guess that it would be something like $N_{A}/N_{B}\approx e^{\frac {S_{A}-S_{B}}{k}}\!$ , where $N_{A}\!$ , $N_{B}\!$ are the counter values and $S_{A}\!$ , $S_{B}\!$ - total entropy (caused by the photon) in the space-time light cone of the observer (entropy 'visible' to the observer).

Proposed experiment realization

Well, WTH. I thought that realizing the experiment would be an interesting pet project, so I've actually bothered to make the apparatus, to perform initial testing and as of now I've started collecting the data. I'm interested in your comments on the setup, as your input may help fixing potential loopholes and bugs.
To keep it simple and inexpensive I've decided on 'off-the shelf' equipment and straightforward experiment realization. I've used of-the-shelf 4Mbit optical QRNG as the {laser/symmetric beam splitter/photo-multipliers} and a regular PC as the {counters/heater}. Here is the diagram of the setup:

Testing showed, that my QRNG unit is not perfectly equidistributed and have an internal bias P('1') = 0.4999762(15). To remove that bias and exclude any possibility of any other systematic bias I've decided to XOR the input sequence with the pseudo random sequence of the MT19937 PRNG generator (Mersenne Twister). To make sure that the MT19937 itself is well equidistributed (MT19937 is known to have a bias, if initialized improperly) and does not introduce systematic bias between the runs the PRNG seed (624*32 bits) is generated for each experiment run by the QRNG (note that 624*32*(0.5-0.4999762) is less than one bit).
The computer's CPU/memory are used as the {heater/counters} (CPU is performing some 'extra' floating point operations for '1's (or '0'), thus creating the asymmetry in the generated entropy).
So pretty much everything, specifying the setup ended up in the following very simple source code (C/GNU):

Source code for "extra entropy for '1's" User:Dc987/res-mt-q-xx.c

Source code for "extra entropy for '0's" User:Dc987/res-mt-q-neg-xx.c

I plan to do a series of experiment runs (3*10^12 bits each) for both 'positive' and 'negative' versions and use the resulting 'mean' value as the single output of each run. The target number of runs is 20 (10 each version, approx 168 days total time). The target level of significant deviation is 3 sigma.
Note/Trick: I've already performed 5 runs of "extra entropy for '1's" version that were part of the initial testing/adjustments/debiasing. The output was P(1) > P(0) for every run (this either favors my initial hypothesis, shows that there is some bias, or it was just a pure coincidence /3% chance/). So now I'm going to discard that initial data and start fresh, but with the "extra entropy for '0's" runs first. I'm going to interrupt the experiment, if first four/five runs of it would end with P(1) > P(0) results (because this would probably mean there is some unexplained bias in the system).
I would very much appreciate your input, comments related to the experiment, methodology, bugs in the code, etc.

Results

Speculations

Consider a simple setup. Laser, half-silvered mirror, detectors, counters:

Let's replace the laser with an optical resonator with initial energy $E_{0}\!$ . The optical resonator now is a passive device, let's assume that we can construct it really well and it does not release any entropy to the environment. $S_{O}=0\!$ . Here is another diagram illustrating the entropy exchange with the environment:

Let's add a really large electric heater to the path A. We connect it in such a way, so a huge amount of free energy is being spend upon arrival of a single photon. This would result in the release of $S_{H}\!$ amount of entropy to the environment. Here is another diagram, with the heater:

Let's assume now that we've made some very nice, very low entropy, high efficiency, almost time reversible photomultiplier+counter. In the limit ( $S_{C}\rightarrow 0\!$ ) we can replace photomultiplier+counter with a quantum harmonic oscillator. And we take a largish electric heater, so in the limit we can simply couple the harmonic oscillator with the Ohmic heat bath. Here is the illustration for the the completely time-reversible limit:

Now the only possible way of interaction with the environment is the heat bath. Everything else is time reversible. With the time flow any passive system can only increase the entropy of the environment (assuming that you take the initial internal energy of the system as zero). So the only logical conclusion here is that, with the time flow, all the photons would chose the path A.

Now lets return $S_{C}\!$ :

Again it is logical, that if ${\frac {S_{C}}{S_{H}}}\rightarrow 0\!$ then almost all the photons would chose the path A.

Now back to the original setup, but with the heater.

Would be something like $N_{A}/N_{B}\approx e^{\frac {S_{A}-S_{B}}{k}}\!$ , where $N_{A}\!$ , $N_{B}\!$ are the counter values and $S_{A}\!$ , $S_{B}\!$ - total entropy (caused by the photon) in the space-time light cone of the observer (entropy 'visible' to the observer)?

Is there any way to derive the answer from the quantum mechanics? My guess that counter values should be proportional to the resulting number of 'decoherence states' for each path. Hence the entropy is in my answer. Is it correct?

Would you think it can be possible to see a statistically significant deviation from the 50/50 distribution in the real experiment? Can this setup be realized in the real experiment in such a way so we get a statistically significant difference between the counters?

Does it bother us that it creates subjective FTL communication paradox, causality paradox and some observer effects?

Consequences

Ok. If $N_{A}/N_{B}\approx e^{\frac {S_{A}-S_{B}}{k}}\!$ is correct, here are the consequences. If we have a quantum random event it would be biased in such a way, so:

For any observer: subjective history where the observer produces the maximum entropy in the environment have the maximum probability;
For any passive system: other factors aside, a history, where this passive system produces the maximum entropy in the environment have the maximum probability;
For any living system: other factors aside, continuing living is more entropic than not and would be more probable;
Generally living systems would be more favorable to non-living as more entropic;

Paradoxes

If Alice, switching the really largish (path A) heater on/off moves the equilibrium and subjectively modifies the $N_{B}\!$ rate:

For spatially separated A and B, subjectively Alice can send information to Bob instantaneously. Alice would never know however, if the information was received.
Bob can receive information from Alice from his subjective future. Most probably the received information would be some random junk. It is not necessarily going to be from Alice from his most probable subjective future history. ;) Alice would never know, if the information was received. Bob's most probable subjective future history would change with the receival of information.

Note: no objective FTL information transfer is happening here. Everything is in accordance to "no FTL signaling theorem". Only a little bit of cheating and adjusting the probability of ending up in the subjective possible future history where the information is already present at the other end.

Conclusion

Reasonable Deviations Sandbox

Well, it is clear to me, that the experiments (2) and (3) are different. And I don't know if the results of these experiments would be different or the same. That is a good enough reason for a question.
(Yes. From the casual/classical point of view the outcome is going to be the same for 2 and 3. Unfortunately this point of view is an approximation and I'm not sure if it would be good enough to analyze this thought experiment. And by the way - consider the regular classical analysis of such an experiment versus the classical analysis based on the action principle. Wouldn't they contradict each other?)
So what I'm really interested in is the answer from the QM perspective.

You are right. I'm not precise enough with the words here. Still, my point was that with the time flow any passive system can only increase the entropy of the environment. And you can consider any system to be a passive system (assuming that you take the initial internal energy of the system as zero).
An example you've given with the fridge. What you have there is a rather complex passive system scattering light :) It takes light in (electromagnetic waves) it spits light out (heat). And with the time flow it increases the entropy of the environment.
Offtopic: As far as I can see this entropy exchange (entanglement with the environment in the quantum world) in fact is what we subjectively call the time flow. But that's only my guess. And I wouldn't bet on it. What I would bet on is that observers are not really different from fridges :) in terms that you can consider an observer to be a passive system.

Yes. The result of the casual/classical (wave function collapse upon the interaction with the classical detector) analysis is trivial. What I'm more interested in is the analysis of the whole system (including the detectors, counters, heater) from the QM perspective (where the classical behavior emerges from the quantum via the decoherence).

Yes. I want to treat the apparatus as "microscopic". Entropic processes would be: "observer doing the measurement", "apparatus generating the entropy in the environment (apparatus entangling with the environment?)". Does that sounds right?
The observer is classical - measurement collapses the wave function, etc.

In the real-life setup that would be an electric heater. But I guess in this thought experiment it can be zeroing the memory (which requires a corresponding increase in the entropy of the environment / Landauer's principle). So there would be some extra decoherence due to the vacuum noise, right? And it could slightly diminish differences between the counters values in the experiment (3)? Well, I don't know if there would be any differences in the first place...

Ok. Let's make things more interesting. Here is my conceptual analysis of the experiment (3). First we should forget that the system is classical and remember that it is really in the superposition of |path A> and |path B>. And what is important here is the resulting number of states.
Now the resulting number of states:

for path A it would be: {a lot of decoherence states of the photomultipliers, counters, heater}

for path B it would be: {a lot of decoherence states of the photomultipliers, counters}

So it looks like the path A would have slightly larger number of states associated with it. As a result we would have slightly larger probability of finding the apparatus in the state A, and if repeated many times slightly larger number in the counter A.
How larger? Well.. that depends of how much decoherence associated with photomultipliers and counters is going on. So "publishing the results of each outcome in a scientific journal." can really screw things :)

How can I derive this answer from quantum mechanics?
No. I can't cite any references. The experiment is too stupid to be in the textbooks. It is obvious that the answer is 50-50. Otherwise there are all kinds of paradoxes: providing that you have a largish heater, subjective FTL communication is possible, you can break causality, speed up quantum computations, there are all kind of observer effects, etc, etc.

That's a very vague statement. And in fact the exact opposite can be true, life is a highly entropic process and thus would be more favorable process.

Well. I have the right to apply the Born rule to the whole apparatus, right?

The independence of the state prepared by the beam splitter of what follows it.
Yes, but only if I'm observing the photons coming out of the beam-splitter. If I am doing that, according to the Copenhagen interpretation the measurement would collapse the wave function and the application of the Born rule will give me probabilities.
But I'm not doing that. I'm observing the counter values. And a lot of scattering photons, generated by photomultipliers, counters, heater.

Let's assume that we've made some nice time-reversible photomultipliers and counters. So: SP = 0; SC = 0;
Now the only possible way of interaction with the environment is the heater. Everything else is time reversible. With the time flow entropy can only increase (it can not even stay the same!). So the only logical conclusion here is that with the time flow all the photons would chose the path A.
A completely reversible photomultiplier+counter can be replaced by a quantum harmonic oscillator. And the light source with an another oscillator with some initial energy. Here is an illustration:

Here is an extreme scenario. Let's assume that we've made some nice time-reversible photomultipliers and counters. So:

S_{P}=0

,

S_{C}=0

. Now the only possible way of interaction with the environment is the heater. Everything else is time reversible. With the time flow entropy can only increase (it can not even stay the same!). So the only logical conclusion here is that with the time flow all the photons would chose the path A.

A completely reversible photomultiplier+counter can be replaced by a quantum harmonic oscillator. And the light source with an another oscillator with some initial energy.

I would think that every system is a little bit reversible. Release of the entropy into the environment (entanglement with the environment) would be the measure of irreversibility.

Interestingly if the switching on the 'heater' affect counters values that creates all kinds of classical paradoxes:

allows subjective FTL communication;

allows communication from the future;

creates nasty observer effects.

It also can speed up quantum computation (switch on the heater upon the 'calculation is ready' condition).

$\sum _{m=0}^{127}sin(42)\!$

http://www.robertnz.net/pdf/xor2.pdf

Sandbox

The maximum probability to find some passive device in some state, would be where this device have done the most damage (highest entropy release to the environment). All devices are passive. That includes observers.
User:Dc987/Observer