# A simple QM experiment analysis question

1. Oct 15, 2009

### dmtr

Can somebody help analyzing the outcome of the following thought
experiment:
1) A simple setup with a photon source, half-silvered mirror and two
photon detectors with counters (on each path - A and B);
2) We adjust the setup to get an equal distribution between the
counters on paths A and B;
3) Now we modify the setup in such a way that upon a detection of a
photon on path A some free energy is being 'spend' (so path A now is
more entropic than B).

The question - would (3) modify the distribution?

-- Dmtr

2. Oct 15, 2009

### cesiumfrog

What are you getting at; why would you expect the result to change?

(Conventionally, what goes on at the macroscopic/classical level isn't going to interfere back here, and you'd probably get a causality paradox otherwise, but this reminds me of interesting questions about why sugar molecules are found in position/chirality eigenstates rather than with definite energies..)

3. Oct 15, 2009

### dmtr

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.

-- Dmtr

4. Oct 15, 2009

### dmtr

Just to be on the same page, here is the schematic illustration of the experiment:

-- Dmtr

5. Oct 15, 2009

### cesiumfrog

How about you try to set up a variation of experiment 2, where you publish the results of counters A&B in a scientific journal. A week afterward, I'll read the journal and smash one vase (or run my space heater for an extra minute, flash a light, or some other entropy-increasing action) for each count you record having got from detector A. In other words, you've unknowingly carried out experiment 3. If there was a difference between experiment 2 & 3, then you would have been able to already tell from the results you yourself later published (and if I happen to be on Mars, then this will have communicated my decision to you far faster than the speed of light; or maybe I'll change my mind, causing history to rewrite). Make sense? (The problem is that the signal leaving a detector is, by definition, macroscopic and classical.)

Entropy might play a part in choosing the preferred basis line along which a quantum state collapses, but it cannot affect which side of that line the state collapses to.

I'm not sure what you mean by that?

Last edited: Oct 15, 2009
6. Oct 15, 2009

### dmtr

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).

-- Dmtr

7. Oct 15, 2009

### cesiumfrog

Do you mean you want to treat more of the apparatus as "microscopic" (i.e., governed by quantum mechanics)?

Then what will your entropic process be? Since QM is time reversible, where will you source the information (randomness) to inject into that stream of energy. (What form of energy could you even define, microscopically, as having high or low entropy?) And still, I don't see anything interfering with the results unless the beam-splitter outputs are provided an opportunity to remix? It seems to me like the apparatus that you're describing isn't sufficient for exploring the effect in question.

Last edited: Oct 15, 2009
8. Oct 16, 2009

### dmtr

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.

-- Dmtr

9. Oct 16, 2009

### Mentz114

dmtr,

how about this variation. The setup is done with two detectors as before, then one is removed. The process of detection and displaying something on the instruments will have different entropic impact from allowing the light to escape.

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10. Oct 16, 2009

### dmtr

Mentz114,
Yes, your version is similar, but I'm not sure if it is simpler to analyze... I'll think about it over the weekend.

-- Dmtr

11. Oct 16, 2009

### f95toli

Maybe I am nitpicking now, but there is a path missing from your schematic. If want to do a full QM analysis you need to include the vacuum "entering" from the top; a beam splitter always has two inputs and two outputs.

It is still not clear what you mean by "entropic process"? Do you simply mean that the system includes some form of ohmic bath?

12. Oct 16, 2009

### dmtr

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...

-- Dmtr

Last edited: Oct 17, 2009
13. Oct 17, 2009

### dmtr

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 :)

Ok. Now can somebody please either refute my argument or give some rough quantitative estimates?

-- Dmtr

Last edited: Oct 17, 2009
14. Oct 19, 2009

### dmtr

Would it be something like:

$$N_{A}/N_{B} = e^{S_{A} - S_{B}}$$

where $$N_{A}$$, $$N_{B}$$ are the counter values and $$S_{A}$$, $$S_{B}$$ - total entropy in the space-time light cone (entropy 'visible' to the observer)?

-- Dmtr

15. Oct 19, 2009

### cesiumfrog

Short answer is no, 50-50. Can you cite any reference that even suggests otherwise?

16. Oct 19, 2009

### dmtr

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.

-- Dmtr

Last edited: Oct 19, 2009
17. Oct 19, 2009

### Mentz114

dmtr:

It seems to me something in the future won't affect the process that takes place at the beam splitter. If all wave equation collapses ( for want of a better term ) were governed by future entropy considerations, life would be non-existent.

18. Oct 19, 2009

### cesiumfrog

Use the Born rule rather than attributing equal probability to every term/"microstate". The state prepared by the beam-splitter is independent of what follows it (as such elements only affect the later evolution of parts of the total quantum state, and in a manner that conserves probability).

19. Oct 19, 2009

### dmtr

Ok. Thank you. I'll need some time to digest that.

I don't think this is true. Remember, the quantum mechanics is time reversible.

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.

20. Oct 20, 2009

### dmtr

Here is another diagram, illustrating the entropy exchange with the environment:

-- Dmtr