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## Wave Function Collapse and Entropy

 Quote by atyy (I guess I'm asking if integrating out is a good enough form of coarse graining.)
I think it's not.

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 Quote by Demystifier If there was nothing resembling the wave function collapse, then one could say that entropy of the subsystem increases, due to decoherence. But the fact is that something resembling the wave function collapse does exist (which decoherence by itself cannot explain, which is why decoherence does not completely solve the measurement problem). For that matter it is not important whether the collapse is related to consciousness (von Neumann), or happens spontaneously (GRW), or is only an illusion (many worlds, Bohmian, etc.), as long as it exists at least in the FAPP sense.
 Quote by Demystifier I think it's not.
I guess it's not obvious to me that the reduced density matrix doesn't involve collapse, since I've seen it said that the Born rule is implicitly used in getting it.

 Quote by Demystifier If there was nothing resembling the wave function collapse, then one could say that entropy of the subsystem increases, due to decoherence.
But in most cases, there is nothing resembling wave function collapse. Decoherence occurs whenever the interaction between systems leads to entanglement and not only during measurements.

I'm still wondering how exactly this is related to the observable entropy increases. If we start with two pure states, I guess that any fundamental interaction that leads to maximal entanglement should begin to disentangle the systems afterwards. So I would expect an oscillating entropy for the systems (in classical mechanics, no entropy change arises from such a situation). Which of course would call for an explanation why our observations always take place in the rising entropy domain.

 Quote by atyy I guess it's not obvious to me that the reduced density matrix doesn't involve collapse, since I've seen it said that the Born rule is implicitly used in getting it.
If you do an experiment, you get one definite outcome for your observable. The density matrix contains the probabilities for all possible outcomes, so it isn't the final state you perceive.

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 Quote by atyy I guess it's not obvious to me that the reduced density matrix doesn't involve collapse, since I've seen it said that the Born rule is implicitly used in getting it.
Mathematically, the reduced density matrix is obtained by partial tracing, which technically does not depend on the Born rule. The Born rule only serves as a motivation for doing the partial trace, but formally you can do the partial trace even without such a motivation.

A Born-rule-independent motivation for doing the partial trace is the fact that the evolution of the resulting object (reduced density matrix) does not depend on the whole Hamiltonian, but only on the Hamiltonian for the subsystem.

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 Quote by kith So I would expect an oscillating entropy for the systems (in classical mechanics, no entropy change arises from such a situation).
In both quantum and classical mechanics (based on deterministic Schrodinger and Newton equations, respectively), it is true that entropy will start to decrease after a certain time, and return arbitrarily closely to the initial state. However, in both cases, the typical time needed for for such a return is many orders of magnitude larger than the age of the Universe.

 Quote by kith Which of course would call for an explanation why our observations always take place in the rising entropy domain.
That's easy to explain. If we happen to live in an entropy-decreasing era, we will naturally redefine the sign of time accordingly, i.e., re-interpret it as an era in which entropy is increasing. That's because our brain, and ability to remember, is also determined by the direction in which entropy increases.

The good question is why the direction in which entropy increases is everywhere the same, i.e., why it is not the case that entropy increases in one subsystem and decreases in another? The answer is that it is interaction between the subsystems which causes them to have the same direction of the entropy increase:
http://arxiv.org/abs/1011.4173v5

 Quote by Demystifier In both quantum and classical mechanics (based on deterministic Schrodinger and Newton equations, respectively), it is true that entropy will start to decrease after a certain time, and return arbitrarily closely to the initial state.
Are you talking about the von Neumann and Liouville entropies or about some course grained entropy? I don't see how the Liouville entropy may increase for a subsystem if it remains constant for the whole system.

 Quote by Demystifier If we happen to live in an entropy-decreasing era, we will naturally redefine the sign of time accordingly, i.e., re-interpret it as an era in which entropy is increasing. That's because our brain, and ability to remember, is also determined by the direction in which entropy increases.
Interesting, I haven't thought along these lines before. Can you recommend a not too technical article which expands on this? Also thanks for the link to your article, it looks quite promising. /edit: I'm skimming it right now and it probably simply is this reference I asked about ;-).

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 Quote by kith Are you talking about the von Neumann and Liouville entropies or about some course grained entropy? I don't see how the Liouville entropy may increase for a subsystem if it remains constant for the whole system.
I am talking about course grained entropy, of course.

 Quote by kith Interesting, I haven't thought along these lines before. Can you recommend a not too technical article which expands on this?
For example, Hawking explains it in his "Brief History of Time". I guess I cannot recommend a less technical literature than that.

A much more refined analysis of that stuff, but still very non-technical, is the book:
H. Price, Time's Arrow and Archimedes Point

Another good related non-technical book is:
D. Z. Albert, Time and Chance

There is also a good non-technical chapter on that in:
R. Penrose, The Emperor's New Mind

Recognitions:
 Quote by Demystifier Mathematically, the reduced density matrix is obtained by partial tracing, which technically does not depend on the Born rule. The Born rule only serves as a motivation for doing the partial trace, but formally you can do the partial trace even without such a motivation. A Born-rule-independent motivation for doing the partial trace is the fact that the evolution of the resulting object (reduced density matrix) does not depend on the whole Hamiltonian, but only on the Hamiltonian for the subsystem.
If the Born rule isn't used, couldn't one just give an arbitrary reweighting of the sum over the environment and still get an object defined only on the subsystem (ie. is the averaging over the environment unique if the Born rule isn't used?)

 Quote by Demystifier The good question is why the direction in which entropy increases is everywhere the same, i.e., why it is not the case that entropy increases in one subsystem and decreases in another? The answer is that it is interaction between the subsystems which causes them to have the same direction of the entropy increase: http://arxiv.org/abs/1011.4173v5
To me, the mystery of entropy is not just that the arrows of time for all parts of the universe are the same, but that the thermodynamic arrow of time is aligned with the cosmological arrow of time. That is, for all parts of the universe, entropy decreases in the direction of the Big Bang.

 Quote by kith If you do an experiment, you get one definite outcome for your observable. The density matrix contains the probabilities for all possible outcomes, so it isn't the final state you perceive.
Well, a nondeterministic theory cannot possibly describe the final outcome, it can only describe the set of possibilities and their associated probabilities.

 Quote by stevendaryl Well, a nondeterministic theory cannot possibly describe the final outcome, it can only describe the set of possibilities and their associated probabilities.
The recipe of using the reduced matrix may not imply the collapse interpretation, but it seems that is as close as you can get.

The collapse interpretation says that initially the system is in some state $\vert \Psi\rangle$. You perform an experiment to measure some observable with eigenvalues $\lambda$ and corresponding eigenstates $\vert \Psi_\lambda\rangle$ (for simplicity, assume non-degeneracy). Then the results are that afterward:

For every value of $\lambda$, there is a probability of $\vert \langle \Psi \vert \Psi_\lambda\rangle \vert^2$ that the system is in state $\vert \Psi_\lambda\rangle$

This is captured by the density matrix formalism as the transition

$\vert \Psi \rangle \langle \Psi \vert \Rightarrow \sum_\lambda \vert \langle \Psi \vert \Psi_\lambda\rangle \vert^2 \vert \Psi_\lambda \rangle \langle \Psi_\lambda \vert$
 Recognitions: Science Advisor I guess thinking about it classically, Demystifier's argument must be right. Measurement gives us more information, which is a reduction in entropy. Entropy increases when we forget, according to Landauer's exorcism of Maxwell's demon. I guess what's not obvious to me is - how much coarse graining do we need, since the partial trace in getting the reduced density matrix is a form of coarse graining?