Wave Function Collapse and Entropy

Demystifier

I think it's not.

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.

kith

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.

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.

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.

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. 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.

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

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.

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 ;-).

Demystifier

I am talking about course grained entropy, of course.

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

atyy

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?)

stevendaryl

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.

stevendaryl

Well, a nondeterministic theory cannot possibly describe the final outcome, it can only describe the set of possibilities and their associated probabilities.

stevendaryl

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 [itex]\vert \Psi\rangle[/itex]. You perform an experiment to measure some observable with eigenvalues [itex]\lambda[/itex] and corresponding eigenstates [itex]\vert \Psi_\lambda\rangle[/itex] (for simplicity, assume non-degeneracy). Then the results are that afterward:

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

This is captured by the density matrix formalism as the transition

[itex]\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[/itex]

atyy

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?