What is the relationship between entanglement and entropy in quantum systems?

[valanna]

This is an observation I'm making that seems like it is wrong but I can't figure out why it would be.

Entanglement links two states together. For example the spin angular momentum of particles. An entangled state of two particles could be where you know if one particle is measured spin up the state collapses and the other one will be measure spin down.
This seems like it would make that state more orderly, however entanglement is often measured by linear entropy and entropy is a measurement of disorder.
How does that work? Am I missing something important in my picture here?

 

[Bystander]

entropy is a measurement of disorder.

First error. Need I continue?

 

[valanna]

First error. Need I continue?

Dictionary definitions of it that I find define it that way but I can understand them being poor sources, could you link me a good source for what entropy really is then?

 

[Bystander]

As a "thermosaur" I prefer defining entropy the "old-fashioned" way, as integral of "q over T" dT, where "q" is heat and "T" absolute temperature; stat-mechers prefer "partition functions," and you can find entropy discussed under both.

 

[valanna]

As a "thermosaur" I prefer defining entropy the "old-fashioned" way, as integral of "q over T" dT, where "q" is heat and "T" absolute temperature; stat-mechers prefer "partition functions," and you can find entropy discussed under both.

Thank you very much,
I was able to find articles on entropy with that definition,
So the change in entropy is
ΔS=∫q/T dT
and entropy itself is defined as
S=kb*lnΩ
with Ω being the number of configurations
So if entropy is related to the number of possible configurations of a state so I can see where the "disorder" idea comes from though see why it isn't simply disorder.
I'm still confused to how it relates to entanglement though.
Entangled states have tied probabilities (I know it's more complex than that), so why would a greater amount of configurations correlate to a more entangled state?

 

[Nugatory]

Entanglement links two states together.

It does not - that is one of the (sadly common) misconceptions that you'll find in pop-sci discussions of QM. Coincidentally we happen to have a live thread going right now about what entanglement is; it should be clear from that thread that there is only a single state involved in entanglement.

@Bystander is also right about entropy. The dictionary definition ("entropy is disorder") is for the way the word is used by the general public, because that is what dictionaries are for. However, in physics the word entropy has a precise mathematical definition (actually two definitions that turn out to be more or less equivalent) and it is not "disorder".

 

[valanna]

Thank you both very much!
Would it be right to say that an entangled state, links the probabilities of the spin (sometimes could be another property) of two particles?

I think part of the misconception is that sometimes the spin of a single particle in the above state is also sometimes referred to as a state?

 

[Truecrimson]

Entanglement does have a connection to the von Neumann entropy which can be thought of as a generalization of Shannon entropy in information theory. It is the Shannon entropy $- \sum_j p_j \log p_j$ of the eigenvalues $\{ p_j \}$ of a density operator $\rho$. In words, it is the amount of uncertainty if you measure the density operator in its eigenbasis.

Now if $\rho$ is a pure state i.e. the system has a wave function, the von Neumann entropy will be zero. This just reflects the fact that you are certain of the measurement outcome if you measure in the basis that contains the wave function. So the von Neumann entropy of a state with a wave function, entangled or not, is zero.

When we have a joint state of two or more systems, we can consider the local density operator of each subsystem. In quantum theory, even if the joint state is pure, the local density operator may not be, and the local von Neumann entropy can be used as a measure of entanglement.

Would it be right to say that an entangled state, links the probabilities of the spin (sometimes could be another property) of two particles?

Entanglement is a form of correlation (but not every correlation is entanglement).

 

[atyy]

This is an observation I'm making that seems like it is wrong but I can't figure out why it would be.

Entanglement links two states together. For example the spin angular momentum of particles. An entangled state of two particles could be where you know if one particle is measured spin up the state collapses and the other one will be measure spin down.
This seems like it would make that state more orderly, however entanglement is often measured by linear entropy and entropy is a measurement of disorder.
How does that work? Am I missing something important in my picture here?

After collapse, the particles will not be entangled.

However, it is true that quantum entanglement provides a new way of getting "disorder" that is not available in classical physics:
http://arxiv.org/abs/1007.3957

It is still speculative, but this may lead to an understanding of why quantum black holes radiate:
http://relativity.livingreviews.org/Articles/lrr-2011-8/
http://people.brandeis.edu/~headrick/talks/EntanglementGeometry.pdf