What is the difference between correlation and entanglement? I have read that not all correlated states are entangled. What is the difference between the two?
Mathematically, it was stated that a system which can be put in the form of
$\sum_{k}p_{k}\hat{\rho}_{k}^{A}\otimes\hat{\rho}_{k}^{B}$
is correlated whereas a system which can't be expressed as above is entangled.
What does this mean?


*

*$\hat{\rho}$ is the density operator.

 A: A system's density matrix can be written in the form you cite if and only if it is a classical mixture of factorisable pure states. A factorisable pure state is, of course, one that can be written as a tensor product $\psi_A\otimes\psi_B$, where $\psi_A$ and $\psi_B$ are pure states in subsystems $A$ and $B$, respectively. Correlations between measurements on subsystems $A$ and $B$ are indistinguishable from any other classical correlation between classical random variables, and, in particular, heed the Bell and CHSH inequalities. The classical probability theory of correlated random variables describes the joint probability densities for measurements on the two subsystems. You can think of these states as classical mixtures of block-diagonal stripes in the full quantum state space.
General entangled states, on the other hand, are those which are classical mixtures of more general, non-factorisable states. Correlations between measurements on the subsystems violate both the Bell and CHSH inequalities.
Classical correlated states have a very special structure. Entangled states are the overwhelmingly likelier case, if you just chose a density matrix for the combined systems at random.
A: The main difference between entanglement and classical correlation can be seen using the proof of Bell's inequality. It all has to do with measuring states using different bases. You can have two entangled spins, but along what axes do you measure those spins? You can measure the spins along the z axis, or the x axis, or any axes. The ability to measure a spin 1/2 particle along any direction, and the fact that, for instance, a particle which is spin up in the z direction can be written as a super position of up and down states in the x direction, allows for a situation which is fundamentally different from classical mechanics.
A: In layman's terms, if subsystem B is measured, it collapses the wave function that belongs to density matrix B. But it does not collapse the wave function of density matrix A, since the two are factored.
So the measurement of B does not determine the state of A.
Remark: I think the sum over the density matrices in the given expression above is not a superposition of states, but lack of some more details about the system.
When a system is entangled, for example of the form $|1,0\rangle + |0.1\rangle$, then if say system A is measured in state 1, then automatically system B is in state 0, since the measurement collapses the combined systems into wavefunction $|1,0\rangle$.
Both states are maximal entangled.
In case there are many more states and systems entangled, one calculates the shannon entropy of respective subsystems from their reduced density matrices, to determine the degree of entanglement.
I recommend to numerically solve such a problem once, in order to understand all the details. A wonderful and interesting problem is to solve the Wigner-Weisskopf model numerically without the born markov approximation, and calculate and display the various entanglements between the subsystems atom, photon and vacuum state, as they evolve over time.
This is the richest and most interesting quantum problem that I know of. Fairly simple, but extremely interesting and understandable results, that explain many features of the transition into the classical world.
