I'm a little confused by some of the answers above, so I will try and help out.

In the question, you specifically mention "generate correlation".  I guess that this is probably a typo, but if not, I would not think of the states as generating correlation.  The correlations are created as the states are created.  This is a subtle, but important distinction in my opinion.

Now to return to what I think you are intended to ask.  

Classical correlations are all around us, and perhaps no better illustrated than the example of Bertlemann's socks, introduced by Bell.  Bertlemann always wears odd socks, so if you see that one of his socks is pink, you don't need to introduce any kind of sophisticated physics to know that the other sock he wears is not pink.  This is an example of a classical correlation - where there is a classical correlation between two, in principle knowable outcomes.

A quantum correlation is where the states of two particles are independently unknown, for example unknown polarisation states of photons, but there is a correlation between the two states - for example it is known that the total angular momentum is equal to some value.  These are the correlations that are observed in EPR type experiments.

Note I have been careful here to talk about classical and quantum correlations, and not entangled or mixed states.

Entangled states are typically those that exhibit quantum correlations, and we would probably want to think of these as 'pure states'.  By this we mean that the state is represented by a unit vector on the appropriate Bloch hypersphere, or to be more colloquial, that there system is isolated from the environment and 'as quantum as possible'.

Mixed states, as has been said above, generalise this result.  They are often defined in terms of ensembles of pure states, however I like to think of them, and explain them, in the limit of maximum mixedness.  In this limit, mixed states describe classical probabilities.  Perhaps the clearest example is to consider a coin that is tossed and the result hidden.  The coin is either heads or tails with equal probabilities.  We describe this state by a mixed state of 50% heads and 50% tails.  There are no quantum correlations whatsoever.  Contrast this with a spin-half system (e.g. electron spin) that is aligned along the X axis, about to be measured by a Stern-Gerlach filter in the z axis.  This will still result in 50% probability of being measured spin up, 50% spin down, but because it is in a pure state, the phase between these probability amplitudes is non-trivial and one can perform other non-trivial rotations.

Now we must also be precise.  There are entangled states that are not pure states, so entanglement can be preserved in the limit of finite mixedness.  There are also non-entangled states that show non-classical correlations (quantum discord).  However I hope that this brief discussion gives you a useful place to start.

Maybe we can say that mixed states are states of subsystems of an entangled state, and these subsystems are entangled with the rest of the appropriate quantum system.

Dear Yaokun Xu,

I understand what a mixed state is: a convex combination

(sum of probabilities must be one)

of certain pure states, which make the mixture.

I do not understand what you mean with an "entanglement state".

Is your question about pure and mixed states?

Mixing two states is a purely classic probabilistic operation. One can create a mixed state by coin and prepare the physical stystem in the first state if the flipping the coin gives head and preapre the system in the second state if the coin gives tail. A measurement on a mixed state gives a result that is a probabilistic mixture of what the measurement would have gives if one had prepared the system in one of the pure states.

Superposition of states (including entanglement) is different. For instance a measurement on a superposition of two states may give a certain result with probability 1 even if the same measurement on any of the states that are part of the superposition lead to results that are probabilistic. 

The notion of entanglement is related to composed systems. The pure entangled states can be obtained as superposition of states that are pure even when they are restricted to any component of the composed system.

To add to what has already been said : We usually refer to "Quantum entanglement" but "mixed states" so that correlation of mixed states causes the quantum entanglement.

SKM, NH, SM

@ Soumitra K Mallick A  (probabilistic) mixture of unentangled states is unentangled, but a superposition of unentangled stated may be entangled.

I presume that by entanglement states you mean entangled states.

The formal difference is that an entangled state may be a pure state. A mixed state is by definition a probabilistic ensemble of pure states. The former is described by a wave function, the latter by a density matrix.

The physical difference is that entanglement may be responsible for observable correlations impossible in classical probabilistic systems (cf. violation of Bell's inequalities).

By "entanglement" we had in mind partition of states and hence the give and take of "partitions" to which particles belong. This is different from give (measure) and take (measure)of particles. Superimposition of "partitions" may cause entanglement because the description of the particles may mean they belong to each partition as well to their union or "meet". This is a notion which is quite often used in Economics as well as information theory or decision theory. I think we can add the physical description for particles with masses because they have quality. This I think would also satisfy Bell's inequalities because the partitions define the state space so the problems themselves are local. This is a "bypass".

SKM, NH, SM

I'm a little confused by some of the answers above, so I will try and help out.

In the question, you specifically mention "generate correlation".  I guess that this is probably a typo, but if not, I would not think of the states as generating correlation.  The correlations are created as the states are created.  This is a subtle, but important distinction in my opinion.

Now to return to what I think you are intended to ask.  

Classical correlations are all around us, and perhaps no better illustrated than the example of Bertlemann's socks, introduced by Bell.  Bertlemann always wears odd socks, so if you see that one of his socks is pink, you don't need to introduce any kind of sophisticated physics to know that the other sock he wears is not pink.  This is an example of a classical correlation - where there is a classical correlation between two, in principle knowable outcomes.

A quantum correlation is where the states of two particles are independently unknown, for example unknown polarisation states of photons, but there is a correlation between the two states - for example it is known that the total angular momentum is equal to some value.  These are the correlations that are observed in EPR type experiments.

Note I have been careful here to talk about classical and quantum correlations, and not entangled or mixed states.

Entangled states are typically those that exhibit quantum correlations, and we would probably want to think of these as 'pure states'.  By this we mean that the state is represented by a unit vector on the appropriate Bloch hypersphere, or to be more colloquial, that there system is isolated from the environment and 'as quantum as possible'.

Mixed states, as has been said above, generalise this result.  They are often defined in terms of ensembles of pure states, however I like to think of them, and explain them, in the limit of maximum mixedness.  In this limit, mixed states describe classical probabilities.  Perhaps the clearest example is to consider a coin that is tossed and the result hidden.  The coin is either heads or tails with equal probabilities.  We describe this state by a mixed state of 50% heads and 50% tails.  There are no quantum correlations whatsoever.  Contrast this with a spin-half system (e.g. electron spin) that is aligned along the X axis, about to be measured by a Stern-Gerlach filter in the z axis.  This will still result in 50% probability of being measured spin up, 50% spin down, but because it is in a pure state, the phase between these probability amplitudes is non-trivial and one can perform other non-trivial rotations.

Now we must also be precise.  There are entangled states that are not pure states, so entanglement can be preserved in the limit of finite mixedness.  There are also non-entangled states that show non-classical correlations (quantum discord).  However I hope that this brief discussion gives you a useful place to start.

Dear all,

        Sorry for my inaccuracy expression. I concern the difference between quantum correlation and classical correlation. Quantum correlation is generated by entangled state and classical correlation is generated by mixed state. Any quantitative methods to distinguish them? like discord.

Dear Yaokun Xu,

Your qualification shows some misunderstanding.  Classical correlations are not, in general, caused by mixed states.  For example, you have a classical correlation in the state |1>|0>, i.e. particle one is in state |1> and particle 2 is in state |0>.  Quantum mechanically this is still a pure state, and there is, by construction, a clear classical correlation - but of course, no entanglement.

There are many quantitative measures for entanglement, see for example Plenio and Virmani, Quant.Inf.Comput.7:1-51,2007, arXiv:quant-ph/0504163. Be careful, however, as many become difficult when mixedness is taken into account, so depending on the system you care about, you might want to use specific metrics to capture the physics that is important.