Example of a classical correlation and a quantum correlation

Question

I'm trying to understand the fundamental differences between classical and quantum correlations through examples of a quantum entangled state and a classically correlated state.

I know that this is an entangled state $|\phi\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)$, so this is a quantum correlation.

Now I'm searching an example of classical correlation, if I chose the following density operator $\rho = \frac{1}{2} \left( |00\rangle \langle 00| + |11\rangle \langle 11| \right)$ When measurements are made on both particles, either both particles are found in state $|00\rangle$ or both in state $|11\rangle$. Is the correlation here classical because it arises from the statistical mixture of the states and not from any quantum superposition or entanglement? Is correct to say that the correlation in the outcomes is due to the preparation of the state? Because if you know the outcome for A, you immediately know the outcome for B, not for any quantum property like entanglement, but simply because the system was prepared in a way that only those particular pairs of states were possible?

Is correct to say that: the difference between classical case and quantum case is that $\rho$ (density operator on classical state) represents our degree of ignorance on the system, we don’t know in if we are in $|00\rangle$ or in $|11\rangle$ while the state $|\phi\rangle$ no contain ignorance because it is a superpoisition, so the randomness of $|\phi\rangle$ have nothing to do with the ignorance of the system but it's an intrinsic characteristic?

Answer 1

Quantum correlation: The one you identify is correct, let's call that |HH>+|VV> for short; H=horizontal, V=vertical. Same as your |00⟩+|11⟩, right?

That state is exactly the same as |DD>+|AA>, where D=diagonal, A=anti-diagonal. Regardless of how you label it, it is an entangled state and cannot be represented in a Product form (as I'm sure you know). The quantum correlation is 1 (completely correlated) at any angle setting chosen for measurement, as long as both are measured alike. So 100% correlation on the H basis or the D basis.

Classical correlation: I wouldn't represent that as you have, but there are a number of different classic pairings that are possible.

a) So here is one we can discuss: |H1⟩|H2⟩. In this state, you have a classical correlation. When you measure both on the H or V basis, you also have 100% correlation. But on any OTHER basis, you will not. For example, on the D or A basis: there will be no (0%) correlation. On other bases, the correlation will vary. And of course, the classical correlation will be represented as a Product. So whether there is 100% correlation or not depends on the basis selected.

b) In the above Classical example, we start out with a known state for 2 otherwise unrelated particles (obviously not entangled). But you could also select a state where there is a mixture, something like: (|H1>+|V1>)(|H2>+|V2>). This is obviously a state which is a Product state and has statistics that generally have no correlation (0%) on any basis. You would expect that result from otherwise unrelated pairs.

Either of the above classical states are possible physically. So the key statistical difference being that Classical states cannot generate perfect (100%) correlations on many/all bases, while Entangled states can. Of course, this is without even talking about the Bell inequalities, which is a quite a subject of its own. :)