In this thesis, section "1.1.4 Quantum Entanglement", page 19. It is mentioned that "for mixed states, entanglement is necessary but not sufficient to ensure the violation of the Bell inequality". I'm finding it hard to understand the meaning of this statement. What I understand is that only the states that violate the Bell inequality are entangled. How can a mixed state be entangled without violating the Bell inequality?
In the thesis, there is an example of this: The Werner state $\rho = p |\psi\rangle\langle \psi| + (1-p) I/4$, $p\in [0,1]$ is entangled for $\frac{1}{3} < p \leq 1$ but violates Bell inequality only when $\frac{1}{\sqrt{2}} < p \leq 1$.
In the case $\frac{1}{3} < p \leq 1$ the only quantum correlation that the system presents is entanglement. In the case $\frac{1}{\sqrt{2}} < p \leq 1$ there is entanglement and another type of quantum correlation (quantum discord, for example). This means that entanglement will always be present in a system that has some type of quantum correlation. Is this statement correct?
I have been reading more and found the hierarchy of entanglement and quantum correlation very confusing. "Entanglement is necessary but not sufficient to ensure the violation of the Bell inequality", this means that for the violation of Bell inequality in mixed states you need quantum correlations. Is not possible to have a system with quantum correlation but without entanglement?