What is the relationship between entanglement and quantum correlation? Are they synonym?

Sometimes, these two words look like synonym; people use one word to explain another,

These states are not a simple tensor product of the states of each qubit but rather a superposition that describes quantum correlations (entanglement) between the qubits.

On the other hand, it seemed we need to distinguish these two words,

If you read a randomly selected nontechnical account of quantum entanglement, you will likely be told that measuring a particle in one place can instantly change another particle elsewhere, no matter the distance between the two. Surprisingly, this is something that Paul Halpern never claims in his new book, Synchronicity. ...to connote an “acausal connecting principle.” Quantum correlations, in Halpern’s view, are a special case of synchronicity.

Excerpted from Quantum correlations at a distance needn’t necessarily be “spooky” 1st paragrah

Moreover, is the following expression correct? (http://memoirs.is.kochi-u.ac.jp/Vol26/MemoirsF26-6.pdf 2.3.2, I translated it into English)

The quantum correlation MUST exist in the entanglement state; however, even if there is a quantum correlation, it cannot be called entanglement. For example, when the relationship between qubits A and B is $|\psi\rangle=\frac{1}{\sqrt{2}}(|0_A0_B\rangle+|1_A1_B\rangle)$, which cannot be separated by the tensor product. In other words, it is an entanglement state, and there exists a quantum correlation.

After measuring, this state changes to $|0_A0_B\rangle$ or $|1_A1_B\rangle$. There is a quantum correlation in $|0_A0_B\rangle$ or $|1_A1_B\rangle$ state, but it is no longer an entanglement state.

  • 1
    $\begingroup$ Honestly, I'm not sure the term quantum correlation has a consistent definition across the literature. In nonlocality it is most often used to mean a probability distribution that could have arisen from measurements on some quantum systems. $\endgroup$
    – Rammus
    May 8, 2021 at 9:30
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    $\begingroup$ handwavily, I'd say entanglement is a property of a bipartite state: $\rho$ is entangled if it cannot be written in a specific way. Correlations are a property of the probability distribution resulting from measuring a state with a given measurement. Quantum correlations refers to correlations which cannot be produced with "classical" states and measurements. However, what precisely "nonclassical/quantum" means might depend on the context. E.g. when studying nonlocality "quantum correlations" are those that cannot be written as local deterministic behaviours. $\endgroup$
    – glS
    May 8, 2021 at 9:34


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