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I wonder if there is a relation between coherence and entanglement of a two qubit system?

I mean do they have a direct or a reverse relation?

Is there any reference about that?

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It depends how formally/quantitative you want to take these terms.

Coherence

Here's a few possible ways people might talk about "coherence" in the context of quantum information:

  • It can loosely refer to how a state is a superposition of other states. To meaningfully talk about coherence in this sense you have to (implicitly or explicitly) use some other set of states as reference, as otherwise any pure state can be written as a superposition of other states in infinitely many ways. In this context, you'd hear about "coherent superposition", meant as the opposite of "mixture". For example, you might say that $|\psi\rangle=|0\rangle+|1\rangle$ is a coherent superposition of $|0\rangle$ and $|1\rangle$, as opposed to a mixture of the same state.

  • When an environment is involved, you might say things like a state "losing coherence", or a "coherent evolution". In this context, this generally means that there is no (or little) loss of information to the environment, and thus the purity of the state is preserved.

There are also quantitative ways to talk about coherence, typically in the context of resource theories. See for example https://arxiv.org/abs/1609.02439 on this topic.

It's worth pointing out that "coherence" is a property that does not require a multipartite state, and does not generally make reference to the multipartite structure of the state, if any is present.

Entanglement

Entanglement is a property that is specifically about multipartite systems. You can roughly speaking think of it as the failure of the parts of the system of being classically correlated (where "uncorrelated" is seen here as a special case of "classically correlated"). You'll find countless discussions about what entanglement is or isn't on this site and on physics.SE, so I won't add much here.

Coherence vs entanglement

Entanglement requires the existence of coherence between specific basis states. For example, the maximally entangled state $|00\rangle+|11\rangle$ is a coherent superposition of $|00\rangle$ and $|11\rangle$. At the same time, not any coherent superposition results in entanglement: for example, a coherent superposition of $|00\rangle$ and $|01\rangle$ gives $|0,+\rangle$ which is not entangled.

To really talk about direct or inverse relations between coherence and entanglement you'd have to precisely define what kind of "coherence" you're referring to, and how you quantify it. Intuitively, a "maximally incoherent state" should be something like a maximally mixed state, which is not entangled, but I'm not sure how standard or sensible this definition is.

Any pure state is arguably "maximally coherent" in some definition of "coherence", but clearly not all pure states are entangled. You can also have entanglement for states that are not pure, and thus not "fully coherent" in some definition of "coherence".

In conclusion, the terms refer to different properties of a multipartite state, which might be related in some situations, but are not in general directly relatable.

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