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In quantum mechanics there are many references to quantum coherences, but I am still unsure of what the precise definition of a coherence is in all cases.

There is clearly a precise sense in which a state can have coherences in terms of the density matrix formalism and the off-diagonal elements of the density matrix, but what does it exactly mean to have coherences between states or eigenstates and what does it mean for an observable to have coherences in a system? By 'coherences between eigenstates' is it meant the phase differences between the different eigenstates in a coherent superposition? In that case, what are coherences between general states, not just eigenstates in a coherent superposition?

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Quantum coherences are, as you said, simply the off-diagonal elements of the density matrix. Note however that the density matrix depends on the choice of basis, and that every density matrix can be diagonalized.

When we say "coherence between (orthogonal) states $|n\rangle$ and $|n'\rangle$", it means that we are looking at the density matrix in a basis containing these two vectors. In other words, the coherence between $|n\rangle$ and $|n'\rangle$ is the matrix element $$ \langle n| \rho |n'\rangle . $$ Without any further specification, "quantum coherences" usually refers to coherences in the eigenbasis of the Hamiltonian.

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  • $\begingroup$ Ah I see. So to talk about coherences of an observable $\hat{A}$ in a system is just referring to coherences in the eigenbasis? When we say 'coherent superposition of states' is it implied that the eigenstates in the superpositon all have some constant phase difference or phase relation to each other? $\endgroup$
    – Tom
    Apr 2, 2020 at 22:36
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    $\begingroup$ @Tom I'm not sure if "coherence of an observable" is very common terminology, but yes, that is how I would understand it. The expression "coherent superposition" stresses that the system is in a pure superposition state and not in a mixed state. Check also the great answer here: physics.stackexchange.com/questions/424578/… $\endgroup$
    – Noiralef
    Apr 2, 2020 at 22:46

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