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I read in a blog

"The coherent state is called “coherent state” because all the states |n⟩ in its expansion have a fixed phase relation to each other. As a result, when we write down the density matrix

$ ρ=|α\rangle \langle α|=e^{-|α|^2∕2} ∑_{n,m=0}^∞\frac{|α|^{2n}}{n!} |n⟩⟨m|$

This matrix is completely filled in the off-diagonals |n⟩⟨m|,n≠m. Those off-diagonal elements are called “coherences” and are generally necessary for interference. Due to this we can say that the coherent state is indeed maximally coherent and maximally classical."

I can't understand this. Any proper explanation?

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  • $\begingroup$ which blog where? BTW $\langle\alpha\vert\alpha\rangle=1$ to you have your bra and key backward to describe $\rho$, and you should have a double sum over $n$ and $m$. $\endgroup$ Commented Sep 24, 2021 at 2:20

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I'm not sure what's meant by "maximally coherent" because all pure states are exactly equally coherent as evident by the fact that all of them have exactly zero von Neumann entropy (which is exactly the measure of how much of the "coherences" in the density matrix are "lost" in some sense). The claim that coherent states are maximally classical is simple to understand, they saturate the Heisenberg uncertainty bound and thus are minimal conjoint uncertainty wavefunctions. So they can be seen as states that are as close to a classical state as possible.

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