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I have seen entropy with several different definitions. Like Von Neumann entropy and Rényi entropy, etc.

So I am curious why there are so many different definitions in quantum mechanics while only one in classical mechanics named after Boltzmann?

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The von Neumann entropy is the analogue of the Boltzmann entropy in quantum mechanics. It really is exactly the same thing. Any density matrix $\rho$ can be written as $\rho = \sum_i p_i |i\rangle\langle i|$, where $p_i = \mbox{probability}(\mbox{state}_i)$ is a probability distribution on state vectors. The von Neumann entropy is the Boltzmann entropy of this distribution. Writing it as $Tr(\rho \operatorname{ln} \rho)$ just makes you look clever.

The Renyi entropy isn't specific to quantum mechanics. It's a concept from information theory and probability theory, a generalization of the usual Boltzmann entropy which allows you to vary the way in which events of low probability contribute to the entropy.

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All the quantum entropies that you cite have a classical analogue. E.g. the Von Neumann entropy $\langle S \rangle = -k_B \mathrm{Tr} (\hat{\rho} \ln \hat{\rho})$ is the quantum version of the Gibbs entropy $\langle S_\mathrm{cl} \rangle = -k_B \int \mathrm{d}p \mathrm{d}x (\rho \ln \rho)$ used in classical statistical mechanics. The Boltzmann entropy is a special case of it.

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