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I am even not sure if there is a good definition of entropy for quantum system. (There could be, I just don't know, and don't remember reading about it.)

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Yes, there is a good definition of entropy in quantum theory. It goes by the name "von Neumann entropy."

Interestingly, the von Neumann entropy was discovered well in advance of the classical information theoretic entropy (which is a more general version of the thermodynamic quantity), which is usually called the "Shannon entropy."

Shannon apparently had a conversation with von Neumann about his information theoretic quantity, where von Neumann was quoted as saying, "Call it entropy. It is already in use under that name and besides, it will give you a great edge in debates because nobody knows what entropy is anyway."

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Yeah, now I remember that we studied it. After all we should have had it at 'Quantum Statistical Mechanics', I've just never had to use it after the university.