It is known that for a classical system the amount of information needed to store its state is the same as the amount of information that can be stored in that system. This amount is equal to Shannon's entropy or its classical generalizations.
On the other hand the amount of information needed to describe the state of a quantum system is greater than the amount of information that can be stored in such quantum system.
My impression is that to describe the state of a quantum system of $N$ qubits one needs $2^N$ bits of information, on the other hand one can only store $N$ bits of information inside such system, which is in my impression equal to von Neumann's entropy of the system.
Thus I have two possible explanations of such discrepancy.
a) The true entropy of a quantum system is exactly von Neumann's entropy. The difference is only due the fact that a classical storage can only independently operate with digits not less than one bit, thus to store, say 4 variables of 1/4 bit each a classical system needs 4 bits (at least one bit for each variable) while a quantum system can use parts of a digit independently and thus store all 4 variables in one binary digit.
b) The true entropy of a quantum system of $N$ qubits is exactly $2^N$ bits but the amount of extractable information is only $N$ bits.
Which explanation do you consider more correct or are there possible other explanations?