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It is well known that the von Neumann entropy is continuous on a subset ofquantum states with bounded energy provided the Hamiltonian $H$ of the systemsatisfies the condition $\Tr\exp(-cH)<+\infty$ for any $c>0$. In this note weconsider the following conjecture: every closed convex subset of quantumstates, on which the von Neumann entropy is continuous, consists of states withbounded energy with respect to a particular Hamiltonian $H$ satisfying theabove condition.  It is shown that the classical analog of this conjecture is valid (i.e. it isvalid for the Shannon entropy). It is also shown that this conjecture holds forsome types of subsets consisting of non-commuting states, but its validity forall subsets of quantum states remains an open question.

On a conjectured property of the von Neumann entropy valid in the commutative case