  In this work, we address the question of the impossibility of certain
single-letter formulas by exploiting the semi-algebraic nature of various
entropy-constrained sets. The focus lies on studying the properties of the
level sets of relative entropy, mutual information, and R\'{e}nyi entropies. We
analyze the transcendental structure of the set of states in which one of the
aforementioned entropy quantities is fixed. Our results rule out
(semi)algebraic single-shot characterizations of these entropy measures with
bounded ancilla for both the classical and quantum cases.
