Transcendental properties of entropy-constrained sets

Vjosa Blakaj, TUM
Herbert-Walther-lecture hall (G 0.25) at MPQ
Wednesday November 23rd, 11:30am (MEZ)

Abstract:

For information-theoretic quantities with an asymptotic operational characterization, the question arises whether an alternative single-shot characterization exists, possibly including an optimization over an ancilla system. When the expressions are algebraic and the ancilla is finite, this leads to semialgebraic level sets. In this talk, I will present a criterion for disproving that a set is semialgebraic, based on an analytic continuation of the Gauss map. Applied to the von Neumann entropy, this shows that its level sets are nowhere semialgebraic in dimension d > 2, ruling out algebraic single-shot characterizations with finite ancilla (e.g., via catalytic transformations). I will also present similar results for related quantities, including relative entropy, and discuss the transcendental nature of the level sets of Rényi entropies and mutual information.