Transcendental properties of entropy-constrained sets

TL;DR

This paper investigates the mathematical nature of entropy-related sets, demonstrating that certain entropy level sets are transcendental and cannot be characterized algebraically with finite ancilla systems, impacting single-shot information theory.

Contribution

It provides a criterion based on the Gauss map to prove that entropy level sets are nowhere semialgebraic in higher dimensions, showing limitations of algebraic single-shot characterizations.

Findings

Von Neumann entropy level sets are nowhere semialgebraic for dimension d>2.

Similar results hold for relative entropy and related quantities.

Entropy values can be transcendental, algebraic, or rational depending on conditions.

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. If the expressions are algebraic and the ancilla is finite, this leads to semialgebraic level sets. In this work, we provide 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). We show similar results for related quantities, including the relative entropy, and discuss under which conditions entropy values are transcendental, algebraic, or rational.