Transcendental properties of entropy-constrained sets: Part II

TL;DR

This paper investigates the complex algebraic structure of entropy-constrained sets, demonstrating the impossibility of simple single-letter formulas for certain entropy measures in classical and quantum information theory.

Contribution

It reveals the transcendental nature of entropy level sets, ruling out algebraic single-shot characterizations with bounded ancilla in both classical and quantum contexts.

Findings

Entropy level sets are transcendental, not algebraic.

Single-shot formulas cannot be semi-algebraic for these entropy measures.

Results apply to both classical and quantum entropy scenarios.

Abstract

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.