Related papers: Transcendental properties of entropy-constrained sets

Transcendental properties of entropy-constrained sets

URL: http://arxiv.org/abs/2111.10363v2
Date: Sun, 19 Dec 2021 17:32:54 GMT
Title: Transcendental properties of entropy-constrained sets
Authors: Vjosa Blakaj, Michael M. Wolf
Abstract summary: We provide a criterion for disproving that a set is semialgebraic based on an analytic continuation of the Gauss map. We show similar results for related quantities, including the relative entropy, and discuss under which conditions entropy values are transcendental, algebraic, or rational.
Score: 0.0
License: http://creativecommons.org/licenses/by-nc-sa/4.0/
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.

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