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Equivalence of Additivity Questions in Quantum Information Theory

Peter W. Shor

TL;DR

The paper proves that four foundational additivity questions in quantum information theory—the additivity of minimum output entropy, the additivity of the Holevo capacity, the additivity of entanglement of formation, and the strong superadditivity of entanglement of formation—are all equivalent: they are either all true or all false. It builds on the Matsumoto–Shimono–Winter correspondence linking constrained Holevo capacity to entanglement of formation and on a linear programming duality for χ_N(ρ) to connect channel capacity with entanglement across tensor products. Through a sequence of implications and carefully constructed auxiliary channels (N′, N′′) and dual functions, the paper shows how proving one conjecture would resolve all. The results imply that focusing efforts on the minimum output entropy additivity could be the most tractable route to settle this entire set of open questions in quantum information theory.

Abstract

We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. We show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.

Equivalence of Additivity Questions in Quantum Information Theory

TL;DR

The paper proves that four foundational additivity questions in quantum information theory—the additivity of minimum output entropy, the additivity of the Holevo capacity, the additivity of entanglement of formation, and the strong superadditivity of entanglement of formation—are all equivalent: they are either all true or all false. It builds on the Matsumoto–Shimono–Winter correspondence linking constrained Holevo capacity to entanglement of formation and on a linear programming duality for χ_N(ρ) to connect channel capacity with entanglement across tensor products. Through a sequence of implications and carefully constructed auxiliary channels (N′, N′′) and dual functions, the paper shows how proving one conjecture would resolve all. The results imply that focusing efforts on the minimum output entropy additivity could be the most tractable route to settle this entire set of open questions in quantum information theory.

Abstract

We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. We show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.

Paper Structure

This paper contains 10 sections, 3 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Background and Results
  3. The correspondence of Matsumoto, Shimono and Winter
  4. Additivity of $\chi$ implies additivity of $E_F$
  5. The linear programming formulation
  6. Additivity of $E_F$ implies strong superadditivity of $E_F$
  7. Additivity of $\min H(N)$ implies additivity of $E_F$.
  8. Implications of strong superadditivity of $E_F$.
  9. Additivity of $\chi_N$ or of $E_F$ implies additivity of $\min H(N)$.
  10. Discussion

Key Result

Theorem 1

The following are equivalent.

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 2
  • Lemma 3