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The asymptotic entanglement cost of preparing a quantum state

Patrick M. Hayden, Michal Horodecki, Barbara M. Terhal

TL;DR

The paper proves that the asymptotic entanglement cost of preparing a bipartite state $\rho$ equals the regularized entanglement of formation $E_f^\infty(\rho)=\lim_{n\to\infty} \frac{E_f(\rho^{\otimes n})}{n}$. It develops a rigorous LOCC formation framework, establishing $E_f^\infty(\rho) \ge E_c(\rho)$ via a typical-subspace construction and $E_f^\infty(\rho) \le E_c(\rho)$ using monotonicity and continuity of $E_f$ under LOCC. The authors further show that alternative definitions like $E_{alt}$ yield the same asymptotic cost and that the results hold under different metrics such as the Bures and trace distances. A key open challenge remains the practical computation of $E_f^\infty(\rho)$, which hinges on the additivity of the entanglement of formation. Overall, the work solidifies the link between formation cost and regularized entanglement of formation, informing both theory and potential applications in quantum information processing.

Abstract

We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a bipartite state ρis equal to the regularized entanglement of formation of ρ.

The asymptotic entanglement cost of preparing a quantum state

TL;DR

The paper proves that the asymptotic entanglement cost of preparing a bipartite state equals the regularized entanglement of formation . It develops a rigorous LOCC formation framework, establishing via a typical-subspace construction and using monotonicity and continuity of under LOCC. The authors further show that alternative definitions like yield the same asymptotic cost and that the results hold under different metrics such as the Bures and trace distances. A key open challenge remains the practical computation of , which hinges on the additivity of the entanglement of formation. Overall, the work solidifies the link between formation cost and regularized entanglement of formation, informing both theory and potential applications in quantum information processing.

Abstract

We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a bipartite state ρis equal to the regularized entanglement of formation of ρ.

Paper Structure

This paper contains 5 sections, 4 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Entanglement cost and entanglement of formation
  3. Alternative definitions of the asymptotic cost
  4. Conclusions
  5. Acknowledgments

Key Result

Theorem 1

The asymptotic entanglement cost of preparing a state $\rho$ is given by where $E_{c}(\rho)$ is defined in Eq. (defent) and $E_f(\rho)$ is the entanglement of formation of $\rho$, defined as and $E(| \psi \rangle\langle \psi |)$ is the von Neumann entropy of the reduced density matrix of $| \psi \rangle$.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3