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Entanglement of Distillation and Conditional Mutual Information

Robert R. Tucci

TL;DR

This work extends the conditional mutual information ($CMI$) framework for quantifying entanglement by expressing the Entanglement of Distillation ($E_D$) in terms of $CMI$. Building on prior results where the Entanglement of Formation ($E_F$) is represented via $CMI$, the paper develops both classical and quantum distillation formulations that rely on a common-ancestor variable and fixed-marginal constraints, and proves data-processing–style bounds linking $E_D$ to sums of $E_F$. The contributions unify entanglement measures under a unified $CMI$ description and provide a tractable information-theoretic route to compare classical and quantum distillation processes. These results sharpen the distinction between quantum and classical correlations and offer a compact characterization that could inform future bounds and protocol design in quantum information processing.

Abstract

In previous papers, we expressed the Entanglement of Formation in terms of Conditional Mutual Information (CMI). In this brief paper, we express the Entanglement of Distillation in terms of CMI.

Entanglement of Distillation and Conditional Mutual Information

TL;DR

This work extends the conditional mutual information () framework for quantifying entanglement by expressing the Entanglement of Distillation () in terms of . Building on prior results where the Entanglement of Formation () is represented via , the paper develops both classical and quantum distillation formulations that rely on a common-ancestor variable and fixed-marginal constraints, and proves data-processing–style bounds linking to sums of . The contributions unify entanglement measures under a unified description and provide a tractable information-theoretic route to compare classical and quantum distillation processes. These results sharpen the distinction between quantum and classical correlations and offer a compact characterization that could inform future bounds and protocol design in quantum information processing.

Abstract

In previous papers, we expressed the Entanglement of Formation in terms of Conditional Mutual Information (CMI). In this brief paper, we express the Entanglement of Distillation in terms of CMI.

Paper Structure

This paper contains 5 sections, 2 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement of Formation
  3. Classical Distillation
  4. Quantum Distillation
  5. Appendix: Some Data Processing Inequalities

Key Result

Lemma A.1

(Data Processing Inequality for Relative Entropy, see Ref.dpi-rel-ent) If $P, Q \in {\rm pd}(S_{\ul{x}})$ and $T= \{T(y|x): y\in S_{\ul{y}}, x\in S_{\ul{x}}\}$ is a matrix of non-negative numbers such that $\sum_y T(y|x)=1$, then where $TP$ should be understood as the matrix product of the column vector $P$ times the matrix $T$.

Figures (3)

  • Figure 1: Classical Bayesian Net that motivates the definition of $E_F$.
  • Figure 2: Classical Bayesian Net that motivates the definition of $E_D$.
  • Figure 3: Classical Bayesian Net that obeys the Data Processing Inequality Eq.(\ref{['eq:dat-pro-cmi']}).

Theorems & Definitions (2)

  • Lemma A.1
  • Lemma A.2