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The entanglement of purification

Barbara M. Terhal, Michal Horodecki, Debbie W. Leung, David P. DiVincenzo

TL;DR

This paper introduces the entanglement of purification as a unified metric for quantum and classical correlations and defines the LOq framework, showing that the entanglement cost under vanishing communication equals the regularized entanglement of purification. It establishes lower bounds on this cost via quantum and classical mutual information, proves continuity properties, and compares E_p to the locally induced Holevo information. The authors provide analytical results for Bell-diagonal states and numerical studies for Werner states, revealing a rich, nontrivial structure of E_p. The main finding is that E_LOq is equal to E_p^∞, with several open questions about additivity and distillation versus formation left for future work, potentially linking to secret-key rates and other correlation measures.

Abstract

We introduce a measure of both quantum as well as classical correlations in a quantum state, the entanglement of purification. We show that the (regularized) entanglement of purification is equal to the entanglement cost of creating a state $ρ$ asymptotically from maximally entangled states, with negligible communication. We prove that the classical mutual information and the quantum mutual information divided by two are lower bounds for the regularized entanglement of purification. We present numerical results of the entanglement of purification for Werner states in ${\cal H}_2 \otimes {\cal H}_2$.

The entanglement of purification

TL;DR

This paper introduces the entanglement of purification as a unified metric for quantum and classical correlations and defines the LOq framework, showing that the entanglement cost under vanishing communication equals the regularized entanglement of purification. It establishes lower bounds on this cost via quantum and classical mutual information, proves continuity properties, and compares E_p to the locally induced Holevo information. The authors provide analytical results for Bell-diagonal states and numerical studies for Werner states, revealing a rich, nontrivial structure of E_p. The main finding is that E_LOq is equal to E_p^∞, with several open questions about additivity and distillation versus formation left for future work, potentially linking to secret-key rates and other correlation measures.

Abstract

We introduce a measure of both quantum as well as classical correlations in a quantum state, the entanglement of purification. We show that the (regularized) entanglement of purification is equal to the entanglement cost of creating a state asymptotically from maximally entangled states, with negligible communication. We prove that the classical mutual information and the quantum mutual information divided by two are lower bounds for the regularized entanglement of purification. We present numerical results of the entanglement of purification for Werner states in .

Paper Structure

This paper contains 10 sections, 6 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Entanglement of Purification
  3. Mutual Information Lower bounds
  4. Proof of Lower bounds
  5. Continuity of Mutual Informations
  6. Other Correlation Measures: The locally Induced Holevo Information
  7. Bell-diagonal states
  8. Werner states
  9. Conclusion
  10. Acknowledgments

Key Result

Lemma 1

Let $\rho$ act on a Hilbert space of dimension $d_{AB}=d_A d_B$. The minimum of Eq. (defep) can always be achieved by a state $\psi$ for which the dimension of $A'$ is $d_{A'}=d_{AB}$ and the dimension of $B'$ is $d_{B'}=d_{AB}^2$ (or vice versa).

Figures (1)

  • Figure 1: Numerical bounds on $E_p$ for Werner states. In the upper curve we restrict to ${\rm dim}(A') = {\rm dim}(B') = 2$; for the next curve, we permit ${\rm dim}(A') = {\rm dim}(B') = 4$. The inset shows the curious behavior of $E_p$ around the point where the eigenvalue of $| \Psi_0 \rangle$ approaches zero. The dotted curve is the $C_A$ lower bound of Sec. \ref{['locin']}. The dashed curve is the entanglement of formation lower bound which vanishes when the eigenvalue is smaller than or equal to $1/2$.

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • Theorem 1: Continuity of the Entanglement of Purification
  • Lemma 2: Monotonicity of the Entanglement of Purification
  • Theorem 2
  • Lemma 3: Monotonicity Properties of Mutual Information
  • Corollary 1
  • Example 1: All correlation is classical correlation
  • Example 2