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Quantifying the unextendibility of entanglement

Kun Wang, Xin Wang, Mark M. Wilde

TL;DR

A state-dependent resource theory is developed to quantify the unextendibility of bipartite entangled states and it is shown that the unExtendible entanglement provides efficiently computable benchmarks for the rate of perfect secret key distillation orEntanglement distillation, as well as for the overhead of probabilistic secret key or entanglements distillation.

Abstract

Entanglement is a striking feature of quantum mechanics, and it has a key property called unextendibility. In this paper, we present a framework for quantifying and investigating the unextendibility of general bipartite quantum states. First, we define the unextendible entanglement, a family of entanglement measures based on the concept of a state-dependent set of free states. The intuition behind these measures is that the more entangled a bipartite state is, the less entangled each of its individual systems is with a third party. Second, we demonstrate that the unextendible entanglement is an entanglement monotone under two-extendible quantum operations, including local operations and one-way classical communication as a special case. Normalization and faithfulness are two other desirable properties of unextendible entanglement, which we establish here. We further show that the unextendible entanglement provides efficiently computable benchmarks for the rate of exact entanglement or secret key distillation, as well as the overhead of probabilistic entanglement or secret key distillation.

Quantifying the unextendibility of entanglement

TL;DR

A state-dependent resource theory is developed to quantify the unextendibility of bipartite entangled states and it is shown that the unExtendible entanglement provides efficiently computable benchmarks for the rate of perfect secret key distillation orEntanglement distillation, as well as for the overhead of probabilistic secret key or entanglements distillation.

Abstract

Entanglement is a striking feature of quantum mechanics, and it has a key property called unextendibility. In this paper, we present a framework for quantifying and investigating the unextendibility of general bipartite quantum states. First, we define the unextendible entanglement, a family of entanglement measures based on the concept of a state-dependent set of free states. The intuition behind these measures is that the more entangled a bipartite state is, the less entangled each of its individual systems is with a third party. Second, we demonstrate that the unextendible entanglement is an entanglement monotone under two-extendible quantum operations, including local operations and one-way classical communication as a special case. Normalization and faithfulness are two other desirable properties of unextendible entanglement, which we establish here. We further show that the unextendible entanglement provides efficiently computable benchmarks for the rate of exact entanglement or secret key distillation, as well as the overhead of probabilistic entanglement or secret key distillation.

Paper Structure

This paper contains 36 sections, 29 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Quantum divergences
  5. Quantifying the unextendibility of entanglement
  6. State-dependent set of free states
  7. Generalized unextendible entanglement
  8. $\alpha$-unextendible entanglement
  9. Relative-entropy-induced unextendible entanglement
  10. Efficiently computable entanglement measures
  11. Max-unextendible entanglement
  12. Min-unextendible entanglement
  13. Unextendible fidelity
  14. Applications for secret key and entanglement distillation
  15. Private states and unextendible entanglement
  16. ...and 21 more sections

Key Result

Lemma 1

Let $\alpha \in (0,\infty)$, and let $\omega$ and $\tau$ be quantum states. Then $D_{\alpha}(\omega\Vert\tau) \geq 0$ and $D_{\alpha}(\omega\Vert\tau) = 0$ if and only if $\omega = \tau$. The same is true for the sandwiched and geometric Rényi relative entropies.

Figures (2)

  • Figure 1: Lower bounds on the overhead of distilling one ebit or one private bit from isotropic states. For the overhead of probabilistic distillation of entanglement or secret key under 1-LOCC operations, our lower bound (solid line) outperforms the previous lower bound (dashed line) based on relative entropy of entanglement.
  • Figure 2: Bounds on the overhead of distilling one ebit or one private bit from S states. For the overhead of probabilistic distillation of entanglement or secret key under 1-LOCC operations, our lower bound (solid line) outperforms the previous lower bound (dashed line) based on relative entropy of entanglement.

Theorems & Definitions (40)

  • Remark 1: Comparison of our results with Kaur2018KDWW21
  • Lemma 1: Faithfulness
  • Definition 1: Generalized unextendible entanglement
  • Remark 2: Connection to joinability
  • Remark 3: Channel representation
  • Theorem 2: Two-extendible monotonicity
  • Proposition 3: Faithfulness
  • Definition 2: $\alpha$-unextendible entanglement
  • Corollary 4: Two-extendible monotonicity
  • Theorem 5: Selective two-extendible monotonicity
  • ...and 30 more