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No-go theorem for heralded exact one-way key distillation

Vishal Singh, Mark M. Wilde

TL;DR

The paper defines the probabilistic, heralded exact one-way secret-key distillation task and identifies a broad class of states, the super two-extendible states, for which this task is provably impossible via min-unextendible entanglement. It proves a no-go theorem: the probabilistic one-way distillable secret key $K_{D}^{\to}$ is zero for every super two-extendible state, including erased and full-rank states. This reveals an extreme gap between probabilistic and approximate one-way distillable keys, since some such states admit nonzero approximate rates, and it extends naturally to one-way entanglement distillation. The results rely on the monotonicity and lower-bounding properties of the min-unextendible entanglement and highlight the necessity of allowing some error in practical key distillation. Overall, the work delineates fundamental limits of probabilistic key (and entanglement) distillation under one-way LOCC and motivates further study of intermediate regimes balancing error and success probability.

Abstract

The heralded exact one-way distillable secret key is equal to the largest expected rate at which perfect secret key bits can be probabilistically distilled from a bipartite state by means of local operations and one-way classical communication. Here we define the set of super two-extendible states and prove that an arbitrary state in this set cannot be used for heralded exact one-way secret-key distillation. This broad class of states includes both erased states and all full-rank states. Comparing the heralded exact one-way distillable secret key with the more commonly studied approximate one-way distillable secret key, our results demonstrate an extreme gap between them for many states of interest, with the approximate one-way distillable secret key being much larger. Our findings naturally extend to heralded exact one-way entanglement distillation, with similar conclusions.

No-go theorem for heralded exact one-way key distillation

TL;DR

The paper defines the probabilistic, heralded exact one-way secret-key distillation task and identifies a broad class of states, the super two-extendible states, for which this task is provably impossible via min-unextendible entanglement. It proves a no-go theorem: the probabilistic one-way distillable secret key is zero for every super two-extendible state, including erased and full-rank states. This reveals an extreme gap between probabilistic and approximate one-way distillable keys, since some such states admit nonzero approximate rates, and it extends naturally to one-way entanglement distillation. The results rely on the monotonicity and lower-bounding properties of the min-unextendible entanglement and highlight the necessity of allowing some error in practical key distillation. Overall, the work delineates fundamental limits of probabilistic key (and entanglement) distillation under one-way LOCC and motivates further study of intermediate regimes balancing error and success probability.

Abstract

The heralded exact one-way distillable secret key is equal to the largest expected rate at which perfect secret key bits can be probabilistically distilled from a bipartite state by means of local operations and one-way classical communication. Here we define the set of super two-extendible states and prove that an arbitrary state in this set cannot be used for heralded exact one-way secret-key distillation. This broad class of states includes both erased states and all full-rank states. Comparing the heralded exact one-way distillable secret key with the more commonly studied approximate one-way distillable secret key, our results demonstrate an extreme gap between them for many states of interest, with the approximate one-way distillable secret key being much larger. Our findings naturally extend to heralded exact one-way entanglement distillation, with similar conclusions.
Paper Structure (17 sections, 10 theorems, 91 equations)

This paper contains 17 sections, 10 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. One-way secret-key distillation
  3. Min-unextendible entanglement
  4. Limitations on probabilistic one-way distillable secret key
  5. Erased private state
  6. Approximate versus probabilistic secret key distillation
  7. Conclusion
  8. Equivalent definitions of probabilistic one-way distillable secret key
  9. The min-unextendible entanglement of bipartite states
  10. Proof of Lemma \ref{['lem:min_unext_ent_2_eras_eq']}
  11. The set of super two-extendible states is convex but not closed
  12. Proof of Proposition \ref{['prop:ads_min_unext_ent_eq_0']}
  13. Proof of Theorem \ref{['theo:prob_distill_ads_eq_0']}
  14. Proof of Proposition \ref{['lem:eras_min_unext_ent_eq_0']}
  15. Proof of Corollary \ref{['cor:dist_full_rank']}
  16. ...and 2 more sections

Key Result

Lemma 1

For all $p \in [0,1]$ and every integer $k \geq 2$, the min-unextendible entanglement of a doubly erased private state $\eta^{p,k}_{AB}$ is bounded from below by the following quantity:

Theorems & Definitions (13)

  • Definition 1
  • Remark 1
  • Lemma 1
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Corollary 1
  • Corollary 2
  • Definition 2: WWW21
  • Lemma 2
  • ...and 3 more