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Limitations on quantum key repeaters for all key correlated states

Leonard Sikorski, Karol Horodecki, Łukasz Pawela

TL;DR

It is shown that the repeated key of the broad class of key correlated states can exceed twice the one-way distillable entanglement of its attacked version by at most the max relative entropy of entanglement of its attacked version.

Abstract

Quantum key repeater is the backbone of the future Quantum Internet. It is an open problem to determine, for an arbitrary mixed bipartite state shared between the stations of a quantum key repeater, how much key can be generated between its two end-nodes. We place a novel bound on the quantum key repeater rate, which uses the relative entropy distance from, in general, entangled quantum states. It allows us to generalize bounds on key repeaters of M. Christandl and R. Ferrara [Phys. Rev. Lett. 119, 220506]. As in the latter article, we consider a scenario used for measurement-device-independent quantum cryptography. The derived bound, although not tighter, holds for a more general class of states, thereby avoiding the NP-hard separability problem. We show that the repeated key of the broad class of key correlated states can exceed twice the one-way distillable entanglement by at most the max relative entropy of entanglement of its attacked version. We also provide a non-trivial upper bound on the amount of private randomness of a generic independent bit - a state containing one bit of ideal private randomness.

Limitations on quantum key repeaters for all key correlated states

TL;DR

It is shown that the repeated key of the broad class of key correlated states can exceed twice the one-way distillable entanglement of its attacked version by at most the max relative entropy of entanglement of its attacked version.

Abstract

Quantum key repeater is the backbone of the future Quantum Internet. It is an open problem to determine, for an arbitrary mixed bipartite state shared between the stations of a quantum key repeater, how much key can be generated between its two end-nodes. We place a novel bound on the quantum key repeater rate, which uses the relative entropy distance from, in general, entangled quantum states. It allows us to generalize bounds on key repeaters of M. Christandl and R. Ferrara [Phys. Rev. Lett. 119, 220506]. As in the latter article, we consider a scenario used for measurement-device-independent quantum cryptography. The derived bound, although not tighter, holds for a more general class of states, thereby avoiding the NP-hard separability problem. We show that the repeated key of the broad class of key correlated states can exceed twice the one-way distillable entanglement by at most the max relative entropy of entanglement of its attacked version. We also provide a non-trivial upper bound on the amount of private randomness of a generic independent bit - a state containing one bit of ideal private randomness.
Paper Structure (12 sections, 14 theorems, 95 equations, 1 figure)

This paper contains 12 sections, 14 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Technical preliminaries
  4. Relaxed bound on one-way private key repeaters
  5. Mutual information bound for the secure content of a random private bit
  6. Localizable private randomness for a generic local independent state
  7. Discussion
  8. Random quantum objects
  9. Ginibre matrices
  10. Wishart matrices
  11. Circular unitary ensemble
  12. Random mixed quantum states

Key Result

Theorem 1

For any key correlated state $\rho$, and its key attacked state $\hat{\sigma}$, it holds If $\hat{\rho}$ is separable then:

Figures (1)

  • Figure 1: Minimal eigenvalue of matrix $2^\lambda \sigma - \rho$ as a function of $\lambda$ plotted for a fixed random $\rho$ of order $d^2$ and 100 randomly selected separable $\sigma$ (top) and a logarithmic plot of the absolute value of the maximum of these eigenvalues over 100 randomly chosen $\sigma$ and $\lambda=2$ as a function of dimension $d$ (bottom). As can be seen, the plots seem to justify the intuition behind the assumption $\lambda=2$ in the proof of Theorem \ref{['thm:first_bound']}.

Theorems & Definitions (29)

  • Definition 1
  • Theorem 1: cf. Theorem 2 in CF17
  • Definition 2
  • Definition 3
  • Theorem 2: Cf. YHW19
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • ...and 19 more