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Randomness compression in communication networks

Yukari Uchibori, Alice Zheng, Anurag Anshu, Jamie Sikora

Abstract

Given a correlation generated by a (possibly quantum) communication network, we study the amount of shared randomness required to generate it. We develop a novel upper bound for approximating distributions generated by arbitrary networks and showcase instances where it significantly outperforms the best-known upper bounds for the exact case. This demonstrates that one can have substantial savings in resources if small perturbations are acceptable. We derive our bound using Hoeffding's inequality and apply it to various commonly-used communication networks such as the Bell scenario and triangle scenario.

Randomness compression in communication networks

Abstract

Given a correlation generated by a (possibly quantum) communication network, we study the amount of shared randomness required to generate it. We develop a novel upper bound for approximating distributions generated by arbitrary networks and showcase instances where it significantly outperforms the best-known upper bounds for the exact case. This demonstrates that one can have substantial savings in resources if small perturbations are acceptable. We derive our bound using Hoeffding's inequality and apply it to various commonly-used communication networks such as the Bell scenario and triangle scenario.
Paper Structure (10 sections, 6 theorems, 33 equations, 5 figures)

This paper contains 10 sections, 6 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our approach
  3. Comparison to existing bounds
  4. Examples with exponential separations
  5. Conclusions and future work
  6. Acknowledgments
  7. Proof of Theorem \ref{['thm:single']}
  8. Proof of Corollary \ref{['cor:multiple']}
  9. Lower bound on scenario (\ref{['eq:scenario-noinput']})
  10. Lower bound on scenario (\ref{['eq:scenario-input']})

Key Result

Theorem 1

Consider a communication network with a protocol generating $p(a \space\mid\space x)$ for $a \in A$ and $x \in X$ using at least one independent randomness source $R$. Given a tolerance level $\epsilon > 0$ and an integer $n$ satisfying there exists a randomness source $Q$ of cardinality at most $n$ that, when sampled by the same protocol instead of $R$ and distributed to the same parties as $R$,

Figures (5)

  • Figure 1: Examples of networks utilizing shared classical resources. The $x_i$s are inputs, the $a_i$s are outputs, and the $r_i$s are sources of shared randomness, each shared between a selected subset of the parties.
  • Figure 2: General communication network with input $x \in X$, output $a \in A$, and $m$ sources of shared randomness $r_i \in R_i$. All other resources such as shared entanglement and classical and quantum communication are within the dotted box.
  • Figure 3: Product network composition.
  • Figure 4: Comparison of the exact upper bound in \ref{['eq:exact']} and our approximate upper bound in \ref{['eq:general']} on the cardinality of each source of randomness, plotted as a function of the size of each party's inputs and outputs. Here we have fixed the error tolerance to be $\epsilon = 0.05$.
  • Figure 5: Minimum value of tolerance level $\epsilon$ such that the approximate bound in \ref{['eq:general']} is tighter than the exact bound in \ref{['eq:exact']}. Thicker lines correspond to the Bell and triangle scenarios.

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 2
  • Lemma C1
  • proof
  • Lemma C2
  • proof
  • Lemma D1
  • proof
  • Lemma D2
  • proof