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Almost-iid information theory

Giulia Mazzola, David Sutter, Renato Renner

Abstract

Information-theoretic techniques are based on the assumption that resources are well characterized by independent and identically distributed (iid) states. This assumption cannot be justified operationally, since, for example, correlations between subsequent systems emitted by a source cannot be detected by any practical tomographic protocol. Operationally motivated symmetry assumptions still imply, via de Finetti theorems, that the resources are described by almost-iid states. This raises the question: Are almost-iid resources as effective as perfect iid resources for information-processing tasks? Here we address this question and prove that the conditional entropy of almost-iid states asymptotically coincides with that of iid states. As an application, this implies that squashed entanglement is robust for almost-iid states, asymptotically matching its value on iid states.

Almost-iid information theory

Abstract

Information-theoretic techniques are based on the assumption that resources are well characterized by independent and identically distributed (iid) states. This assumption cannot be justified operationally, since, for example, correlations between subsequent systems emitted by a source cannot be detected by any practical tomographic protocol. Operationally motivated symmetry assumptions still imply, via de Finetti theorems, that the resources are described by almost-iid states. This raises the question: Are almost-iid resources as effective as perfect iid resources for information-processing tasks? Here we address this question and prove that the conditional entropy of almost-iid states asymptotically coincides with that of iid states. As an application, this implies that squashed entanglement is robust for almost-iid states, asymptotically matching its value on iid states.
Paper Structure (19 sections, 17 theorems, 150 equations)

This paper contains 19 sections, 17 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Almost-iid states
  3. Quantum vs. classical exponential de Finetti theorem
  4. Conditional entropy of almost-iid states
  5. Entropic quantities
  6. Asymptotic equipartition property for almost-iid states
  7. Proof of \ref{['prop_strong_AEP']}
  8. Proof of \ref{['thm_entropy_of_almost_iid']}
  9. Robustness of information measures for almost-iid states
  10. Robustness of squashed entanglement
  11. Robustness of entanglement distillation and entanglement cost
  12. Robustness of relative entropy of entanglement
  13. Acknowledgements
  14. Justification of \ref{['rmk_fernando_def']}
  15. Properties of almost-iid states
  16. ...and 4 more sections

Key Result

Lemma 2.5

Let $| \Theta \rangle_{AH} \in \mathcal{H}_{AH}$ for some composite Hilbert space $\mathcal{H}_{AH}$, and let $\mathcal{V}_A \subseteq \mathcal{H}_A$ be any subset of vectors such that $\,\mathrm{span}\,\mathcal{V}_A$ admits an orthonormal basis $\{ | \Psi_j \rangle_A\}_{j \in \mathcal{T}}$ with vec In addition, if $| \Theta \rangle_{AH}$ is normalized and satisfies $(ii)$, we have $1 = \langle \T

Theorems & Definitions (37)

  • Definition 2.1: Almost-iid states
  • Example 2.2
  • Remark 2.3
  • Remark 2.4: Properties of almost-iid states
  • Lemma 2.5: Purification in a preferred basis
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7: Statistics of almost-iid states
  • Theorem 3.1: Exponential de Finetti renner07
  • ...and 27 more