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Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels

Touheed Anwar Atif, S. Sandeep Pradhan, Andreas Winter

TL;DR

A one-shot quantum covering lemma is proved in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory, and this covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion.

Abstract

We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Secondly, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author.

Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels

TL;DR

A one-shot quantum covering lemma is proved in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory, and this covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion.

Abstract

We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Secondly, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author.
Paper Structure (15 sections, 19 theorems, 59 equations, 3 figures)

This paper contains 15 sections, 19 theorems, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Useful definitions
  4. Useful lemmas
  5. Quantum soft covering
  6. Lossy quantum source coding
  7. Quantum channel resolvability
  8. Identification via quantum channels
  9. Proofs of quantum source coding results
  10. Proof of Proposition \ref{['prop:petz_reference']}
  11. Proof of Lemma \ref{['lem:coveringAndcompression']}
  12. Proof of the converse in Theorem \ref{['thm:CoveringOneShotLossy']}
  13. Proof of the converse in Theorem \ref{['thm:mainResult_lossy']}
  14. Conclusions and future work
  15. Proof of Lemma \ref{['lemma:H-max-min']} (courtesy of Frédéric Dupuis)

Key Result

Lemma 7

For any two states $\rho,\sigma \in \mathcal{D}(A)$, we have

Figures (3)

  • Figure 1: Figure demonstrating the construction of the posterior reference map $W$ from the isometry $V$ (the Stinespring's dilation of $\mathcal{N}_V$) and the source state $\rho^B$.
  • Figure 2: Illustration of Lossy Quantum Compression protocol
  • Figure 3: A One-shot lossy quantum source coding protocol and the associated CPTP maps and their Stinespring dilations.

Theorems & Definitions (41)

  • Definition 1: Coherent information
  • Definition 2: Holevo information
  • Definition 3: Quantum information variance
  • Definition 4: Min- and max-entropy tomamichel2015quantum
  • Definition 5: Smoothed entropies tomamichel2015quantum
  • Definition 6: Smoothed max-relative entropy tomamichel2013hierarchy
  • Lemma 7: Fuchs/van de Graaffuchs1999cryptographic, see also wilde_arxivBook
  • Lemma 8: Canonical purification atif2023lossy, winter
  • Lemma 9: AEP tomamichel2013hierarchy
  • Lemma 10: Min- vs. max-entropy vitanov2013chain, tomamichel2012framework
  • ...and 31 more