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Classical-Quantum Channel Resolvability Using Matrix Multiplicative Weight Update Algorithm

Koki Takahashi, Shun Watanabe

TL;DR

The paper addresses classical-quantum channel resolvability and proposes a deterministic coding approach based on the matrix MWU (MMWU) algorithm. By interpreting resolvability as a two-player game and employing quantum hypergraph soft covering, it derives single-shot, fixed-type, and i.i.d. resolvability results, showing that the achievable rate equals the Holevo capacity $C(W)$. It provides explicit codebook size scalings, establishing a deterministic alternative to random coding for C-Q resolvability with provable error guarantees. The work strengthens the connection between resolvability and quantum soft-covering techniques and has implications for secure and efficient quantum communications.

Abstract

We study classical-quantum (C-Q) channel resolvability. C-Q channel resolvability has been proved by only random coding in the literature. In our previous study, we proved channel resolvability by deterministic coding, using multiplicative weight update algorithm. We extend this approach to C-Q channels and prove C-Q channel resolvability by deterministic coding, using the matrix multiplicative weight update algorithm. This is the first approach to C-Q channel resolvability using deterministic coding.

Classical-Quantum Channel Resolvability Using Matrix Multiplicative Weight Update Algorithm

TL;DR

The paper addresses classical-quantum channel resolvability and proposes a deterministic coding approach based on the matrix MWU (MMWU) algorithm. By interpreting resolvability as a two-player game and employing quantum hypergraph soft covering, it derives single-shot, fixed-type, and i.i.d. resolvability results, showing that the achievable rate equals the Holevo capacity . It provides explicit codebook size scalings, establishing a deterministic alternative to random coding for C-Q resolvability with provable error guarantees. The work strengthens the connection between resolvability and quantum soft-covering techniques and has implications for secure and efficient quantum communications.

Abstract

We study classical-quantum (C-Q) channel resolvability. C-Q channel resolvability has been proved by only random coding in the literature. In our previous study, we proved channel resolvability by deterministic coding, using multiplicative weight update algorithm. We extend this approach to C-Q channels and prove C-Q channel resolvability by deterministic coding, using the matrix multiplicative weight update algorithm. This is the first approach to C-Q channel resolvability using deterministic coding.
Paper Structure (11 sections, 12 theorems, 76 equations, 1 algorithm)

This paper contains 11 sections, 12 theorems, 76 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation of C-Q channel resolvability
  3. Matrix multiplicative weight algorithm
  4. Quantum hypergraph soft covering
  5. Before applying assumptions for edges
  6. Under assumptions for edges
  7. Classical-Quantum channel resolvability
  8. Proof of Lemma \ref{['lem:MMWU_Algorithm']}
  9. Proof of nontrivial inequality \ref{['eq:appe:nontrivial_1']}
  10. Proof of nontrivial inequality \ref{['eq:appe:nontrivial_2']}
  11. Proof of Lemma \ref{['lem:Tr_Q_bound']}

Key Result

Lemma 1

(MMWU algorithm performance) Fix an arbitrary $\varepsilon\in (0, \frac{1}{2})$. The mixed states $F(1),F(2),\ldots,F(L)$ which ${\cal H}$-player obtains by Algorithm algo:MMWU satisfies that for every state $\ket{\psi}\in \mathcal{H}$,

Theorems & Definitions (19)

  • Lemma 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 9 more