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Capacity Theorems for Quantum Multiple Access Channels: Classical-Quantum and Quantum-Quantum Capacity Regions

Jon Yard, Igor Devetak, Patrick Hayden

TL;DR

The paper derives regularized, multi-letter capacity regions for a quantum two-user multiple-access channel, addressing both classical-quantum and quantum-quantum information transmission. It develops achievability and converse proofs, yielding a general framework and establishing that in some channels (e.g., quantum erasure MAC, collective phase-flip channel) the regions reduce to single-letter expressions. A new concavity property of coherent information for degradable channels is proved, and the work demonstrates equivalences among cq/qq capacity, entanglement generation, entanglement transmission, and strong subspace transmission. The results illuminate how classical and quantum data interact in multiterminal quantum networks and discuss extensions to scenarios where users transmit both classical and quantum information.

Abstract

We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multi-letter characterizations of two different two-dimensional capacity regions. The first region is comprised of the rates at which it is possible for one sender to send classical information, while the other sends quantum information. The second region consists of the rates at which each sender can send quantum information. For each region, we give an example of a channel for which the corresponding region has a single-letter description. One of our examples relies on a new result proved here, perhaps of independent interest, stating that the coherent information over any degradable channel is concave in the input density operator. We conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information.

Capacity Theorems for Quantum Multiple Access Channels: Classical-Quantum and Quantum-Quantum Capacity Regions

TL;DR

The paper derives regularized, multi-letter capacity regions for a quantum two-user multiple-access channel, addressing both classical-quantum and quantum-quantum information transmission. It develops achievability and converse proofs, yielding a general framework and establishing that in some channels (e.g., quantum erasure MAC, collective phase-flip channel) the regions reduce to single-letter expressions. A new concavity property of coherent information for degradable channels is proved, and the work demonstrates equivalences among cq/qq capacity, entanglement generation, entanglement transmission, and strong subspace transmission. The results illuminate how classical and quantum data interact in multiterminal quantum networks and discuss extensions to scenarios where users transmit both classical and quantum information.

Abstract

We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multi-letter characterizations of two different two-dimensional capacity regions. The first region is comprised of the rates at which it is possible for one sender to send classical information, while the other sends quantum information. The second region consists of the rates at which each sender can send quantum information. For each region, we give an example of a channel for which the corresponding region has a single-letter description. One of our examples relies on a new result proved here, perhaps of independent interest, stating that the coherent information over any degradable channel is concave in the input density operator. We conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information.

Paper Structure

This paper contains 36 sections, 8 theorems, 197 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Classical-Quantum (cq) protocols
  4. I - Entanglement generation
  5. II - Entanglement transmission
  6. Quantum-Quantum protocols
  7. I - Entanglement generation
  8. II - Entanglement transmission
  9. Main results
  10. Proofs of Theorems 1 and 2
  11. Proof of Theorem \ref{['theo:cq']} (converse)
  12. Proof of Theorem \ref{['theo:cq']} (achievability)
  13. Proof of Theorem \ref{['theo:qq']} (converse)
  14. Remark
  15. Proof of Theorem \ref{['theo:qq']} (achievability)
  16. ...and 21 more sections

Key Result

Theorem 1

Given a quantum multiple access channel ${\cal N}\colon A'B'\rightarrow C$, its cq capacity region ${\cal C}{\cal Q}({\cal N})$ is given by the closure of where ${\cal C}{\cal Q}^{(1)}({\cal M})$ equals the pairs of nonnnegative rates $(r,S)$ satisfying for some pure state ensemble $\{p(x), | \phi_x \rangle^{A'}\}_{x\in{\cal X}}$ and a bipartite pure state $| \Psi \rangle^{BB'}$ giving rise to

Figures (1)

  • Figure 1: ${\cal C}{\cal Q}($erasure channel$)$

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3: Continuity of coherent information
  • Lemma 4: Gentle measurement
  • Proposition 1
  • Proposition 2: HSW Theorem
  • proof
  • proof
  • ...and 1 more