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An additive refinement of quantum channel capacities

D. -S. Wang

TL;DR

This paper addresses the longstanding challenge of additivity in quantum channel capacities by proposing a refined set of quantum communication settings that closely mirror standard practice while enabling single-letter, additive capacity formulas. Central to the approach is the channel-state duality embodied by Choi states and the coherent information I(Φ) evaluated at the maximally mixed input π_d, which leads to the results Q(Φ) = Q_ED(Φ) and the additive, orthogonal-ensemble classical capacity C(Φ) = χ_ort(Φ). The authors further derive an additive private capacity P(Φ) and introduce an entanglement-assisted refinement with EA orthogonal ensembles, yielding C_eao(Φ) = log d + I(Φ) and Q_eao(Φ) = (log d + I(Φ))/2, thereby connecting capacity additivity to resource-theoretic considerations. These findings highlight von Neumann entropy as a core metric and suggest that additive capacities can simplify quantum Shannon-theorem proofs and capacity computations, with potential extensions to quantum resource theory and beyond.

Abstract

Capacities of quantum channels are fundamental quantities in the theory of quantum information. A desirable property is the additivity for a capacity. However, this cannot be achieved for a few quantities that have been established as capacity measures. Asymptotic regularization is generically necessary making the study of capacities notoriously hard. In this work, by a proper refinement of the physical settings of quantum communication, we prove additive quantities for quantum channel capacities that can be employed for quantum Shannon theorems. This refinement, only a tiny step away from the standard settings, is consistent with the principle of quantum theory, and it further demonstrates von Neumann entropy as the cornerstone of quantum information.

An additive refinement of quantum channel capacities

TL;DR

This paper addresses the longstanding challenge of additivity in quantum channel capacities by proposing a refined set of quantum communication settings that closely mirror standard practice while enabling single-letter, additive capacity formulas. Central to the approach is the channel-state duality embodied by Choi states and the coherent information I(Φ) evaluated at the maximally mixed input π_d, which leads to the results Q(Φ) = Q_ED(Φ) and the additive, orthogonal-ensemble classical capacity C(Φ) = χ_ort(Φ). The authors further derive an additive private capacity P(Φ) and introduce an entanglement-assisted refinement with EA orthogonal ensembles, yielding C_eao(Φ) = log d + I(Φ) and Q_eao(Φ) = (log d + I(Φ))/2, thereby connecting capacity additivity to resource-theoretic considerations. These findings highlight von Neumann entropy as a core metric and suggest that additive capacities can simplify quantum Shannon-theorem proofs and capacity computations, with potential extensions to quantum resource theory and beyond.

Abstract

Capacities of quantum channels are fundamental quantities in the theory of quantum information. A desirable property is the additivity for a capacity. However, this cannot be achieved for a few quantities that have been established as capacity measures. Asymptotic regularization is generically necessary making the study of capacities notoriously hard. In this work, by a proper refinement of the physical settings of quantum communication, we prove additive quantities for quantum channel capacities that can be employed for quantum Shannon theorems. This refinement, only a tiny step away from the standard settings, is consistent with the principle of quantum theory, and it further demonstrates von Neumann entropy as the cornerstone of quantum information.
Paper Structure (13 sections, 6 theorems, 37 equations, 3 figures, 1 table)

This paper contains 13 sections, 6 theorems, 37 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum channels
  3. Entropy
  4. Quantum capacity
  5. Classical capacity
  6. Private capacity
  7. Entanglement-assisted setting
  8. Conclusion
  9. Acknowledgement
  10. Examples
  11. Unital channels
  12. Entanglement-breaking channels
  13. Amplitude-damping channels

Key Result

Theorem 1

Let $\Phi\in C(\mathcal{X}, \mathcal{Y})$ be a channel, then $Q(\Phi)=Q_{ED}(\Phi)$.

Figures (3)

  • Figure 1: The capacities of quantum channels that are studied in this work which are all additive.
  • Figure 2: The capacity of the qutrit amplitude-damping channel.
  • Figure 3: The capacity of the qubit extreme channel.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • Definition 3
  • Theorem 4
  • proof
  • ...and 4 more