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On small perturbations of coherent information

Zhen Wu, Zhihao Ma, James Fullwood

TL;DR

The paper introduces a perturbative framework for analyzing coherent information to identify suboptimal input states and derive three general criteria that guarantee positive gains in coherent information under small perturbations. These criteria yield sufficient conditions for superadditivity of the one-shot quantum capacity and for a positive gap between one-shot private and quantum capacities, with concrete results for depolarizing, Pauli, dephrasure, and platypus channels, including a qutrit platypus–amplitude-damping example. The approach provides analytic tools to detect nonadditivity in quantum channel capacities without solving full nonconvex optimizations, advancing understanding of when quantum channels exhibit holistic capacity behavior and how private capacity can exceed quantum capacity in the one-shot setting.

Abstract

Quantum capacity quantifies the amount of quantum information that can be transmitted by a quantum channel with an arbitrary small probability of error. Mathematically, the quantum capacity is given by an asymptotic formula involving the one-shot quantum capacity of the associated channel, which, due to purely quantum effects such as superadditivity of one-shot quantum capacity, is rarely computable. The one-shot quantum capacity is mathematically characterized in terms of optimizing an entropic quantity referred to as coherent information over all possible input states of a channel, the computation of which also tends to be intractable due to the difficulty of optimizing the coherent information. In this work, we develop perturbative methods for analyzing the behavior of coherent information of a quantum channel with respect to small perturbations of the input state. By doing so, we are able to derive three general criteria for determining whether an input state yields suboptimal coherent information. We then show how our criteria yield sufficient conditions for superadditivity of one-shot quantum capacity, and also for detecting a positive gap between one-shot private capacity and one-shot quantum capacity. The utility of our criteria is illustrated through examples, which yield new results regarding the one-shot quantum capacity of qubit depolarizing channels, Pauli channels and dephrasure channels.

On small perturbations of coherent information

TL;DR

The paper introduces a perturbative framework for analyzing coherent information to identify suboptimal input states and derive three general criteria that guarantee positive gains in coherent information under small perturbations. These criteria yield sufficient conditions for superadditivity of the one-shot quantum capacity and for a positive gap between one-shot private and quantum capacities, with concrete results for depolarizing, Pauli, dephrasure, and platypus channels, including a qutrit platypus–amplitude-damping example. The approach provides analytic tools to detect nonadditivity in quantum channel capacities without solving full nonconvex optimizations, advancing understanding of when quantum channels exhibit holistic capacity behavior and how private capacity can exceed quantum capacity in the one-shot setting.

Abstract

Quantum capacity quantifies the amount of quantum information that can be transmitted by a quantum channel with an arbitrary small probability of error. Mathematically, the quantum capacity is given by an asymptotic formula involving the one-shot quantum capacity of the associated channel, which, due to purely quantum effects such as superadditivity of one-shot quantum capacity, is rarely computable. The one-shot quantum capacity is mathematically characterized in terms of optimizing an entropic quantity referred to as coherent information over all possible input states of a channel, the computation of which also tends to be intractable due to the difficulty of optimizing the coherent information. In this work, we develop perturbative methods for analyzing the behavior of coherent information of a quantum channel with respect to small perturbations of the input state. By doing so, we are able to derive three general criteria for determining whether an input state yields suboptimal coherent information. We then show how our criteria yield sufficient conditions for superadditivity of one-shot quantum capacity, and also for detecting a positive gap between one-shot private capacity and one-shot quantum capacity. The utility of our criteria is illustrated through examples, which yield new results regarding the one-shot quantum capacity of qubit depolarizing channels, Pauli channels and dephrasure channels.

Paper Structure

This paper contains 11 sections, 7 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Perturbative analysis
  4. Detecting superadditivity of one-shot quantum capacity
  5. One-shot quantum capacity vs. one-shot private capacity
  6. discussion
  7. Acknowledgments
  8. Perturbation Theory and The Derivatives of Entropy
  9. The proof of Criterion \ref{['cri1']}
  10. The proof of Criterion \ref{['cri2']}
  11. The proof of Criterion \ref{['cri3']}

Key Result

Proposition 2

Let $\rho_1$ and $\rho_2$ be optimal states for quantum channels $\mathcal{N}_1$ and $\mathcal{N}_2$, let $\rho(\varepsilon)$ be a perturbative expansion of $\rho=\rho_1\otimes \rho_2$ as in perstate, and let $f(\varepsilon)$ be the function given by FNCXN71 with $\mathcal{N}=\mathcal{N}_1\otimes \m

Figures (1)

  • Figure 1: Super-additivity of one-shot quantum capacity for the qutrit platypus channel $\mathcal{N}_s$ and the qubit amplitude damping channel $\mathcal{A}_{1/2}$. Here $\mathcal{A}_{1/2}$ is anti-degradable with zero quantum capacity, and the perturbative expansion $\rho(\varepsilon)$ is given in \ref{['pertstateNs']} with $a^2 = 1-\varepsilon^2$.

Theorems & Definitions (19)

  • Remark 1
  • Proposition 2: Singular
  • proof
  • Remark 3
  • Example 1
  • Proposition 4
  • proof
  • Remark 5
  • Example 2
  • Proposition 6
  • ...and 9 more