Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Detecting positive quantum capacities of quantum channels

Satvik Singh, Nilanjana Datta

TL;DR

An elementary perturbative technique to solve the problem of determining whether a noisy quantum channel can be used to reliably transmit quantum information in a wide variety of circumstances, and reveals that a channel’s ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces.

Abstract

Determining whether a noisy quantum channel can be used to reliably transmit quantum information at a non-zero rate is a challenging problem in quantum information theory. This is because it requires computation of the channel's coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel's ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels' input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.

Detecting positive quantum capacities of quantum channels

TL;DR

An elementary perturbative technique to solve the problem of determining whether a noisy quantum channel can be used to reliably transmit quantum information in a wide variety of circumstances, and reveals that a channel’s ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces.

Abstract

Determining whether a noisy quantum channel can be used to reliably transmit quantum information at a non-zero rate is a challenging problem in quantum information theory. This is because it requires computation of the channel's coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel's ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels' input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.

Paper Structure

This paper contains 18 sections, 25 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Main results
  4. Applications
  5. Depolarizing and transpose depolarizing channels
  6. Generalized Pauli (or Weyl Covariant) Channels
  7. Multi-level amplitude damping channels
  8. (Conjugate) diagonal unitary covariant channels
  9. Generalized dephasing or Hadamard channels
  10. A family of channels with equal output and environment dimensions
  11. Structure theorems for more capable quantum channels
  12. Discussion
  13. Methods
  14. Minimal output and environment dimensions
  15. Analytic perturbation theory and the proof of Theorem \ref{['theorem:main']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Phi$ and $\Phi_c$ be complementary quantum channels. For a pure state $\ketbra{\psi}$, denote the orthogonal projections onto $\ker \Phi(\ketbra{\psi}{\psi})$ and $\ker \Phi_c(\ketbra{\psi}{\psi})$ by $K_\psi$ and $K^c_\psi$, respectively. Then,

Figures (2)

  • Figure 1: Watanabe2012capable The set of more capable (MC) quantum channels, i.e. channels with zero complementary quantum capacity. Degradable (DG) and transpose degradable (TDG) channels lie strictly within this set. The relationship between the sets of DG and TDG channels is currently unestablished.
  • Figure 2: Smith2012incapacity The set of quantum channels with zero quantum capacity: PPT and anti-degradable (ADG) channels are the only kinds of channels which are currently known to belong in this set. Entanglement-breaking (EB) channels form a strict subset of the intersection of the PPT and anti-degradable sets of channels. The question mark indicates that it is not currently known if there exist quantum channels which are neither anti-degradable nor PPT but still have zero quantum capacity.

Theorems & Definitions (59)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Theorem 3.2
  • ...and 49 more