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Information transmission under Markovian noise

Satvik Singh, Nilanjana Datta

Abstract

We consider an open quantum system undergoing Markovian dynamics, the latter being modelled by a discrete-time quantum Markov semigroup $(Φ^n)_{n \in {\mathbb{N}}}$, resulting from the action of sequential uses of a quantum channel $Φ$, with $n \in {\mathbb{N}}$ being the discrete time parameter. We find upper and lower bounds on the one-shot $ε$-error information transmission capacities of $Φ^n$ for a finite time $n\in \mathbb{N}$ and $ε\in [0,1)$ in terms of the structure of the peripheral space of the channel $Φ$. We consider transmission of $(i)$ classical information (both in the unassisted and entanglement-assisted settings); $(ii)$ quantum information and $(iii)$ private classical information.

Information transmission under Markovian noise

Abstract

We consider an open quantum system undergoing Markovian dynamics, the latter being modelled by a discrete-time quantum Markov semigroup , resulting from the action of sequential uses of a quantum channel , with being the discrete time parameter. We find upper and lower bounds on the one-shot -error information transmission capacities of for a finite time and in terms of the structure of the peripheral space of the channel . We consider transmission of classical information (both in the unassisted and entanglement-assisted settings); quantum information and private classical information.
Paper Structure (15 sections, 12 theorems, 89 equations)

This paper contains 15 sections, 12 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical communication
  4. Private classical communication
  5. Entanglement-assisted classical communication
  6. Quantum communication
  7. Spectral properties of quantum channels
  8. Main result
  9. Infinite-time capacities
  10. Rate of convergence
  11. IID noise
  12. Conclusion
  13. Acknowledgements
  14. Technical results
  15. Justification for the assumption that $\mathcal{H}_0=\{0\}$

Key Result

Theorem 3.1

Let $\Phi:\mathcal{L}({\mathcal{H}_A})\to \mathcal{L}({\mathcal{H}_A})$ be a quantum channel, $(\Phi^n)_{n\in \mathbb{N}}$ be the associated dQMS, and $\varepsilon\in [0,1)$. Then, for all $n\in\mathbb{N}$, the one-shot $\varepsilon-$error capacities satisfy: Moreover, for $n$ large enough, the following converse bounds hold: Here, $d_k=\dim \mathcal{H}_{k,1}$ for $k\in \{1,2,\ldots ,K \}$ are t

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • ...and 16 more