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Capacities of quantum Markovian noise for large times

Omar Fawzi, Mizanur Rahaman, Mostafa Taheri

Abstract

Given a quantum Markovian noise model, we study the maximum dimension of a classical or quantum system that can be stored for arbitrarily large time. We show that, unlike the fixed time setting, in the limit of infinite time, the classical and quantum capacities are characterized by efficiently computable properties of the peripheral spectrum of the quantum channel. In addition, the capacities are additive under tensor product, which implies in the language of Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the same. We also provide an improved algorithm for computing the structure of the peripheral subspace of a quantum channel, which might be of independent interest.

Capacities of quantum Markovian noise for large times

Abstract

Given a quantum Markovian noise model, we study the maximum dimension of a classical or quantum system that can be stored for arbitrarily large time. We show that, unlike the fixed time setting, in the limit of infinite time, the classical and quantum capacities are characterized by efficiently computable properties of the peripheral spectrum of the quantum channel. In addition, the capacities are additive under tensor product, which implies in the language of Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the same. We also provide an improved algorithm for computing the structure of the peripheral subspace of a quantum channel, which might be of independent interest.
Paper Structure (14 sections, 11 theorems, 85 equations, 1 figure, 1 table, 7 algorithms)

This paper contains 14 sections, 11 theorems, 85 equations, 1 figure, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Results
  4. Infinite-dimensional Hilbert spaces
  5. Examples
  6. Conclusion
  7. Infinite-time capacities
  8. Proof of Theorem \ref{['thm:classical and quantum capacity of repeated channel']}
  9. Quantum Markov semi-group's infinite-time capacities
  10. Additivity of infinite-time capacities
  11. Proof of Theorem \ref{['thm:additivity']}
  12. Proof of Proposition \ref{['prop: Shannon theory capacity']}
  13. Algorithm
  14. Infinite dimension

Key Result

Theorem 2.3

Let $\mathcal{T}: \mathcal{M}_d \rightarrow \mathcal{M}_ d$ be a quantum channel and let $K$ and $\{d_k\}_{k=1}^K$ be the integers from the peripheral subspace decomposition in eq:peripheral-decomposition. For any $\delta \in [0,1)$, the infinite time classical capacity of $\mathcal{T}$ is given by and the quantum capacity can be bounded as

Figures (1)

  • Figure 1: (a) Passive error correction: The system evolves under noise ($\mathcal{T}$) over time, starting with encoding ($\mathcal{E}$) and ending with decoding ($\mathcal{D}$). (b) Active error correction: Noise ($\mathcal{N}$) and recovery ($\mathcal{R}$) maps are applied periodically to maintain system integrity, from encoding ($\mathcal{E}$) to decoding ($\mathcal{D}$).

Theorems & Definitions (34)

  • Definition 2.1: Capacity
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Example 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 24 more