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Zero-error communication under discrete-time Markovian dynamics

Satvik Singh, Mizanur Rahaman, Nilanjana Datta

TL;DR

The paper investigates how information can be stored and retrieved with zero error in open quantum systems evolving under discrete-time quantum Markov dynamics. It introduces one-shot zero-error classical and quantum capacities and scrambling times, revealing that classical zero-error memory vanishes under mixing dynamics while quantum zero-error memory vanishes under asymptotically entanglement-breaking dynamics; when dynamics are not scrambling, optimal encoding resides in the peripheral space $\chi(\Phi)$, with long-time capacities given by the peripheral block dimensions in the decomposition $\chi(\Phi)=0\oplus\bigoplus_{k}(\mathcal{L}(\mathcal{H}_{k,1})\otimes\rho_k)$ and the capacities stabilizing after a universal time $N\le d^2$. The work establishes universal bounds on scrambling times, exact asymptotic capacity formulas, and connections to invariant subspace structures and Wielandt indices, offering a framework for zero-error storage in quantum memories and pointing to extensions to non-Markovian and approximate settings.

Abstract

Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics 'scrambles' the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.

Zero-error communication under discrete-time Markovian dynamics

TL;DR

The paper investigates how information can be stored and retrieved with zero error in open quantum systems evolving under discrete-time quantum Markov dynamics. It introduces one-shot zero-error classical and quantum capacities and scrambling times, revealing that classical zero-error memory vanishes under mixing dynamics while quantum zero-error memory vanishes under asymptotically entanglement-breaking dynamics; when dynamics are not scrambling, optimal encoding resides in the peripheral space , with long-time capacities given by the peripheral block dimensions in the decomposition and the capacities stabilizing after a universal time . The work establishes universal bounds on scrambling times, exact asymptotic capacity formulas, and connections to invariant subspace structures and Wielandt indices, offering a framework for zero-error storage in quantum memories and pointing to extensions to non-Markovian and approximate settings.

Abstract

Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics 'scrambles' the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.
Paper Structure (22 sections, 45 theorems, 129 equations)

This paper contains 22 sections, 45 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof ideas
  4. Auxiliary Results
  5. Discussion and Outlook
  6. Acknowledgements
  7. Preliminaries
  8. Spectral and ergodic properties
  9. Fixed points and multiplicative domains
  10. Contraction coefficient
  11. Zero error communication
  12. Main results
  13. Characterization of all eventually c-scrambling dQMS
  14. Zero-error classical encodings for non-mixing dQMS
  15. Zero-error quantum communication through dQMS
  16. ...and 7 more sections

Key Result

Theorem 1.1

A dQMS $\{\Phi^n \}_{n\in \mathbb N}$ governing the dynamics of an open quantum system is

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1
  • Theorem A.1
  • Theorem A.2
  • Theorem A.3
  • ...and 74 more