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Spectrum Broadcast Structures from von Neumann type interaction Hamiltonians

Alberto Acevedo, Janek Wehr, Jarek Korbicz

Abstract

In this paper, we contribute to the mathematical foundations of the recently established theory of Spectrum Broadcast Structures (SBS). These are multipartite quantum states, encoding an operational notion of objectivity and exhibiting a more advanced form of decoherence. We study SBS in the case of a central system interacting with N environments via the von Neumann-type measurement interactions, ubiquitous in the theory of open quantum systems. We state and prove a novel sufficient condition for SBS to arise dynamically for finite-dimensional systems. The condition is based on the Gram-Schmidt orthogonalization rather than on the Knill-Barnum error estimation used before.

Spectrum Broadcast Structures from von Neumann type interaction Hamiltonians

Abstract

In this paper, we contribute to the mathematical foundations of the recently established theory of Spectrum Broadcast Structures (SBS). These are multipartite quantum states, encoding an operational notion of objectivity and exhibiting a more advanced form of decoherence. We study SBS in the case of a central system interacting with N environments via the von Neumann-type measurement interactions, ubiquitous in the theory of open quantum systems. We state and prove a novel sufficient condition for SBS to arise dynamically for finite-dimensional systems. The condition is based on the Gram-Schmidt orthogonalization rather than on the Knill-Barnum error estimation used before.
Paper Structure (13 sections, 11 theorems, 111 equations)

This paper contains 13 sections, 11 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem and existing results
  3. Estimates for the Related Quantum State Discrimination Optimization Problem
  4. Dynamical Monitoring for discrete variables
  5. Quantum-Measurement Limit
  6. Partial Tracing
  7. Monitoring the Process Information Broadcasting by the System
  8. A New Bound for the Trace Distance of a Multipartite State and an Approximating SBS State
  9. Mixed Environmental States
  10. How general may the $\mathbf{\hat{B}}_{k}$ be?
  11. Conclusions
  12. Acknowledgements
  13. Quantum State Discrimination

Key Result

Lemma 1

Telescopic inequality JKthree: Let $\mathbf{\hat{A}}^{k}$ and $\mathbf{\hat{B}}^{k}$ be trace class operators for all $k$. Then,

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Definition 5
  • Lemma 2
  • ...and 14 more