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Manipulating heterogeneous quantum resources over a network

Ray Ganardi, Jeongrak Son, Jakub Czartowski, Seok Hyung Lie, Nelly H. Y. Ng

TL;DR

A unified framework for composite quantum resource theories that describes distributed networks of locally constrained parties is developed, and natural axioms derive fundamental bounds on resource manipulation that hold universally, independent of the particular network characteristics.

Abstract

Quantum information processing relies on a variety of resources, including entanglement, coherence, non-Gaussianity, and magic. In realistic settings, protocols run on networks of parties with heterogeneous local resource constraints, so different resources coexist and interact. Yet, resource theories have mostly treated each resource in isolation, and a general theory for manipulation in such distributed settings has been lacking. We develop a unified framework for composite quantum resource theories that describes distributed networks of locally constrained parties. We formulate natural axioms a composite theory should satisfy to respect the local structure, and from these axioms derive fundamental bounds on resource manipulation that hold universally, independent of the particular network characteristics. We apply our results to central operational tasks, including resource conversion and assisted distillation, and introduce new methods to construct new resource monotones from this setup. Our framework further reveals previously unexplored phenomena in the remote certification of quantum resources. Together, these results establish foundational laws for distributed quantum resource manipulation across diverse physical platforms.

Manipulating heterogeneous quantum resources over a network

TL;DR

A unified framework for composite quantum resource theories that describes distributed networks of locally constrained parties is developed, and natural axioms derive fundamental bounds on resource manipulation that hold universally, independent of the particular network characteristics.

Abstract

Quantum information processing relies on a variety of resources, including entanglement, coherence, non-Gaussianity, and magic. In realistic settings, protocols run on networks of parties with heterogeneous local resource constraints, so different resources coexist and interact. Yet, resource theories have mostly treated each resource in isolation, and a general theory for manipulation in such distributed settings has been lacking. We develop a unified framework for composite quantum resource theories that describes distributed networks of locally constrained parties. We formulate natural axioms a composite theory should satisfy to respect the local structure, and from these axioms derive fundamental bounds on resource manipulation that hold universally, independent of the particular network characteristics. We apply our results to central operational tasks, including resource conversion and assisted distillation, and introduce new methods to construct new resource monotones from this setup. Our framework further reveals previously unexplored phenomena in the remote certification of quantum resources. Together, these results establish foundational laws for distributed quantum resource manipulation across diverse physical platforms.
Paper Structure (26 sections, 15 theorems, 41 equations, 2 figures)

This paper contains 26 sections, 15 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Composite resource theory
  3. Results
  4. Extremal composite theories
  5. Single-shot transformations
  6. Asymptotic transformations
  7. Multi-copy vs multi-partite setting
  8. Upper bounds on asymptotic composite transformation rates
  9. A universal upper bound for assisted distillation
  10. Assisted distillation as a witness of correlations
  11. Resource certification
  12. Remote resource certification
  13. A case study for coherence and imaginarity
  14. Discussions and Conclusion
  15. Acknowledgements
  16. ...and 11 more sections

Key Result

Lemma 1

$S \subseteq S'$ does not imply $\mathop{\mathrm{RNG}}\nolimits(S) \subseteq \mathop{\mathrm{RNG}}\nolimits(S')$ nor $\mathop{\mathrm{RNG}}\nolimits(S') \subseteq \mathop{\mathrm{RNG}}\nolimits(S)$.

Figures (2)

  • Figure 1: Rules of compatibility between the local resource theories and the composite resource theory (see \ref{['definition: composite RTs']} for the formal statement): (a) tensor product of locally free states are free, (b) tensor product of locally free operations are free, (c) all reduced states of a composite free state are locally free, and (d) for any free input, all of the effective reduced operations of a composite free operation are locally free. Heuristically, rules (a) and (b) can be thought as product rules, while rules (c) and (d) as marginal rules.
  • Figure 2: Resource certification: (a) standard resource certification, (b) remote resource certification.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • ...and 25 more