Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Magic Resource Can Enhance the Quantum Capacity of Channels

Kaifeng Bu, Arthur Jaffe

TL;DR

This work analyzes how non-stabilizer (magic) resources influence the quantum capacity of a discrete beam splitter channel $\Lambda_{s,\sigma}$ in prime-dimension qudit systems. Using a DV Weyl phase-space formalism and the Lloyd–Shor–Devetak coherent-information framework, it shows that environmental states that are convex mixtures of stabilizer states lead to zero capacity, while certain magic states enable a nonzero capacity that scales linearly with the number of magic copies, and the capacity is bounded above by the magic content via $Q(\Lambda_{\sigma})\le \mathrm{MRM}(\sigma)$. A phase-space inversion symmetry can, in some cases, force zero capacity even for magic environments, indicating magic is necessary but not sufficient for enhancement. The results link magic-resource theory to channel capacity, guiding the search for optimal magic states and suggesting extensions to other capacities and non-Clifford settings.

Abstract

We investigate the role of magic resource in the quantum capacity of channels. We consider the quantum channel of the recently proposed discrete beam splitter with the fixed environmental state. We find that if the fixed environmental state is a stabilizer state, then the quantum capacity is zero. Moreover, we find that the quantum capacity is nonzero for some magic states, and the quantum capacity increases linearly with respect to the number of single-qudit magic states in the environment. We also bound the maximal quantum capacity of the discrete beam splitter in terms of the amount of magic resource in the environmental states. These results suggest that magic resource can increase the quantum capacity of channels; it sheds new insight into the role of stabilizer and magic states in quantum communication.

Magic Resource Can Enhance the Quantum Capacity of Channels

TL;DR

This work analyzes how non-stabilizer (magic) resources influence the quantum capacity of a discrete beam splitter channel in prime-dimension qudit systems. Using a DV Weyl phase-space formalism and the Lloyd–Shor–Devetak coherent-information framework, it shows that environmental states that are convex mixtures of stabilizer states lead to zero capacity, while certain magic states enable a nonzero capacity that scales linearly with the number of magic copies, and the capacity is bounded above by the magic content via . A phase-space inversion symmetry can, in some cases, force zero capacity even for magic environments, indicating magic is necessary but not sufficient for enhancement. The results link magic-resource theory to channel capacity, guiding the search for optimal magic states and suggesting extensions to other capacities and non-Clifford settings.

Abstract

We investigate the role of magic resource in the quantum capacity of channels. We consider the quantum channel of the recently proposed discrete beam splitter with the fixed environmental state. We find that if the fixed environmental state is a stabilizer state, then the quantum capacity is zero. Moreover, we find that the quantum capacity is nonzero for some magic states, and the quantum capacity increases linearly with respect to the number of single-qudit magic states in the environment. We also bound the maximal quantum capacity of the discrete beam splitter in terms of the amount of magic resource in the environmental states. These results suggest that magic resource can increase the quantum capacity of channels; it sheds new insight into the role of stabilizer and magic states in quantum communication.
Paper Structure (10 sections, 17 theorems, 114 equations, 1 figure)

This paper contains 10 sections, 17 theorems, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic Framework
  3. Main results
  4. Conclusion and future work
  5. Acknowledgements
  6. Properies of the discrete Beam splitter
  7. Discrete beam splitter with stabilizer environment state
  8. Discrete beam splitter with environment state being a magic state
  9. Discrete beam splitter with $s^2\equiv t^2\mod d$
  10. Entanglement fidelity of the discrete beam splitter

Key Result

Theorem 2

Let nontrivial $s,t\in \mathbb{Z}_d$ satisfy $s^2+t^2\equiv 1\mod d$, and the environmental state $\sigma$ be a convex combination of stabilizer states. Then

Figures (1)

  • Figure 1: The diagram to show the equivalence of $\Lambda^c_{s,\sigma}$ and $\mathcal{A}\circ\Lambda_{t, \mathcal{A}(\sigma)}$.

Theorems & Definitions (31)

  • Remark 1
  • Theorem 2: Stabilizer environments yield zero quantum capacity of discrete beam splitters
  • Theorem 3: Magic resource can enhance quantum capacity of discrete beam splitters
  • Theorem 4: Magic bound on quantum capacity for discrete beam splitters
  • Theorem 5: Symmetry can limit quantum capacity of balanced beam splitters
  • Example 6
  • Proposition 7: Proposition 35 in BGJ23b
  • Definition 8: Quantum convolution defined by discrete beam splitter
  • Definition 9: Mean state
  • Lemma 10: Gross06
  • ...and 21 more