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Emulation Capacity between Idempotent Channels

Idris Delsol, Omar Fawzi, Li Gao, Mizanur Rahaman

TL;DR

The paper analyzes interconversion (emulation) rates between idempotent quantum channels, establishing a single-letter zero-error capacity expressed through shape vectors $\lambda(\cdot)$ as $C(\mathcal{G} \mapsto \mathcal{F}) = \inf_{p \in [1,+\infty]} \frac{\log(\|\lambda(\mathcal{G})\|_p)}{\log(\|\lambda(\mathcal{F})\|_p)}$. It shows additivity under tensor powers and that emulation is generally not reversible, with exact equalities in certain cases (identity or dephasing channels). It provides achievability via $*$-algebra embeddings (Kuperberg) and a corresponding reduction to reduced channels, and a converse based on injective homomorphisms between fixed-point algebras. The work extends to errorful emulation, proving a strong converse bound and discussing finite-blocklength behavior using Holevo bounds and approximate $C^*$-algebras, thereby connecting operator-algebraic structure with quantum channel interconversion.

Abstract

We study the optimal rates of emulation (also called interconversion) between quantum channels. When the source and the target channels are idempotent, we give a single-letter expression for the zero-error emulation capacity in terms of structural properties of the range of the two channels. This expression shows that channel emulation is not reversible for general idempotent channels. Furthermore, we establish a strong converse rate that matches with the zero-error emulation capacity when the source or the target channel is either an identity or a completely dephasing channel.

Emulation Capacity between Idempotent Channels

TL;DR

The paper analyzes interconversion (emulation) rates between idempotent quantum channels, establishing a single-letter zero-error capacity expressed through shape vectors as . It shows additivity under tensor powers and that emulation is generally not reversible, with exact equalities in certain cases (identity or dephasing channels). It provides achievability via -algebra embeddings (Kuperberg) and a corresponding reduction to reduced channels, and a converse based on injective homomorphisms between fixed-point algebras. The work extends to errorful emulation, proving a strong converse bound and discussing finite-blocklength behavior using Holevo bounds and approximate -algebras, thereby connecting operator-algebraic structure with quantum channel interconversion.

Abstract

We study the optimal rates of emulation (also called interconversion) between quantum channels. When the source and the target channels are idempotent, we give a single-letter expression for the zero-error emulation capacity in terms of structural properties of the range of the two channels. This expression shows that channel emulation is not reversible for general idempotent channels. Furthermore, we establish a strong converse rate that matches with the zero-error emulation capacity when the source or the target channel is either an identity or a completely dephasing channel.

Paper Structure

This paper contains 11 sections, 27 theorems, 214 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Decomposition properties of idempotent channels
  4. Emulation of idempotent channels with zero error
  5. Achievability in Theorem \ref{['thm:zero_error_case']}
  6. Converse of Theorem \ref{['thm:zero_error_case']}
  7. Emulation of idempotent channels with errors
  8. An application of approximate $C^*$-algebras
  9. The Error Reduction Theorem
  10. Approximate Multiplicative Domains
  11. Proof of Theorem \ref{['thm:dimension_dependent_error']}

Key Result

Theorem 1

Let $\mathcal{F} = \mathcal{F} \circ \mathcal{F}$ and $\mathcal{G} = \mathcal{G} \circ \mathcal{G}$ be two idempotent channels with $\lambda(\mathcal{F}) \neq (1)$If $\lambda(\mathcal{F}) = (1)$, then $\mathcal{F}$ is basically a replacer channel (see Proposition prop:structure_idempotent_channel) a In particular, when $\mathcal{F}$ or $\mathcal{G}$ is either the identity channel or the completely

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2: Strong converse rate
  • Proposition 1: Structure of the range of idempotent channels, see Wolf12
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof : Proof of Proposition \ref{['prop:structure_idempotent_channel']}
  • Proposition 2
  • proof
  • ...and 46 more