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Quantum Channel Simulation in Fidelity is no more difficult than State Splitting

Michael X. Cao, Rahul Jain, Marco Tomamichel

TL;DR

This paper shows that, in fidelity, the quantum channel simulation can be directly achieved via quantum state splitting without using a technique known as the de~Finetti reduction, and thus provides a pair of tighter one-shot bounds.

Abstract

Characterizing the minimal communication needed for the quantum channel simulation is a fundamental task in the quantum information theory. In this paper, we show that, in fidelity, the quantum channel simulation can be directly achieved via quantum state splitting without using a technique known as the de~Finetti reduction, and thus provide a pair of tighter one-shot bounds. Using the bounds, we also recover the quantum reverse Shannon theorem in a much simpler way.

Quantum Channel Simulation in Fidelity is no more difficult than State Splitting

TL;DR

This paper shows that, in fidelity, the quantum channel simulation can be directly achieved via quantum state splitting without using a technique known as the de~Finetti reduction, and thus provides a pair of tighter one-shot bounds.

Abstract

Characterizing the minimal communication needed for the quantum channel simulation is a fundamental task in the quantum information theory. In this paper, we show that, in fidelity, the quantum channel simulation can be directly achieved via quantum state splitting without using a technique known as the de~Finetti reduction, and thus provide a pair of tighter one-shot bounds. Using the bounds, we also recover the quantum reverse Shannon theorem in a much simpler way.
Paper Structure (5 sections, 5 theorems, 26 equations, 3 figures)

This paper contains 5 sections, 5 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quantum Channel Simulation and Quantum State Splitting
  3. Quantum Channel Simulation via State Splitting
  4. First-Order Analysis
  5. An Alternative Proof to Lemma \ref{['lemma:f:convex']}

Key Result

Lemma 1

The function $f$ defined in eq:channel:distance is convex in $\rho_{\mathsf{A}'}\in\mathcal{D}(\mathcal{H}_{\mathsf{A}'})$ for each fixed $\widetilde{\mathcal{N}}_{\mathsf{A}\to\mathsf{B}}\in\mathfrak{P}^{(M)}_{\mathsf{A}\to\mathsf{B}}$.

Figures (3)

  • Figure 1: The task of quantum channel simulation with fidelity at least $1-\epsilon^2$. The goal is to have $\widetilde{\rho}_{\mathsf{A}'\mathsf{B}} \approx_{\epsilon}\rho_{\mathsf{A}'\mathsf{B}} \coloneqq\mathop{\mathrm{id}}\nolimits_{\mathsf{A}'}\otimes\mathcal{N}_{\mathsf{A}\to\mathsf{B}}(\left\lvert\rho\middle\rangle\!\middle\langle\rho\right\rvert_{\mathsf{A}'\mathsf{A}})$for all input states $\rho_{\mathsf{A}'}$, where $\left\lvert\rho\middle\rangle\!\middle\langle\rho\right\rvert_{\mathsf{A}'\mathsf{A}}$ is the canonical purification of $\rho_{\mathsf{A}'}$.
  • Figure 2: The task of quantum state splitting with high fidelity (at least $1-\epsilon^2$). The goal is to have $\widetilde{\rho}_\mathsf{RSP}\approx_\epsilon \left\lvert\rho\middle\rangle\!\middle\langle\rho\right\rvert_\mathsf{RSP}$ where $\rho_{\mathsf{SP}}$ is fixed and known prior to the operations, and $\mathsf{R}$ is some reference system purifying $\mathsf{SP}$.
  • Figure 3: A quantum channel simulation protocol constructed from a state splitting protocol. Here, $U_{\mathsf{A}\to\mathsf{EB}}$ is the isometry representation of the original channel $\mathcal{N}_{\mathsf{A}\to\mathsf{B}}$. Note that we used the state splitting protocol on systems $\mathsf{E}$ and $\mathsf{B}$, and then discarded system $\mathsf{E}$.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Theorem 2: Sion's minimax theorem sion1958general
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof