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All $S_p$ notions of quantum expansion are equivalent

Francisco Escudero Gutiérrez, Garazi Muguruza

TL;DR

The paper proves that all $S_p$-based quantum expansion notions are equivalent for $p\in[1,\infty)$. It establishes this via two inequalities $h_{S_p}(\mathbf B) \le d^{(p-q)/2} h_{S_q}(\mathbf B)$ and $h_{S_p}(\mathbf B) \ge [h_{S_q}(\mathbf B)]^{p/q}$, showing a unified $S_p$-expansion landscape. It then discusses the metric-embedding perspective, recalling the exact $\Theta(\log n/p)$ distortion for $\ell_p$ and outlining an open program to obtain analogous lower bounds for $S_p$ through a graph-expansion proxy $\tilde{h}_{S_p}(G)$, with a natural upper bound inherited from isometric embedding of $\ell_p$ into $S_p$. The work clarifies the structural relations among quantum expansion notions and frames concrete open questions about embedding arithmetic into quantum-norm settings, potentially guiding future research on quantum-analogous embedding phenomena.

Abstract

In a recent work Li, Qiao, Wigderson, Wigderson and Zhang introduced notions of quantum expansion based on $S_p$ norms and posed as an open question if they were all equivalent. We give an affirmative answer to this question.

All $S_p$ notions of quantum expansion are equivalent

TL;DR

The paper proves that all -based quantum expansion notions are equivalent for . It establishes this via two inequalities and , showing a unified -expansion landscape. It then discusses the metric-embedding perspective, recalling the exact distortion for and outlining an open program to obtain analogous lower bounds for through a graph-expansion proxy , with a natural upper bound inherited from isometric embedding of into . The work clarifies the structural relations among quantum expansion notions and frames concrete open questions about embedding arithmetic into quantum-norm settings, potentially guiding future research on quantum-analogous embedding phenomena.

Abstract

In a recent work Li, Qiao, Wigderson, Wigderson and Zhang introduced notions of quantum expansion based on norms and posed as an open question if they were all equivalent. We give an affirmative answer to this question.
Paper Structure (6 sections, 1 theorem, 14 equations)

This paper contains 6 sections, 1 theorem, 14 equations.

Table of Contents

  1. Introduction
  2. All $S_p$ notions of expansion are equivalent
  3. How much distortion is needed to embed metric spaces into $S_p$?
  4. Embeddings into $\ell_p$ spaces
  5. What about $S_p$?
  6. Acknowledgements.

Key Result

Theorem 1

Let $\mathbf{B}\in M_n(\mathbb C)^d$ be a bistochastic tuple. Let $p\geq q\geq 1$. Then, In particular, $h_{S_p}$ and $h_{S_q}$ are equivalent for every $p,q\in [1,\infty)$.

Theorems & Definitions (1)

  • Theorem 1