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Linear independence of quantum translates

Mansi Mishra

TL;DR

The paper addresses whether quantum translates of a nonzero operator in the Schatten class $S^p(\mathcal{H})$ remain linearly independent. It develops a Weyl-analytic framework, translating translation relations into exponential-polynomial identities via the Weyl and Fourier-Wigner transforms, and then applies Rosenblatt-type growth arguments to obtain a sharp threshold. The main result shows independence for $1\le p\le \frac{4n}{2n-1}$ and constructs a nonzero operator in $S^p(\mathcal{H})$ with dependent translates for $p>\frac{4n}{2n-1}$, with an explicit counterexample built from a hypersurface measure to demonstrate sharpness. This extends classical results on translates in $L^p$ spaces to a noncommutative quantum setting, bridging harmonic analysis, operator theory, and representation theory with potential implications for quantum harmonic analysis.

Abstract

If $A$ is in the $p$-Schatten class on $\mathbb{R}^n$, $1\leq p \leq \frac{4n}{2n-1}$, then the quantum translates of $A$ are linearly independent. Moreover, there exists a non-zero operator in the $p$-Schatten class on $\mathbb{R}^n$, $p>\frac{4n}{2n-1}$ whose quantum translates are linearly dependent.

Linear independence of quantum translates

TL;DR

The paper addresses whether quantum translates of a nonzero operator in the Schatten class remain linearly independent. It develops a Weyl-analytic framework, translating translation relations into exponential-polynomial identities via the Weyl and Fourier-Wigner transforms, and then applies Rosenblatt-type growth arguments to obtain a sharp threshold. The main result shows independence for and constructs a nonzero operator in with dependent translates for , with an explicit counterexample built from a hypersurface measure to demonstrate sharpness. This extends classical results on translates in spaces to a noncommutative quantum setting, bridging harmonic analysis, operator theory, and representation theory with potential implications for quantum harmonic analysis.

Abstract

If is in the -Schatten class on , , then the quantum translates of are linearly independent. Moreover, there exists a non-zero operator in the -Schatten class on , whose quantum translates are linearly dependent.

Paper Structure

This paper contains 4 sections, 6 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Weyl Transform
  3. Proof of the main result
  4. Example

Key Result

Theorem 1.1

Let $A \in S^p({\mathcal{H}})$, $1\le p \le 4n/(2n-1)$, be a non-zero operator and let $(x_1,y_1), \dots, (x_k,y_k)$ be distinct elements of ${\mathbb{R}}^{2n}$. Then $\{(x_1,y_1) \cdot A, \dots, (x_k,y_k) \cdot A\}$ is a linearly independent set.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3