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Triangle Inequality for a Quantum Wasserstein Divergence

Melchior Wirth

TL;DR

This work resolves the conjecture that the quantum Wasserstein divergence $W_{2,\mathrm{DPT}}$ satisfies the triangle inequality by proving it in full generality. It introduces a novel integral representation of the transport cost, derived via complex-analytic methods, first for bounded cost operators and then extended to unbounded costs through finite second moments and interpolation. The integral representation implies cost subadditivity under composition of normal ucp maps and, consequently, establishes the triangle inequality for $W_{2,\mathrm{DPT}}$, linking the divergence to a robust transport metric. The results pave the way for further connections with completely Dirichlet forms and derivations, and set the stage for operator-algebraic extensions to broader quantum OT frameworks and unbounded-cost scenarios.

Abstract

We resolve a conjecture of De Palma and Trevisan by proving the triangle inequality for a quantum 2-Wasserstein distance. The proof relies on complex analysis methods to establish a new integral representation of the cost in the optimal transport problem.

Triangle Inequality for a Quantum Wasserstein Divergence

TL;DR

This work resolves the conjecture that the quantum Wasserstein divergence satisfies the triangle inequality by proving it in full generality. It introduces a novel integral representation of the transport cost, derived via complex-analytic methods, first for bounded cost operators and then extended to unbounded costs through finite second moments and interpolation. The integral representation implies cost subadditivity under composition of normal ucp maps and, consequently, establishes the triangle inequality for , linking the divergence to a robust transport metric. The results pave the way for further connections with completely Dirichlet forms and derivations, and set the stage for operator-algebraic extensions to broader quantum OT frameworks and unbounded-cost scenarios.

Abstract

We resolve a conjecture of De Palma and Trevisan by proving the triangle inequality for a quantum 2-Wasserstein distance. The proof relies on complex analysis methods to establish a new integral representation of the cost in the optimal transport problem.

Paper Structure

This paper contains 3 sections, 14 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Triangle inequality for bounded cost operators
  3. Extension to unbounded costs

Key Result

Proposition 2.1

Let $H$, $K$ be Hilbert spaces. A linear map $\Phi\colon \mathbb B(H)\to \mathbb B(K)$ is a normal ucp map if and only if there exists a family $(v_j)_{j\in J}$ in $\mathbb B(K;H)$ such that $\sum_{j\in J}v_j^\ast v_j=1$ in the strong operator topology and in the strong operator topology for all $x\in\mathbb B(H)$. Moreover, if $\rho\in S^1(H)$, $\sigma\in S^1(K)$, then $\Phi_\ast(\sigma)=\rho$ i

Theorems & Definitions (33)

  • Conjecture : De Palma, Trevisan DPT24
  • Proposition 2.1: Kraus
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 23 more