# Quantum-inspired probability metrics define a complete, universal space for statistical learning

## TL;DR

This paper introduces quantum probability metrics (QPMs), a new class of tools for comparing probability distributions that are more complete and sensitive, especially in high-dimensional, non-compact spaces, with practical applications in generative modeling.

## Contribution

The paper develops quantum probability metrics by embedding probability measures into quantum state space, extending kernel methods, and overcoming MMD's limitations on non-compact spaces.

## Key findings

- QPMs are complete and universal for bounded, uniformly continuous functions.
- QPMs outperform MMD in high-dimensional generative modeling tasks.
- Eigenvalue-based computation allows practical implementation of QPMs.

## Abstract

Comparing probability distributions is a core challenge across the natural, social, and computational sciences. Existing methods, such as Maximum Mean Discrepancy (MMD), struggle in high-dimensional and non-compact domains. Here we introduce quantum probability metrics (QPMs), derived by embedding probability measures in the space of quantum states: positive, unit-trace operators on a Hilbert space. This construction extends kernel-based methods and overcomes the incompleteness of MMD on non-compact spaces. Viewed as an integral probability metric (IPM), QPMs have dual functions that uniformly approximate all bounded, uniformly continuous functions on $\mathbb{R}^n$, offering enhanced sensitivity to subtle distributional differences in high dimensions. For empirical distributions, QPMs are readily calculated using eigenvalue methods, with analytic gradients suited for learning and optimization. Although computationally more intensive for large sample sizes ($O(n^3)$ vs. $O(n^2)$), QPMs can significantly improve performance as a drop-in replacement for MMD, as demonstrated in a classic generative modeling task. By combining the rich mathematical framework of quantum mechanics with classical probability theory, this approach lays the foundation for powerful tools to analyze and manipulate probability measures.