Comparing probability distributions: application to quantum states of light

TL;DR

This paper explores the use of various probability distribution metrics, especially the Wasserstein distance, to compare quantum states of light in quantum optics, providing new insights into their differences.

Contribution

It introduces a novel application of probability distribution distances, notably the Wasserstein distance, to analyze and compare quantum states of light.

Findings

Wasserstein distance $W_{1}$ is effective for comparing quantum states.

The study demonstrates the applicability of distribution metrics in quantum optics.

New insights into quantum state differences using probability distances.

Abstract

Probability distributions play a central role in quantum mechanics, and even more so in quantum optics with its rich diversity of theoretically conceivable and experimentally accessible quantum states of light. Quantifiers that compare two different states or density matrices in terms of `distances' between the respective probability distributions include the Kullback-Leibler divergence $D_{\rm KL}$, the Bhattacharyya distance $D_{\rm B}$, and the $p$-Wasserstein distance $W_{p}$. We present a novel application of these notions to a variety of photon states, focusing particularly on the $p=1$ Wasserstein distance $W_{1}$ as it is a proper distance measure in the space of probability distributions.