Comparing probability distributions: application to quantum states of light

TL;DR

The paper investigates how probability-distribution distances can quantify differences between quantum states of light, focusing on the $p=1$ Wasserstein distance $W_{1}$ as a genuine metric. It systematically derives and contrasts $D_{ m KL}$, $D_{ m B}$, and $W_{p}$ for both continuous and discrete (photon-number) distributions, including explicit results for particle-in-a-box spectra, coherent and squeezed states, thermal and Glauber–Lachs states, and blackbody radiation. A key finding is that $W_{1}$ often captures distinctions more sensitively than $D_{ m KL}$ or $D_{ m B}$ (e.g., between vacuum and excited photon-number states, or between vacuum and high-number states in the $x$-representation), highlighting its practical utility in quantum-state tomography and data-driven state discrimination. The work points to tomographic approaches and machine-learning pipelines that leverage Wasserstein and related distances to analyze evolving quantum states without full state reconstruction.

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.

Figures (10)

• Figure 1: Illustrating the earth-mover's distance in one variable.
• Figure 2: $W_{1}$ between $G_{n}$ and the classical CDF $F_{\rm cl}(x) = x$ for the particle in a box (red squares). The $2$-Wasserstein distance $W_{2}$ is also plotted (black circles).
• Figure 3: PDFs for $n=1$ (ground state; black) and $n = 10$ (blue) for the particle in a box.
• Figure 4: CDFs for $n=1$ (ground state; black) and $n = 10$ (blue) for the particle in a box.
• Figure 5: Top to bottom: $W_{1}(G_{m}, G_{n})$ for the particle in a box versus $n$ with (a)$m=1$ and (b)$m=2$. (a) Shaded circles, $n = 3,5,7\dotsc$; hollow circles, $n = 2,4,6,\dotsc$. The red dashed line denotes the value $1/\pi^{2}$. (b) Hollow circles, $n = 4, 8, 12, \dotsc$; shaded triangles, $n = 3, 5, 7, \dotsc$; shaded squares, $n = 6, 10, 14, \dotsc$. The red dashed line denotes the value $1/({2\pi^{2}})$.