Photon State Comparison Using Probability Distributions and Wasserstein Distance.

Research demonstrates the application of probability distribution quantifiers, including Kullback-Leibler divergence, Bhattacharyya distance, and the Wasserstein distance, to characterise photon states. The Wasserstein distance, a robust metric for comparing probability distributions, proves particularly useful in analysing distinctions between various states of light.

The characterisation of quantum states of light relies fundamentally on the precise comparison of their underlying probability distributions, a task with implications for quantum communication, imaging, and sensing. Researchers now apply established mathematical tools – specifically, the Kullback-Leibler divergence, the Bhattacharyya distance, and the Wasserstein distance – to analyse a range of photon states, with a particular focus on the Wasserstein distance as a robust metric for quantifying differences in probability distributions.

This work, detailed in a recent publication, originates from Soumyabrata Paul, V. Balakrishnan, S. Ramanan, and S. Lakshmibala, all affiliated with the Department of Physics and the Center for Quantum Information, Communication, and Computing (CQuICC) at the Indian Institute of Technology Madras. Their article, entitled “Comparing probability distributions: application to quantum states of light”, presents a novel application of these established mathematical concepts to the nuanced field of quantum optics.

The comparison of quantum states represents a fundamental challenge in quantum optics, and established distance metrics offer a computationally efficient means of quantifying their dissimilarity. This work investigates the utility of the Kullback-Leibler divergence, the Bhattacharyya distance, and the Wasserstein distance in characterising differences between states of light, specifically by analysing photon number distributions rather than undertaking full quantum state reconstruction. Photon number distribution describes the probability of detecting a specific number of photons in a given time interval.

Quantum mechanics describes systems using probability distributions, necessitating robust methods for comparing these distributions to characterise diverse quantum states. The Kullback-Leibler divergence, a measure of how one probability distribution diverges from a second, expected probability distribution, provides one such tool. The Bhattacharyya distance, related to the Bhattacharyya coefficient, quantifies the overlap between two probability distributions, with smaller distances indicating greater similarity. The Wasserstein distance, also known as the Earth Mover’s Distance, calculates the minimum ‘work’ required to transform one probability distribution into another, offering a different perspective on their dissimilarity.

Results demonstrate the particular sensitivity of the Wasserstein distance (W1) to variations in photon number distributions, allowing it to distinguish between states that other metrics might classify as similar. This is because W1 considers the ‘cost’ of moving probability mass between distributions, capturing subtle differences in their shapes. This capability has practical relevance in areas such as quantum machine learning, where these metrics facilitate the classification of quantum states, and state tomography, a process used to reconstruct the quantum state of a system from experimental measurements. By providing a robust means of quantifying state similarity, these tools contribute to advancements in quantum information processing and characterisation.

Future research will focus on extending these methods to higher-dimensional quantum states and exploring their application to more complex quantum systems. Scientists aim to refine these mathematical tools to provide even more precise and insightful characterisation of quantum states of light, enabling further advancements in quantum technologies and unlocking new possibilities for quantum information processing and communication. A deeper understanding of these fundamental principles will be crucial for designing and building the next generation of quantum devices and harnessing the full potential of quantum technologies.