Observing geometry of quantum states in a three-level system

TL;DR

This paper experimentally measures the geometry of a three-level quantum system by observing joint numerical ranges of observables, revealing insights into quantum phases and degeneracies in higher-dimensional systems.

Contribution

First experimental measurement of joint numerical ranges in a three-level quantum system, extending geometric analysis beyond qubits.

Findings

Successfully classified joint numerical ranges of a triple of observables.

Linked geometry of observables to ground-state degeneracies and quantum phases.

Demonstrated a versatile geometric approach for higher-dimensional quantum systems.

Abstract

In quantum mechanics, geometry has been demonstrated as a useful tool for inferring non-classical behaviors and exotic properties of quantum systems. One standard approach to illustrate the geometry of quantum systems is to project the quantum state space to the Euclidean space via measurements of observables on the system. Despite the great success of this method in studying two-level quantum systems (qubits) with the celebrated Bloch sphere representation, there is always the difficulty to reveal the geometry of multi-dimensional quantum systems. Here we report the first experiment measuring the geometry of such projections beyond the qubit. Specifically, we observe the joint numerical ranges (JNRs) of a triple of observables in a three-level photonic system, providing complete classification of the JNRs. We further show that the geometry of different classes reveal ground-state degeneracies of a Hamiltonian as a linear combination of the observables, which is related to quantum phases in the thermodynamic limit. Our results offer a versatile geometric approach for exploring the properties of higher-dimensional quantum systems.