Measuring Mixed-State Topological Invariant in Open Photonic Quantum Walk

TL;DR

This paper presents an experimental protocol to measure the topological invariant of mixed quantum states using photonic quantum walks, extending geometric phase concepts to open quantum systems.

Contribution

It introduces a novel method to directly measure the Uhlmann topological invariant in mixed states via engineered nonunitary dynamics and machine learning techniques.

Findings

Successfully prepared topologically nontrivial mixed states from trivial states.

Direct measurement of the quantized geometric phase in mixed states.

Framework extends topological invariants to open quantum systems.

Abstract

Pure-state manifestations of geometric phase are well established and have found applications across essentially all branches of physics, yet their generalization to mixed-state regimes remains largely unexplored experimentally. The Uhlmann geometric phase offers a natural extension of pure-state paradigms and can exhibit a topological character. However, observation of this invariant is impeded by the incompatibility between Uhlmann parallel transport and Hamiltonian dynamics, as well as the difficulty of preparing topologically nontrivial mixed states. To address this challenge, we report an experimentally accessible protocol for directly measuring the mixed-state topological invariant. By engineering controlled nonunitary dynamics in a photonic quantum walk, we prepare topologically nontrivial mixed states from a trivial initial state. Furthermore, by machine-learning the full density matrix in momentum space, we directly extract the quantized geometric phase of the nontrivial mixed states. These results highlight a geometric phase framework that naturally extends to open quantum systems both in and out of thermal equilibrium.