Researchers Information, University of Science and Technology of China have directly measured a topological invariant within an open photonic quantum walk, addressing a long-standing challenge in extending geometric phase principles to mixed quantum states. While pure-state geometric phases are well established across physics, their generalization to mixed states, systems where quantum properties are not fully defined has remained largely unexplored experimentally.
The team prepared topologically nontrivial mixed states from a trivial initial state by engineering controlled dynamics in a photonic quantum walk, then extracted the quantized geometric phase using machine learning. These results demonstrate a geometric phase framework applicable to open quantum systems, even those not in thermal equilibrium.
Uhlmann Phase Extension to Open Quantum Systems
This measurement addresses a key impediment to observing topological invariants, characteristics that help understand the behavior of quantum systems, and opens new avenues for exploring topological matter beyond idealized conditions. The experimental protocol relies on full tomographic access to the density matrix in momentum space, achieved through machine-learning techniques. This allowed for the precise construction of the Uhlmann connection, a mathematical tool essential for calculating the Uhlmann geometric phase, and ultimately, the direct extraction of the quantized geometric phase of the nontrivial mixed states.
The measured Uhlmann geometric phases align with theoretical predictions, demonstrating a jump from 0 to π near a critical time where the effective Hamiltonian changes its topology. This agreement validates the method’s ability to accurately capture the topological properties of the open quantum system, even with finite dephasing strength causing transient oscillations in the instantaneous eigenbasis.
The team’s method uses the principles of open quantum systems, where interactions with the environment are not neglected, but rather incorporated into the description of the system’s behavior. This is particularly relevant as the exploration of topological matter extends beyond zero-temperature regimes, where thermal fluctuations and dissipative effects become increasingly important.
“We observe that the measured mixed-state topological invariant of the dephased final state agrees with the nontrivial topology of the target Hamiltonian,” the researchers report, resolving the topological mismatch often encountered in unitary dynamics. The implications of this work extend beyond the specific system studied.
The underlying mechanism, the restoration of dynamically broken symmetries through noise, is broadly applicable to other symmetry classes of topological insulators and superconductors driven out of equilibrium. The researchers suggest that their method for measuring mixed-state topological invariants, and for preparing topologically nontrivial mixed states via nonunitary dynamics, can be extended to open 2D topological systems. In these systems, the relevant invariant is the topological Uhlmann number, defined as the winding number of the Uhlmann geometric phase.
The work was supported by grants from the National Key Research Program of China (No. 2021ZD0301200), the National Natural Science Foundation of China (Nos. 12474494, 12204468), the Fundamental Research Funds for the Central Universities (No. 2024M753083, BX20240353). Shaojun Dong of the Hefei Comprehensive National Science Center collaborated in developing the neural-network tomographic codes used in the experiment, highlighting the interdisciplinary nature of the research. The successful measurement of the Uhlmann geometric phase in an open quantum walk represents a step towards a more complete understanding of topological phenomena in complex quantum systems.
Nonunitary Dynamics Prepare Nontrivial Mixed States
Engineering controlled noise into a photonic quantum walk allows preparation of topologically nontrivial mixed states, a feat previously hindered by fundamental incompatibilities between established measurement techniques and the dynamics of open quantum systems. The team’s protocol directly measures a mixed-state topological invariant, circumventing limitations inherent in traditional Hamiltonian dynamics and offering a new approach to characterizing these complex states. The researchers achieved this preparation by intentionally introducing a noisy term into the quantum walk’s evolution, effectively suppressing coherence between energy levels while preserving overall band occupation.
This nonunitary dynamic, applied during a near-adiabatic quench, a slow transition between initial and final Hamiltonians, restores dynamically broken symmetries and yields a topologically nontrivial state from a trivial starting point. The resulting quantized Uhlmann geometric phase of the dephased final state corresponds to the Berry phase of the final Hamiltonian, effectively creating a pseudoground state of the nontrivial Hamiltonian.
The team varied a rotation parameter, θ₁, across 21 settings within an interval of 0.2, averaging over many realizations to produce the desired nonunitary dynamics and mixed state. Reconstructing the full mixed state, however, presented a significant challenge. While 2N+1 photon-counting collection is sufficient for pure-state tomography, it proves inadequate for directly determining the density matrix of a mixed state. The ability to prepare and characterize these states opens new avenues for exploring fundamental quantum phenomena and potentially developing novel quantum technologies.
Photonic Quantum Walk Simulates Su-Schrieffer-Heeger Model
This approach bypasses the need for auxiliary systems previously required to approximate the Uhlmann process, offering a more direct route to observing mixed-state geometric phases. The team’s protocol relies on a split-step quantum walk, a discrete simulation of particle movement on a lattice, to mimic the behavior of electrons in the Su-Schrieffer-Heeger model. This model, a bipartite lattice, exhibits two distinct topological phases, and the researchers successfully induced a transition to a nontrivial phase through carefully controlled dissipation.
This contrasts with traditional methods that often struggle to reconcile the geometric phase with the system’s inherent dynamics, as the exact Uhlmann parallel-transport process is generally incompatible with Hamiltonian evolution. Full reconstruction of the density matrix in momentum space, a detailed map of the quantum state, was critical to extracting the quantized geometric phase. This was achieved through machine learning techniques, allowing for precise characterization of the mixed state’s topological properties.
This agreement validates the method and confirms the successful preparation of a topologically protected state. The ability to prepare these states from a trivial initial condition represents an advancement. Previously, creating topologically nontrivial mixed states demanded complex initial preparations, limiting experimental accessibility. The introduction of dephasing noise, a controlled form of dissipation, into the slow-quench dynamics proved instrumental in achieving this transition. This technique effectively “restores” broken symmetries, allowing the system to evolve into a topologically ordered phase.
The team’s work extends the understanding of geometric phases to open quantum systems, those interacting with their environment, and demonstrates that these phases can exist even when the system is not in thermal equilibrium. The experimental platform employed by Qin-Qin Wang, Xiao-Ye Xu, Yong-Jian Han, Chuan-Feng Li, Guang-Can Guo and colleagues grants access to the full density matrix in momentum space, a level of control essential for precise measurement.
Quantized Geometric Phase Reveals Topological Invariant
This measurement confirms a topological invariant in a mixed quantum state, extending geometric phase principles beyond the traditionally studied pure states. Researchers achieved this by engineering controlled, non-unitary dynamics, effectively simulating dissipation within the quantum walk. This approach bypasses a long-standing incompatibility between Uhlmann parallel transport and standard Hamiltonian dynamics. The experimental setup uses a split-step version of a quantum walk to model the Su-Schrieffer-Heeger model, a bipartite lattice exhibiting two distinct topological phases.
A quantum walker, defined by its internal spin state, hops across a one-dimensional discrete lattice, its movement governed by the system’s Hamiltonian. The measured Uhlmann geometric phases, represented as green opaque bars in their data, closely match theoretical predictions shown as transparent bars.
Immediately following tc, the measured phase displays transient oscillations, attributed to residual coherence within the instantaneous eigenbasis, a consequence of the finite dephasing strength applied at each time step. The ability to accurately measure this phase in an open quantum system represents an advancement in the field.
The team’s work extends beyond simply demonstrating the measurement. The direct measurement of the mixed-state topological invariant, achieved through the combination of engineered non-unitary dynamics and machine learning, offers a powerful tool for characterizing and manipulating complex quantum systems.
See today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals.
