Quantum Operations Exhibit Strong-to-Weak Symmetry Breaking Phases in Steady States, Demonstrating Spontaneous Symmetry Breaking

The behaviour of quantum systems often exhibits symmetries, but these symmetries can exist at different levels, and transitions between them represent a fundamental challenge in quantum physics. Niklas Ziereis, Sanjay Moudgalya, and Michael Knap, all from the Technical University of Munich and the Munich Center for Quantum Science and Technology, investigate how strong symmetries in quantum systems spontaneously break down into weaker forms, a process known as Strong-to-Weak Spontaneous Symmetry Breaking. Their research demonstrates that this symmetry breaking is not merely a fleeting phenomenon, but a robust feature present even in noisy quantum operations, such as those found in random circuits with measurements. Importantly, the team reveals that this breakdown can be driven to a point where the system regains strong symmetry with sufficient control, offering a potential pathway for designing quantum systems with tailored properties based on their inherent symmetries and providing a framework for creating diverse mixed-state quantum phases.

Symmetries exist at varying levels, either directly within individual states, known as strong symmetry, or across the entire collection of states, referred to as weak symmetry. Scientists established that maximally mixed symmetric density matrices, commonly found as steady states in random quantum circuits, undergo SW-SSB when the symmetry represents a compact Lie or finite group. This breaking of symmetry isn’t a simple loss of order, but a shift in how symmetry is expressed within the quantum state.

Z2 Symmetry and Random Quantum Circuits

This research investigates the emergence of symmetry-protected topological (SPT) phases and symmetry breaking in random quantum circuits possessing on-site symmetries. Specifically, the team focuses on circuits exhibiting Z2 symmetry, a property related to spin flips, and aims to understand how these circuits evolve and whether they exhibit special phases or undergo transitions where symmetry is broken. Researchers demonstrate that the fidelity correlator, a measure of similarity to a reference state, behaves similarly to the Rényi-2 correlator, a measure of entanglement. This connection allows them to apply knowledge from the well-understood Transverse Field Ising Model to analyze the random circuits. The team showed this SW-SSB isn’t an isolated occurrence, but a stable phase that persists even under more complex quantum operations, specifically measurements followed by weak postselection. Remarkably, applying sufficiently strong postselection can drive the system through a second-order transition, restoring strong symmetry to the steady state. Analytical and numerical results detail these transitions for both abelian Z2 and non-abelian S3 symmetries within Brownian random quantum circuits, providing concrete examples of this behavior.

Further investigation revealed a crucial constraint: continuous SW-SSB transitions are absent in the steady states of general strongly symmetric, trace-preserving quantum channels, including Lindbladian and unital dynamics. Analysis of steady-state degeneracies confirms this, demonstrating that these channels maintain a lower bound on symmetry, preventing the observed transitions. The research establishes that robust SW-SSB phases and their transitions occur specifically in the steady states of noisy quantum operations, offering a framework for realizing diverse mixed-state quantum phases based on their inherent symmetries. The team shows that systems initially possessing strong symmetry can transition to a state of weak symmetry, where only the ensemble as a whole maintains symmetry. This transition is identified through specific correlation functions sensitive to the density matrix, offering a way to characterise these mixed-state phases. The findings reveal that SW-SSB occurs robustly even in the presence of noise and continuous measurements, suggesting its relevance to realistic quantum systems.

Importantly, the researchers demonstrate that this type of symmetry breaking is not present in all symmetric quantum channels, highlighting specific conditions required for its emergence. The authors acknowledge that their analysis focuses on specific types of random circuits and channels, and future work could explore SW-SSB in more complex systems and different physical settings. They suggest that further investigation into these mixed-state phases could provide a deeper understanding of quantum systems far from equilibrium and their potential for information processing.