Dynamical purification phase transitions induced by quantum measurements (1905.05195v5)

Abstract: Continuously monitoring the environment of a quantum many-body system reduces the entropy of (purifies) the reduced density matrix of the system, conditional on the outcomes of the measurements. We show that, for mixed initial states, a balanced competition between measurements and entangling interactions within the system can result in a dynamical purification phase transition between (i) a phase that locally purifies at a constant system-size-independent rate, and (ii) a "mixed" phase where the purification time diverges exponentially in the system size. The residual entropy density in the mixed phase implies the existence of a quantum error-protected subspace where quantum information is reliably encoded against the future non-unitary evolution of the system. We show that these codes are of potential relevance to fault-tolerant quantum computation as they are often highly degenerate and satisfy optimal tradeoffs between encoded information densities and error thresholds. In spatially local models in 1+1 dimensions, this phase transition for mixed initial states occurs concurrently with a recently identified class of entanglement phase transitions for pure initial states. The mutual information of an initially completely-mixed state in 1+1 dimensions grows sublinearly in time due to the formation of the error protected subspace. The purification transition studied here also generalizes to systems with long-range interactions, where conventional notions of entanglement transitions have to be reformulated. Purification dynamics is likely a more robust probe of the transition in experiments, where imperfections generically reduce entanglement and drive the system towards mixed states. We describe the motivations for studying this novel class of non-equilibrium quantum dynamics in the context of advanced quantum computing platforms and fault-tolerant quantum computation.

• The paper demonstrates that continuous measurements induce a dynamical phase transition between constant-rate purification in the pure phase and exponentially slow purification in the mixed phase.
• Using numerical simulations of monitored random circuits, the authors reveal the emergence of a quantum error-protected subspace that supports fault-tolerant quantum computation.
• The study highlights that purification dynamics can serve as robust experimental probes and inspire novel quantum error correction schemes in many-body systems.

Overview of "Dynamical Purification Phase Transitions Induced by Quantum Measurements"

The paper "Dynamical Purification Phase Transitions Induced by Quantum Measurements" by Michael J. Gullans and David A. Huse investigates the interplay between quantum measurements and entangling interactions in quantum many-body systems. At its core, the paper reveals how monitoring the environment of these systems induces phase transitions between different purification regimes.

Key Concepts and Findings

The authors examine the purification dynamics of a quantum system in terms of its reduced density matrix. They establish that continuous monitoring can lead to two distinct phases: a pure phase, where purification occurs at a constant rate regardless of system size, and a mixed phase, characterized by an exponentially diverging purification time with system size. This strongly suggests the existence of a dynamical purification phase transition akin to phase transitions in statistical mechanics, but driven by quantum measurements.

1. Purification Transition: The paper reveals the emergence of a subtle and fascinating phase transition induced by quantum measurements, leading to a bifurcation between a pure phase and a mixed phase. In the mixed phase, the quantum system's residual entropy density indicates the presence of a quantum error-protected subspace, which robustly encodes quantum information against future non-unitary evolutions.
2. Implications for Quantum Information: This work has profound implications for quantum error correction and fault-tolerant quantum computation. The emergence of a quantum error-protected subspace potentially paves the way for new families of error correction codes. These codes show promise because they are highly degenerate and satisfy optimal tradeoffs between encoded information densities and error thresholds, extending the applicability to a range of quantum information processing tasks.
3. Numerical Exploration: Through numerical simulations, the authors explore models like monitored random quantum circuits in 1+1 dimensions and all-to-all coupled models, providing evidence of the dynamical purification transition. The behavior of mutual information within these models, particularly the growth of mutual information in initially mixed states, is shown to be sublinear in time due to the formation of an error-protected subspace.
4. Experiments and Robust Probes: The authors suggest purification dynamics may serve as a more robust experimental probe of such transitions. Under realistic experimental conditions, where imperfections generically reduce entanglement, purification dynamics still reliably indicate transition phases.

Theoretical Implications and Future Directions

The identification of these dynamical phase transitions is theoretically intriguing, as it challenges and extends our understanding of phase transitions beyond traditional thermal and quantum criticality to encompass non-equilibrium dynamics driven by measurement processes. The paper remarkably bridges concepts from open quantum systems, error correction, and quantum chaos.

The potential generalization to systems with long-range interactions, where traditional entanglement transitions must be reformulated, presents a rich avenue for future research. This work suggests that similar phase transitions could occur in different quantum systems and architectures, offering a new lens to examine quantum dynamics and error correction methodologies.

Concluding Thoughts

"Dynamical Purification Phase Transitions Induced by Quantum Measurements" provides a rigorous and detailed exploration of the measurement-induced purification in quantum systems. It opens up new frontiers in understanding the role of measurements in quantum dynamics and points to valuable implications for developing robust quantum computing technologies. The paper is a critical addition to the growing efforts in understanding quantum dynamics in noisy intermediate-scale quantum (NISQ) devices and contributes significantly to the theoretical foundation of quantum error correction schemes. Future work in this area could further unravel new quantum phases and error correction strategies, enhancing the capabilities and understanding of quantum technologies.