Continuous Monitoring of One-Dimensional Free Fermions Yields Volume-Law Entanglement for Α = 0

The continuous observation of quantum systems fundamentally alters their behaviour, and recent research explores how this impacts one-dimensional quantum materials. Clemens Niederegger, Tatiana Vovk, and Elias Starchl, working alongside Lukas M. Sieberer at the University of Innsbruck and the Austrian Academy of Sciences, investigate whether constant monitoring induces critical behaviour, such as increased entanglement, in these systems. Their work focuses on free fermions, a fundamental building block of many materials, and demonstrates that while monitoring can mimic critical behaviour over certain distances, it ultimately fails to create genuine long-range entanglement. This finding, achieved through a combination of theoretical modelling and numerical simulation, clarifies the nature of monitored quantum systems and establishes a crucial distinction between apparent and true criticality in these materials.

The research investigates many-body criticality, specifically examining how entanglement grows, correlations develop, and conformal invariance emerges within systems that are not in equilibrium. A central question concerns whether these signatures indicate a genuine new phase of quantum matter or merely persist over limited distances. To address this, scientists study a chain of free fermions subjected to continuous monitoring of each lattice site. The monitoring process involves choosing a measurement scheme, which interpolates between different ways of unraveling the quantum state, each corresponding to a different stochastic outcome of the same underlying quantum evolution.

Monitored Fermions, Measurement and Open Quantum Systems

This work provides a comprehensive understanding of quantum measurement, open quantum systems, many-body physics, and entanglement, particularly within monitored free fermion systems. Scientists employ Keldysh field theory, a powerful tool for analyzing systems driven by external forces and interacting with their environment, to understand the impact of measurement and dissipation. Lindblad master equations are used to describe how open quantum systems evolve, accounting for both their natural dynamics and the effects of measurement and decoherence. Researchers also investigate entanglement entropy, a measure of quantum correlations, and how it can be used to probe the properties of quantum systems, including the effects of interactions and disorder, connecting it to particle number cumulants to calculate entanglement in interacting systems.

A key focus is understanding measurement-induced phase transitions, where continuous measurement drives a system into a new phase of matter. Recent work explores the role of random quantum circuits and the effects of disorder on quantum systems, drawing parallels to Anderson localization. The research relies on mathematical tools including conformal field theory and Weingarten calculus, employing differential operators and matrix integrals for calculations. The replica trick is used to calculate entanglement entropy in disordered systems, and the Keldysh formalism provides a powerful approach for analyzing non-equilibrium quantum systems. Continuous quantum measurement can act as a driving force, inducing phase transitions and altering the properties of quantum systems, with entanglement serving as a powerful tool for probing behaviour and a potential order parameter for measurement-induced phase transitions.

Entanglement Area Law Beyond Exponential Scales

Scientists investigated a one-dimensional system of free fermions subjected to continuous monitoring, revealing detailed insights into entanglement behaviour and the absence of a measurement-induced phase transition. The work demonstrates that entanglement ultimately obeys an area law, meaning that the amount of entanglement grows with the area of the system’s boundary, but only beyond an exponentially large scale proportional to the inverse of the hopping amplitude and the measurement rate. Experiments confirmed this prediction, establishing that no measurement- or unraveling-induced entanglement transition occurs in this model. The research team employed replica Keldysh field theory to derive a nonlinear sigma model describing the long-wavelength physics of the system, providing a theoretical framework for understanding entanglement dynamics.

Analysis of this model shows that while entanglement initially grows logarithmically, it transitions to an area law beyond a scale of approximately the inverse measurement rate, indicating a crossover rather than a true phase transition. Numerical simulations corroborated these findings, confirming the absence of a critical phase and validating the theoretical predictions regarding the exponential scaling of the crossover length. Further investigations explored different unraveling schemes, including unitary random noise, finding that these also yield volume-law steady-state entanglement, but ultimately adhere to the area law at sufficiently large scales. Tuning the unraveling phase does not induce an entanglement transition, reinforcing the conclusion that the observed phenomena are crossovers rather than genuine phase transitions. Measurements confirm that the logarithmic growth of entanglement entropy is cut off at a length scale exponentially dependent on the inverse measurement rate, aligning with expectations from disordered electronic systems and Anderson localization.

Entanglement Growth and Area Law Violation

This research establishes a comprehensive understanding of entanglement growth in continuously monitored free fermionic systems, resolving a long-standing question regarding the emergence of critical-like behaviour. Scientists investigated how different measurement schemes affect entanglement, demonstrating that the choice of measurement protocol significantly influences the resulting quantum state. Through a combination of replica Keldysh field theory and numerical simulations, the team showed that while certain measurement settings can mimic criticality, entanglement ultimately obeys an area law beyond an exponentially large scale determined by the system’s hopping amplitude and measurement rate. Importantly, the study reveals that the observed critical-like behaviour appears below a more accessible crossover scale that grows algebraically, allowing for detailed numerical verification of the theoretical predictions. The team confirmed the absence of measurement-induced entanglement transitions in this model, clarifying that the system does not exhibit a genuine phase transition to a critical state.