Measurement-Induced Phase Transitions in the Dynamics of Entanglement

Abstract: We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate $p$ for each degree of freedom, we show that the system has two dynamical phases: entangling' anddisentangling'. The former occurs for $p$ smaller than a critical rate $p_c$, and is characterized by volume-law entanglement in the steady-state and ballistic' entanglement growth after a quench. By contrast, for $p > p_c$ the system can sustain only area-law entanglement. At $p = p_c$ the steady state is scale-invariant and, in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth R\'{e}nyi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher R\'{e}nyi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains, and show that the phenomenology of the two phases is similar to that of the toy model, but with distinctquantum' critical exponents, which we calculate numerically in $1+1$D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.

• The paper introduces a framework combining numerical simulations with theoretical models to reveal critical measurement thresholds for transitioning between entangling and disentangling phases.
• It employs random projective measurements in quantum circuits to trigger a phase transition from volume-law to area-law entanglement, highlighting distinct universality classes.
• The study's insights on entanglement dynamics have important implications for quantum computing, particularly in assessing simulation complexity for quantum systems.

Measurement-Induced Phase Transitions in the Dynamics of Entanglement

This paper explores the dynamics of quantum entanglement under the influence of random, local measurements—a dynamic area of inquiry within quantum information theory and many-body physics. It provides a nuanced exploration of quantum systems experiencing entanglement phase transitions induced by measurements. The nuance lies in employing random projective measurements, which give rise to two distinct dynamical phases: an "entangling" phase characterized by volume-law entanglement, and a "disentangling" phase where systems only sustain area-law entanglement.

Core Findings and Methodology

The authors introduce a compelling framework that draws upon both numerical simulations and theoretical models. They begin with a conceptual toy model that simplifies these dynamics to the zeroth Renyi entropy in discrete time, elucidating the dynamics by mapping them onto classical percolation problems. In this manner, they derive a heuristic understanding that is robustly corroborated with numerical simulations.

For the unitary dynamics, the paper examines both Floquet dynamics for non-integrable Ising models and random circuit dynamics. These allow the authors to demonstrate consistent universal properties, irrespective of the specifics of the dynamical model chosen. In the entangling phase, the dynamics lead to volume-law entanglement in the steady state and enables 'ballistic' entanglement growth following a quench. Conversely, the disentangling phase consistently results in area-law entanglement.

Key Results and Implications

The study provides quantitative insights, particularly around critical measurement rates and their relation to entropy scaling. Strong numerical results show a striking critical measurement rate $p_c$ that delineates the transition from volume-law to area-law scaling. Interestingly, they observe a set of critical exponents that distinguish this transition from classical percolation, shedding light on a new universality class for these transitions.

One intriguing implication of the study is its relevance to quantum computation, particularly in understanding the computational complexity involved in simulating quantum systems influenced by measurements. The entangling-disentangling transition indicates a bounds shift in simulation difficulty—volume-law states complicating simulation exponentially while area-law states offer more computational tractability.

Future Directions

The research holds profound implications for the development of quantum simulators and suggests potential pathways for further exploration. Theoretical extensions can probe deeper into the links between entanglement structure and computational complexity, and further experimental work can verify the non-intuitive behaviors observed numerically. Additionally, exploring analogous transitions in higher-dimensional systems could yield new insights into the universality and diversity of quantum entanglement dynamics.

In conclusion, this paper offers a comprehensive analysis of measurement-induced phase transitions in quantum systems, combining innovative theoretical models with robust numerical simulations, and opening pathways for new explorations in the landscape of quantum entanglement dynamics. The study contributes to a more nuanced understanding of entanglement's role in quantum information processing and many-body quantum physics.