# Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition

## TL;DR

This paper investigates how entanglement entropy evolves in open quantum systems with measurements, revealing a phase transition driven by the balance between scrambling and measurement rate, linked to quantum error correction.

## Contribution

It introduces a random circuit model with independent control of scrambling and measurement, connecting entanglement phases to quantum error correction and channel capacity.

## Key findings

- Identifies volume-law and area-law entanglement phases in the model.
- Discovers a phase transition governed by measurement rate and scrambling.
- Provides a phase diagram illustrating the transition conditions.

## Abstract

We analyze the dynamics of entanglement entropy in a generic quantum many-body open system from the perspective of quantum information and error corrections. We introduce a random unitary circuit model with intermittent projective measurements, in which the degree of information scrambling by the unitary and the rate of projective measurements are independently controlled. This model displays two stable phases, characterized by the volume-law and area-law scaling entanglement entropy in steady states. The transition between the two phases is understood from the point of view of quantum error correction: the chaotic unitary evolution protects quantum information from projective measurements that act as errors. A phase transition occurs when the rate of errors exceeds a threshold that depends on the degree of information scrambling. We confirm these results using numerical simulations and obtain the phase diagram of our model. Our work shows that information scrambling plays a crucial role in understanding the dynamics of entanglement in an open quantum system and relates the entanglement phase transition to changes in quantum channel capacity.