# Topological transition in measurement-induced geometric phases

## TL;DR

This paper explores how a sequence of weak quantum measurements induces a stochastic geometric phase whose topological properties can undergo a phase transition, revealing new insights into measurement-induced quantum control.

## Contribution

It demonstrates a topological transition in the geometric phase induced by measurement strength, linking measurement sequences to topological phenomena in quantum systems.

## Key findings

- Measurement strength controls the stochastic geometric phase.
- A topological transition occurs in the phase mapping.
- Implications for quantum control and measurement foundations.

## Abstract

The state of a quantum system, adiabatically driven in a cycle, may acquire a measurable phase depending only on the closed trajectory in parameter space. Such geometric phases are ubiquitous, and also underline the physics of robust topological phenomena such as the quantum Hall effect. Equivalently, a geometric phase may be induced through a cyclic sequence of quantum measurements. We show that the application of a sequence of weak measurements renders the closed trajectories, hence the geometric phase, stochastic. We study the concomitant probability distribution and show that, when varying the measurement strength, the mapping between the measurement sequence and the geometric phase undergoes a topological transition. Our finding may impact measurement-induced control and manipulation of quantum states---a promising approach to quantum information processing. It also has repercussions on understanding the foundations of quantum measurement.