# Dynamical purification phase transitions induced by quantum measurements

## TL;DR

This paper investigates a phase transition in quantum systems where measurements and interactions compete, leading to a state where quantum information can be reliably stored despite ongoing non-unitary evolution, with implications for quantum computing.

## Contribution

It introduces the concept of a dynamical purification phase transition in mixed states, linking measurement-induced purification to error-protected quantum subspaces and fault-tolerant computation.

## Key findings

- Identification of a phase transition between constant-rate purification and diverging purification time.
- Existence of a quantum error-protected subspace with residual entropy density.
- Growth of mutual information in mixed states indicating error protection.

## 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.