Dynamical Purification Phase Transition Induced by Quantum Measurements

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 nonunitary 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 trade-offs between encoded information densities and error thresholds. In spatially local models in1+1dimensions, this phase transition for mixed initial states occurs concurrently with a recently identified class of entanglement phase transitions for pure initial states. The purification transition studied here also generalizes to systems with long-range interactions, where conventional notions of entanglement transitions have to be reformulated. We numerically explore this transition for monitored random quantum circuits in1+1dimensions and all-to-all models. Unlike in pure initial states, the mutual information of an initially completely mixed state in1+1dimensions grows sublinearly in time due to the formation of the error-protected subspace. 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 nonequilibrium quantum dynamics in the context of advanced quantum computing platforms and fault-tolerant quantum computation.

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Plain Language Summary

The paradoxes of entanglement and quantum measurement are ubiquitous in quantum physics. Although many of the most puzzling features are now well understood in systems of just a few particles, the advent of experimental quantum information science has the potential to bring these issues to bear in systems containing many particles, raising new (and old) conceptual issues at the heart of quantum physics. To explore these issues, we theoretically investigate how a generic many-body system, subject to continuous monitoring of its environment, can be brought into a pure quantum state.

In our study, we examine a class of random quantum circuit models subject to measurements. We find that as one changes the rate of measurements, the system undergoes a phase transition from a disordered phase, which steps toward a pure quantum state at a constant rate, and an ordered phase, which takes an exponentially long time to purify. This transition can be identified as a type of quantum error-correction threshold between a phase that retains memory of initial conditions for exponentially long times and a phase that quickly forgets initial conditions. We therefore obtain precise estimates of many critical properties of the transition and generate new families of quantum error-correcting codes with potentially useful applications to fault-tolerant quantum computation.

We expect our predictions to be experimentally testable in advanced quantum computing platforms, which might provide a useful application of near-term quantum computers.