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 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 in 1+1 dimensions and all-to-all models. Unlike 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. 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.

中文翻译：

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量子测量引起的动态纯化相变

连续监测量子多体系统的环境，以测量结果为条件，可以减少（纯化）系统密度矩阵的熵。我们表明，对于混合的初始状态，系统之间的测量和纠缠相互作用之间的平衡竞争可能导致（i）以恒定的系统尺寸独立速率进行局部纯化的相之间的动态纯化相变，以及（ii ）是一个“混合”相，其中净化时间在系统尺寸上呈指数变化。混合相中的残留熵密度意味着存在一个受量子错误保护的子空间，在该子空间中，量子信息被可靠地编码，以对抗未来的非单位演化。系统。我们表明，这些代码与容错量子计算潜在相关，因为它们通常高度退化，并满足编码信息密度和错误阈值之间的最佳折衷。在空间局部模型中$\mathrm{1个}+\mathrm{1个}$在尺寸上，混合初始状态的这种相变与纯净初始状态的最近确定的纠缠相变类同时发生。此处研究的纯化跃迁也可以推广到具有长距离相互作用的系统，在该系统中，必须重新构造纠缠跃迁的常规概念。我们以数值形式探索了用于监视的1 + 1维随机量子电路和所有模型的过渡。与纯初始状态不同，由于受错误保护的子空间的形成，初始完全混合状态在1 + 1维中的互信息在时间上亚线性增长。净化动力学可能是实验中过渡的更可靠方法，其中缺陷通常会减少纠缠并使系统趋向混合态。