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 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 in 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 nonequilibrium 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，系统之间的度量和纠缠相互作用之间的平衡竞争可能会导致（i）以恒定的系统尺寸独立速率进行局部纯化的相与（ii）之间的动态纯化相变。 “混合”阶段，纯​​化时间在系统大小上呈指数变化。混合相中的剩余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，可靠地对量子信息进行了编码，以对抗系统未来的非单位演化。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。我们用数值方法探索了这种转变，用于监测中的随机量子电路$\mathrm{1个}+\mathrm{1个}$尺寸和所有模型。与纯初始状态不同，初始完全混合状态的互信息$\mathrm{1个}+\mathrm{1个}$由于形成了受错误保护的子空间，尺寸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。我们描述了在先进的量子计算平台和容错量子计算的背景下研究这类新型的非平衡量子动力​​学的动机。