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Theory of Ergodic Quantum Processes
Physical Review X ( IF 11.6 ) Pub Date : 2021-10-01 , DOI: 10.1103/physrevx.11.041001
Ramis Movassagh , Jeffrey Schenker

The generic behavior of quantum systems has long been of theoretical and practical interest. Any quantum process is represented by a sequence of quantum channels. We consider general ergodic sequences of stochastic channels with arbitrary correlations and non-negligible decoherence. Ergodicity includes and vastly generalizes random independence. We obtain a theorem which shows that the composition of such a sequence of channels converges exponentially fast to a replacement (rank-one) channel. Using this theorem, we derive the limiting behavior of translation-invariant channels and stochastically independent random channels. We then use our formalism to describe the thermodynamic limit of ergodic matrix product states. We derive formulas for the expectation value of a local observable and prove that the two-point correlations of local observables decay exponentially. We then analytically compute the entanglement spectrum across any cut, by which the bipartite entanglement entropy (i.e., Rényi or von Neumann) across an arbitrary cut can be computed exactly. Other physical implications of our results are that most Floquet phases of matter are metastable and that noisy random circuits in the large depth limit will be trivial as far as their quantum entanglement is concerned. To obtain these results, we bridge quantum information theory to dynamical systems and random matrix theory.

中文翻译:

遍历量子过程理论

量子系统的一般行为长期以来一直受到理论和实践的关注。任何量子过程都由一系列量子通道表示。我们考虑具有任意相关性和不可忽略退相干性的随机通道的一般遍历序列。遍历性包括并极大地概括了随机独立性。我们得到了一个定理,该定理表明这样一个通道序列的组成以指数方式快速收敛到一个替换(秩一)通道。使用这个定理,我们推导出平移不变通道和随机独立随机通道的限制行为。然后我们使用我们的形式来描述遍历矩阵乘积状态的热力学极限。我们推导出局部可观察量的期望值的公式,并证明局部可观察量的两点相关性呈指数衰减。然后我们分析计算任意切割的纠缠谱,通过它可以精确计算任意切割的二分纠缠熵(即 Rényi 或 von Neumann)。我们的结果的其他物理含义是,物质的大多数 Floquet 相都是亚稳态的,并且就它们的量子纠缠而言,大深度限制中的嘈杂随机电路将是微不足道的。为了获得这些结果,我们将量子信息理论与动力系统和随机矩阵理论联系起来。Rényi 或 von Neumann) 可以精确计算任意切割。我们的结果的其他物理含义是,物质的大多数 Floquet 相都是亚稳态的,并且就它们的量子纠缠而言,大深度限制中的嘈杂随机电路将是微不足道的。为了获得这些结果,我们将量子信息理论与动力系统和随机矩阵理论联系起来。Rényi 或 von Neumann) 可以精确计算任意切割。我们的结果的其他物理含义是,物质的大多数 Floquet 相都是亚稳态的,并且就它们的量子纠缠而言,大深度限制中的嘈杂随机电路将是微不足道的。为了获得这些结果,我们将量子信息理论与动力系统和随机矩阵理论联系起来。
更新日期:2021-10-01
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