Echo state property (ESP) is a fundamental property that allows an input-driven dynamical system to perform information processing tasks. Recently, extensions of ESP to potentially nonstationary systems and subsystems, that is, nonstationary ESP and subset/subspace ESP, have been proposed. In this paper, we theoretically and numerically analyze the sufficient and necessary conditions for a quantum system to satisfy nonstationary ESP and subset/subspace nonstationary ESP. Based on extensive usage of the Pauli transfer matrix (PTM) form, we find that (1) the interaction with a quantum-coherent environment, termed $\textit{coherence influx}$, is indispensable in realizing nonstationary ESP, and (2) the spectral radius of PTM can characterize the fading memory property of quantum reservoir computing (QRC). Our numerical experiment, involving a system with a Hamiltonian that entails a spin-glass/many-body localization phase, reveals that the spectral radius of PTM can describe the dynamical phase transition intrinsic to such a system. To comprehensively understand the mechanisms under ESP of QRC, we propose a simplified model, multiplicative reservoir computing (mRC), which is a reservoir computing (RC) system with a one-dimensional multiplicative input. Theoretically and numerically, we show that the parameters corresponding to the spectral radius and coherence influx in mRC directly correlates with its linear memory capacity (MC). Our findings about QRC and mRC will provide a theoretical aspect of PTM and the input multiplicativity of QRC. The results will lead to a better understanding of QRC and information processing in open quantum systems.

翻译：回波状态特性（ESP）是使输入驱动动力系统能够执行信息处理任务的基本特性。最近，ESP已被扩展至可能非平稳的系统与子系统，即非平稳ESP和子集/子空间ESP。本文从理论与数值上分析了量子系统满足非平稳ESP及子集/子空间非平稳ESP的充分必要条件。基于对泡利转移矩阵（PTM）形式的广泛运用，我们发现：（1）与量子相干环境的相互作用——称为“相干性流入”——是实现非平稳ESP不可或缺的条件；（2）PTM的谱半径能够表征量子储备池计算（QRC）的渐消记忆特性。我们的数值实验涉及一个具有自旋玻璃/多体局域化相哈密顿量的系统，结果表明PTM的谱半径能够描述此类系统固有的动力学相变。为全面理解QRC中ESP的机制，我们提出了一个简化模型——乘法储备池计算（mRC），即具有一维乘法输入的储备池计算（RC）系统。通过理论与数值分析，我们证明mRC中对应于谱半径和相干性流入的参数与其线性记忆容量（MC）直接相关。关于QRC和mRC的研究结果将为PTM的理论层面及QRC的输入乘法性提供新的见解。这些成果将有助于深化对QRC以及开放量子系统中信息处理机制的理解。