标题： 自由独立性与随机矩阵乘积幺正的酉设计 显示英文标题

标题： Free Independence and Unitary Design from Random Matrix Product Unitaries

摘要： 理解复杂量子系统如何模拟随机性是量子混沌、热化和信息论的核心问题。 在一个场景中，最近发现时间有序关联函数（OTOCs）可以探测海森堡算符之间的渐近自由性：统计独立性的非交换推广。 在另一个研究方向中，近似酉设计的概念导致了根据正向时间协议看起来随机的酉操作的有效构造。 将这两个概念联系起来，在本工作中我们研究了从随机矩阵乘积酉操作（RMPU）集合中出现的自由性。 我们证明，仅需要多项式维度的bond维度，这些酉操作就能再现局部有限迹可观测量的高阶OTOC的哈尔值——正是根据本征态热化假说在混沌多体系统中导致热相关性的可观测量。 RMPU集合确实构成了一个酉设计，但我们认为这并不能解释平均OTOC行为，因此也不能解释自由性的出现。 我们进一步精确计算了该集合的框架潜力到二阶，显示其收敛到哈尔值，且具有多项式偏差，表明对于全局可观测量，自由性也达到了平均意义上的实现。 另一方面，为了再现局部无迹可观测量的类似哈尔的OTOC值，所考虑的集合需要体积律的算符纠缠。 因此，这种相关性超出了可以高效复制的随机酉特征范式。 我们的结果强调了在算符动力学背景下改进先前的酉设计概念的必要性，引导我们走向真正的量子优势协议，同时揭示了混沌多体系统中出现的复杂性。 显示英文摘要

摘要： Understanding how complex quantum systems emulate randomness is central to quantum chaos, thermalization, and information theory. In one setting, out-of-time-ordered correlators (OTOCs) have recently been shown to probe asymptotic freeness between Heisenberg operators: the non-commutative generalization of statistical independence. In a distinct research direction, the concept of approximate unitary designs have led to efficient constructions of unitaries that look random according to forward-in-time protocols. Bridging these two concepts, in this work we study the emergence of freeness from a random matrix product unitary (RMPU) ensemble. We prove that, with only polynomial bond dimension, these unitaries reproduce Haar values of higher-order OTOCs for local, finite-trace observables -- precisely the observables that lead to thermal correlations in chaotic many-body systems according to the eigenstate thermalization hypothesis. The RMPU ensemble provably forms a unitary design, but, we argue, this does not account for average OTOC behavior and therefore the emergence of freeness. We further compute the ensemble's frame potential exactly to second order, showing convergence to Haar values also with polynomial deviations, indicating that freeness is also reached on-average for global observables. On the other hand, to reproduce the Haar-like OTOC value for local, traceless observables, the considered ensemble requires volume-law operator entanglement. Such correlations therefore lie beyond the paradigm of random unitary features which can be replicated efficiently. Our results highlight the need to refine previous notions of unitary designs in the context of operator dynamics, guiding us towards protocols for genuine quantum advantage while shedding light on the emergent complexity of chaotic many-body systems.