标题： 探索具有对称性的系统中量子态设计的出现 显示英文标题

标题： Unraveling the emergence of quantum state designs in systems with symmetry

摘要： 量子态设计通过高效采样随机量子态，在设计和基准测试各种具有广泛应用的量子协议（从电路设计到黑洞物理）中起着至关重要的作用。 另一方面，对称性预计会降低态的随机性。 尽管对称性无处不在，但其对量子态设计的影响仍然是一个悬而未决的问题。 最近引入的投影系综框架通过依赖投影测量和多体量子混沌生成了有效的近似态$t$-设计。 在这项工作中，我们研究了表现出对称性的随机生成态中态设计的出现。 利用平移对称性，我们解析地确立了导致态$t$-设计的测量基的充分条件。 然后，通过利用迹距离度量，我们数值研究了向设计收敛的过程。 随后，我们检查了充分条件的违反情况以识别未能收敛的基。 我们进一步通过研究具有平移对称性的混沌倾斜场伊辛链的动力学，展示了物理系统中态设计的出现。 我们发现，在早期时间演化过程中，迹距离的收敛速度比对称性被破坏的情况更快。 为了描述我们的结果的一般适用性，我们将分析扩展到其他对称性。 我们期望我们的发现能为封闭和开放量子多体系统的深度热化和平衡的进一步探索铺平道路。 显示英文摘要

摘要： Quantum state designs, by enabling an efficient sampling of random quantum states, play a quintessential role in devising and benchmarking various quantum protocols with broad applications ranging from circuit designs to black hole physics. Symmetries, on the other hand, are expected to reduce the randomness of a state. Despite being ubiquitous, the effects of symmetry on quantum state designs remain an outstanding question. The recently introduced projected ensemble framework generates efficient approximate state $t$-designs by hinging on projective measurements and many-body quantum chaos. In this work, we examine the emergence of state designs from the random generator states exhibiting symmetries. Leveraging on translation symmetry, we analytically establish a sufficient condition for the measurement basis leading to the state $t$-designs. Then, by making use of the trace distance measure, we numerically investigate the convergence to the designs. Subsequently, we inspect the violation of the sufficient condition to identify bases that fail to converge. We further demonstrate the emergence of state designs in a physical system by studying the dynamics of a chaotic tilted field Ising chain with translation symmetry. We find faster convergence of the trace distance during the early time evolution in comparison to the cases when the symmetry is broken. To delineate the general applicability of our results, we extend our analysis to other symmetries. We expect our findings to pave the way for further exploration of deep thermalization and equilibration of closed and open quantum many-body systems.