The Trotter product formula and the quantum Zeno effect are both indispensable tools for constructing time-evolutions using experimentally feasible building blocks. In this work, we discuss assumptions under which quantitative bounds can be proven in the strong operator topology on Banach spaces and provide natural bosonic examples. Specially, we assume the existence of a continuously embedded Banach space, which relatively bounds the involved generators and creates an invariant subspace of the limiting semigroup with a stable restriction. The slightly stronger assumption of admissible subspaces is well-recognized in the realm of hyperbolic evolution systems (time-dependent semigroups), to which the results are extended. By assuming access to a hierarchy of continuously embedded Banach spaces, Suzuki-higher-order bounds can be demonstrated. In bosonic applications, these embedded Banach spaces naturally arise through the number operator, leading to a diverse set of examples encompassing notable instances such as the Ornstein-Uhlenbeck semigroup and multi-photon driven dissipation used in bosonic error correction.

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关于 Trotter 和 Zeno 产品公式与玻色子应用的强界限


Trotter 乘积公式和量子芝诺效应都是使用实验上可行的构建块构建时间演化不可或缺的工具。在这项工作中，我们讨论了可以在 Banach 空间上的强算子拓扑中证明定量界限的假设，并提供自然的玻色子示例。特别地，我们假设存在一个连续嵌入的 Banach 空间，它相对限制了所涉及的生成元，并创建了一个具有稳定限制的极限半群的不变子空间。容许子空间稍强的假设在双曲演化系统（时间相关半群）领域得到了广泛认可，其结果也扩展到了该领域。通过假设访问连续嵌入的 Banach 空间的层次结构，可以演示铃木高阶界限。在玻色子应用中，这些嵌入式巴拿赫空间自然地通过数字算子产生，从而产生了一系列不同的示例，其中包括值得注意的实例，例如玻色子误差校正中使用的奥恩斯坦-乌伦贝克半群和多光子驱动耗散。