希尔伯特空间中的玻尔兹曼计数
Boltzmann counting in Hilbert space
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中文总结 AI 辅助
本文将玻尔兹曼计数视角扩展至量子场景，提出基于希尔伯特空间体积对数的量子制备几何熵，分析三类约束并得到相关体积的标度律与闭式表达式，补充了量子熵的度量维度。
中文摘要 AI 辅助
我们为量子态制备引入了一种几何熵，其定义为与给定约束集相容的纯态的希尔伯特空间体积的对数。该构造将玻尔兹曼的计数视角扩展至量子场景，其中相容态无需正交，且“态的数量”的相关概念自然被态空间中的体积所取代。我们分析三类约束：限制到子空间、固定期望值、粗粒化子系统描述。针对子空间投影、自旋期望值、部分迹、不完善探测器映射等代表性示例，我们得到了相关体积的显式标度律与闭式表达式。该框架在制备层面提供了量子无知的几何度量，是对基于密度矩阵与粗粒化的熵概念的补充。
英文摘要
We introduce a geometric entropy for quantum preparations, defined as the logarithm of the Hilbert-space volume of pure states compatible with a given set of constraints. This construction extends Boltzmann's counting perspective to the quantum setting, where compatible states need not be orthogonal and the relevant notion of "number of states" is naturally replaced by a volume in state space. We analyze three classes of constraints: restriction to a subspace, fixed expectation values, and coarse-grained subsystem descriptions. For representative examples, including subspace projection, spin expectation values, partial trace, and an imperfect detector map, we obtain explicit scaling laws and closed-form expressions for the associated volumes. The resulting framework provides a geometric measure of quantum ignorance at the level of the preparation and complements entropy notions based on density matrices and coarse graining.
