Husimi 表示的微分层级
Differential hierarchy of the Husimi representation
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中文总结 AI 辅助
本文通过将 Husimi 表示的正定性条件局部化，建立了基于导数的微分层级，并给出等价的代数行列式判据，揭示了算子秩与正定性在相空间中的局部性。
中文摘要 AI 辅助
相空间表示揭示了量子力学的几何结构，并将量子特征与经典特征区分开来。然而，密度算子的正定性是量子态的性质中相空间表示最不直接捕获的性质之一：判断一个相空间函数是否描述一个物理量子态，迄今为止需要耦合相空间中遥远点的全局条件。在这里，我们表明这些条件实际上可以变得局部化，通过考虑相空间中的 Husimi 表示。我们建立了一组完整的条件，用于捕获相空间函数 $f$ 何时是量子态的 Husimi 函数，每个条件仅涉及函数 $f$ 在单个相空间点处求值的导数。我们进一步表明，每个条件都有一个等价的代数表征，其形式为在位移 Fock 基中底层算子的行列式的非负性。因此，这些条件满足一个递推关系，将它们组织成一个微分层级，该层级根据算子的秩终止。我们的结果因此表明算子秩和正定性是相空间中的局部性质，强化了相空间表示作为量子力学几何描述的作用。
英文摘要
Phase-space representations reveal the geometry of quantum mechanics and distinguish quantum features from classical ones. However, the positivity of a density operator is among the properties of quantum states that phase-space representations capture least directly: deciding whether a phase-space function describes a physical quantum state has so far required global conditions coupling distant points in phase space. Here we show that these conditions can in fact be made local, by considering the Husimi representation in phase space. We establish a complete set of conditions which captures when a phase-space function $f$ is the Husimi function of a quantum state, with each condition involving only the derivatives of the function $f$ evaluated at a single phase-space point. We further show that each condition has an equivalent algebraic characterization, which takes the form of the nonnegativity of a determinant of the underlying operator in a displaced Fock basis. As a result, these conditions obey a recurrence relation organizing them into a differential hierarchy, which terminates based on the rank of the operator. Our results thus show that operator rank and positivity are local properties in phase space, reinforcing the role of phase-space representations as a geometric description of quantum mechanics.
发表机构
- DIENS, École Normale Supérieure, PSL University, CNRS, INRIA (QAT)（高等师范学院，巴黎文理研究大学，法国国家科学研究中心，法国国家信息与自动化研究所）
机构由 AI 辅助整理，请以论文原文为准。
