Dias Nuno Costa, de Gosson Maurice A, Prata João Nuno
1Escola Superior Náutica Infante D. Henrique, Av. Eng. Bonneville Franco, 2770-058 Paço d'Arcos, Portugal.
2Grupo de Física Matemática, Departamento de Matemática, Faculdade de Cências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa, Portugal.
J Fourier Anal Appl. 2019;25(1):210-241. doi: 10.1007/s00041-018-9602-x. Epub 2018 Feb 22.
We propose a refinement of the Robertson-Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state . We prove sharp estimates which critically depend on the purity of and which are saturated in the Gaussian case.
我们提出了一种使用维格纳分布对罗伯逊 - 薛定谔不确定性原理（RSUP）的改进。这个新原理比RSUP更强。特别是，与RSUP不同，RSUP可以被许多相空间函数饱和，而改进后的RSUP只能被纯高斯维格纳函数饱和。此外，新原理在技术上与标准RSUP一样简单。另外，它与现代调和分析有直接联系，因为它涉及维格纳变换及其辛傅里叶变换，即雷达模糊函数。作为改进后的RSUP的一个副产品，我们推导出了涉及维格纳分布的熵和协方差矩阵的不等式。这些不等式改进了混合量子态的维格纳分布的香农不等式和赫希曼不等式。我们证明了精确的估计，这些估计严重依赖于量子态的纯度，并且在高斯情况下是饱和的。
