Symplectic-dilation mixed wavelet transform and its correspondence in quantum optics

Symplectic Dilations, Gaussian States and Gaussian Channels

By elementary matrix algebra we show that every real $2n \times 2n$ matrix admits a dilation to an element of the real symplectic group $Sp (2(n+m))$ for some nonnegative integer $m.$ Our methods do not yield the minimum value of $m,$ for which such a dilation is possible. After listing some of the main properties of Gaussian states in $L^2 (\mathbb{R}^n),$ we analyse the implications of symplectic dilations in the study of quantum Gaussian channels which lead to some interesting open problems, particularly, in the context of the work of Heinosaari, Holevo and Wolf \cite{3}.

Quantum Channels, Wavelets, Dilations and Representations of $O_n$

We show that the representations of the Cuntz C$^\ast$-algebras $O_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, an application in quantum information theory is obtained; namely, a structure theorem for the fixed point set of a unital quantum channel. We also include some open problems motivated by this work.

Quantum Optical Version of Classical Optical Transformations and Beyond

By the newly developed technique of integration within an ordered product (IWOP) of operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We can not only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theo…

量子线性时平移不变系统：共轭辛结构、不确定性界限与层析成像

Quantum Linear Time-Translation-Invariant Systems: Conjugate Symplectic Structure, Uncertainty Bounds, and Tomography

针对多模态经典LTI系统提出一种通用量子化方案，利用李群技术揭示共轭辛结构与复压缩现象，推导出更严格的海森堡不确定性界限并提出辛差检测法，建立了任意量子LTI系统分析、合成与测量的完整框架。

经典与量子随机电路的电阻和忆阻电路模型

Classical and Quantum Stochastic Models of Resistive and Memristive Circuits

本文研究了电路相空间中随机马尔可夫模型，分析电阻和忆阻电路的耗散特性，并提出对称噪声概念，构建量子力学类比模型。