Star-product quantization and symplectic tomography

部分辛量子层析方案：可观测量、演化方程与定态方程

Partial symplectic quantum tomography schemes. Observables, evolution equations, and stationary states equations

本文研究量子力学的部分辛条件与联合概率表示，利用量子化子-去量子化子形式体系推导算符对偶符号和对应规则，并给出演化方程与定态方程。

Comment on ``Symplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty''

In Phys. Rev. A 70, 032104 (2004), M. Montesinos and G. F. Torres del Castillo consider various symplectic structures on the classical phase space of the two-dimensional isotropic harmonic oscillator. Using Dirac's quantization condition, the authors investigate how these alternative symplectic forms affect this system's quantization. They claim that these symplectic structures result in mutually inequivalent quantum theories. In fact, we show here that there exists a unitary map between the two representation spaces so that the various quantizations are equivalent.

Reconstruction of Markovian Master Equation parameters through symplectic tomography

In open quantum systems, phenomenological master equations with unknown parameters are often introduced. Here we propose a time-independent procedure based on quantum tomography to reconstruct the potentially unknown parameters of a wide class of Markovian master equations. According to our scheme, the system under investigation is initially prepared in a Gaussian state. At an arbitrary time t, in order to retrieve the unknown coefficients one needs to measure only a finite number (ten at maximum) of points along three time-independent tomograms. Due to the limited amount of measurements requi…

Symplectic areas, quantization, and dynamics in electromagnetic fields

A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: a choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the qua…

Symplectic tomography of ultracold gases in tight-waveguides

The phase space is the natural ground to smoothly extrapolate between local and non-local correlation functions. With this objective, we introduce the symplectic tomography of many-body quantum gases in tight-waveguides, and concentrate on the reduced single-particle symplectic tomogram (RSPST) whose marginals are the density profile and momentum distribution. We present an operational approach to measure the RSPST from the time evolution of the density profile after shutting off the interactions in a variety of relevant situations: free expansion, fall under gravity, and oscillations in a har…

On the goodness of "quantum blobs" in phase space quantization

We replace the usual notion of quantum cell from statistical mechanics by that of "quantum blob". A quantum blob is the transform, by a linaer symplectic transformation, of a phase space ball with radius equal to the square root of h-bar. The intersection of a quantum blob with any symplectic plane is an ellipse with area one half of h. This very special property, which is closed related to the principle of the symplectic camel, leads to a symplectic invariant statement of the uncertainty principle. We moreover prove that the average of a phase space Gaussian over a quantum blob is the Wigner …

Geometric quantization of symplectic foliations

Geometric quantization of a Poisson manifold need not imply quantization of its symplectic leaves. We provide the leafwise geometric quantization of a Poisson manifold, seen as a foliated one, whose quantum algebra restricted to each leaf is quantized.

Dynamic Symmetries and Entropic Inequalities in the Probability Representation of Quantum Mechanics

The probability representation of quantum and classical statistical mechanics is discussed. Symplectic tomography, center-of-mass tomography, and spin tomography are studied. The connection of tomographic probabilities with dynamic symmetries like symplectic group is considered. Entropic uncertainty relations and inequalities for spin tomograms are reviewed.

Symplectic Cuts and Projection Quantization

The recently proposed projection quantization, which is a method to quantize particular subspaces of systems with known quantum theory, is shown to yield a genuine quantization in several cases. This may be inferred from exact results established within symplectic cutting.

Nonexistence of Finite-dimensional Quantizations of a Noncompact Symplectic Manifold

We prove that there is no faithful finite-dimensional representation by skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^\infty(M) containing B.

On the information completeness of quantum tomograms

We address the problem of information completeness of quantum measuremets in connection to quantum state tomography and with particular concern to quantum symplectic tomography. We put forward some non-trivial situations where informationally incomplete set of tomograms allows as well the state reconstruction provided to have some a priori information on the state or its dynamics. We then introduce a measure of information completeness and apply it to symplectic quantum tomograms.

一种新的正则、协变与辛哈密顿密度的量子化

Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

本文将Koopman-von Neumann经典力学推广至多辛场并恢复De Donder-Weyl理论，提出具有洛伦兹协变性和辛几何的新哈密顿形式，通过经典与量子对易子代数间的算符映射实现一次和二次量子化。

Path Integral Quantization and Riemannian-Symplectic Manifolds

We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed.

Wigner分布和拉格朗日层析成像的广义边际分布

Generalized Marginals of tie -Wigner Distribution nand Lagrangian Tomography

该研究基于辛群传递作用建立拉格朗日Radon变换的辛形式，将Wigner变换边际扩展至任意拉格朗日子空间，统一处理辛旋转关联的位置与动量测量，解决Pauli重构问题并建立密度算子重构公式。

Symplectic quantization of multi-field Generalized Proca electrodynamics

We explicitly carry out the symplectic quantization of a family of multi-field Generalized Proca (GP) electrodynamics theories. In the process, we provide an independent derivation of the so-called secondary constraint enforcing relations -- consistency conditions that significantly restrict the allowed interactions in multi-field settings already at the classical level. Additionally, we unveil the existence of quantum consistency conditions, which apply in both single- and multi-field GP scenarios. Our newly found conditions imply that not all classically well-defined (multi-)GP theories are …

量化低秩量子态层析：双曲量化与黎曼最小二乘恢复

Quantized Low-Rank Quantum State Tomography: Hyperbolic Quantization and Riemannian Least-Squares Recovery

该研究针对有限比特泡利批量响应的低秩量子态层析问题，提出HyperQuant量化器与QuantRGD方法，建立相关恢复保证与比特- shots权衡，通过实验验证其有效性。

Symplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty

We analyze the quantum dynamics of the non-relativistic two-dimensional isotropic harmonic oscillator in Heisenberg's picture. Such a system is taken as toy model to analyze some of the various quantum theories that can be built from the application of Dirac's quantization rule to the various symplectic structures recently reported for this classical system. It is pointed out that that these quantum theories are inequivalent in the sense that the mean values for the operators (observables) associated with the same physical classical observable do not agree with each other. The inequivalence do…

辛几何与电路量子化

Symplectic geometry and circuit quantization

针对含非线性元件的电路无法用标准拉格朗日方法量子化的问题，研究结合辛几何与图论提出无耗散电路的哈密顿量子化方案，推导兼容已知结果并给出可处理更广泛电路的高效算法。

Quantization of symplectic vector spaces over finite fields

In this paper, we construct a quantization functor, associating a complex vector space H(V) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V). The main new technical result is a proof of a stronger form of the Stone-von Neumann property for the Heisenberg group. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group…

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。

玻色子的第三量子化：辛对角化、非厄米哈密顿量与对称性

Third quantization for bosons: symplectic diagonalization, non-Hermitian Hamiltonian, and symmetries

本文提出一种基于单一辛变换的第三量子化方法，用于求解二次玻色子开放系统的刘维算符，并利用非厄米有效哈密顿量分析系统对称性及推导光子电流等可观测量的累积量生成函数。

Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization

We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our a…

Hilbert 双模作为辛对偶对的经典极限

Classical Limits of Hilbert Bimodules as Symplectic Dual Pairs

本文证明严格形变量子化不仅能从辛对偶对构造 Hilbert 双模，还能反向函子地取 Hilbert 双模的经典极限来重构辛对偶对。

Photon-number tomography and fidelity

The scheme of photon-number tomography is discussed in the framework of star-product quantization. The connection of dual quantization scheme and observables is reviewed. The quantizer and dequantizer operators and kernels of star product of tomograms in photon-number tomography scheme and its dual one are presented in explicit form. The fidelity and state purity are discussed in photon{number tomographic scheme, and the expressions for fidelity and purity are obtained in the form of integral of the product of two photon-number tomograms with integral kernel which is presented in explicit form…

Berezin-Toeplitz quantization and the least unsharpness principle

We show that compatible almost-complex structures on symplectic manifolds correspond to optimal quantizations.

Geometric Quantization and Epistemically Restricted Theories: The Continuous Case

It is possible to reproduce the quantum features of quantum states, starting from a classical statistical theory and then limiting the amount of knowledge that an agent can have about an individual system [5, 18].These are so called epistemic restrictions. Such restrictions have been recently formulated in terms of the symplectic geometry of the corresponding classical theory [19]. The purpose of this note is to describe, using this symplectic framework, how to obtain a C*-algebraic formulation for the epistemically restricted theories. In the case of continuous variables, following the groupo…