S. Mancini, V. I. Man'ko, P. Tombesi
By using a generalization of the optical tomography technique we describe the dynamics of a quantum system in terms of equations for a purely classical probability distribution which contains complete information about the system.
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Quantum tomography for continuous variables is based on the symplectic transformation group acting in the phase space. A particular case of symplectic tomography is optical tomography related to the action of a special orthogonal group. In the tomographic description of spin states, the connection between special unitary and special orthogonal groups is used. We analyze the representation for spin tomography using the Cayley-Klein parameters and discuss an analog of symplectic tomography for discrete variables. We propose a representation for tomograms of discrete variables through quaternions…
Squeezed and rotated quadrature of an ion in a Paul trap is discussed in connection with reconstructing its quantum state using symplectic-tomography method. Marginal distributions of the quadrature for squeezed and correlated states and for nonlinear coherent states of a trapped ion are obtained and the density matrices in the Fock basis are expressed explicitly in terms of these marginal distributions.
The marginal distribution of squeezed and rotated quadrature for two types of nonclassical states of trapped ion -- for squeezed and correlated states and for squeezed even and odd coherent states (squeezed Schrödinger cat states) is studied. The obtained marginal distribution for the two types of states is shown to satisfy classical dynamical equation equivalent to standard quantum evolution equation for density matrix (wave function) derived in symplectic tomography scheme.
A review of the symplectic tomographic approaches within the framework of star-product quantization is presented. The classical statistical mechanics within the framework of the tomographic representation is considered. The kernels of star-product of functions - symbols of operators in classical and quantum mechanics are presented.
The dynamical equation of quantum mechanics are rewritten in form of dynamical equations for the measurable, positive marginal distribution of the shifted, rotated and squeezed quadrature introduced in the so called "symplectic tomography". Then the possibility of a purely classical description of a quantum system as well as a reinterpretation of the quantum measurement theory is discussed and a comparision with the well known quasi-probabilities approach is given. Furthermore, an analysis of the properties of this marginal distribution, which contains all the quantum information, is performed…
The marginal distribution of squeezed, rotated and shifted quadrature for two types of nonclassical states of a trapped ion - squeezed correlated states and squeezed even and odd coherent states (squeezed Schrodinger cat states) - is studied.
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。
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…
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…
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.
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.
A possibility of describing two-level atom states in terms of positive probability distributions (analog to the symplectic tomography scheme) is considered. As a result the basis of the irreducible representation of a rotation group can be realized by a family of the probability distributions of the spin projection parametrized by points on the sphere. Furthermore the tomography of rotational states of molecules and nuclei which can be described by the model of a symmetric top is discussed.
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。
Tomography on f-oscillators
该研究围绕f-振子展开，回顾辛层析成像技术，介绍非线性f-振子概念性质，探讨其相干态的层析概率表示与纠缠问题，并通过线性熵评估奇偶f-相干态的纠缠情况。
The mechanism of describing quantum states by standard probability (tomographic one) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of the identity decompositions in the Hilbert space of hermitian linear operators, with trace formula as scalar product of operators. Decompositions of identity are considered with respect to over-complete families of projectors labeled by extra parameters and containing a measure, depending on these parameters. It plays the role of a Gram-Schmidt orthonormalization kernel…
We propose a quantum process tomography scheme that utilizes two-mode squeezed vacuum to realize the parameter estimation with Heisenberg scaling. The objective is to estimate a rotating angle of polarization and parity detection is used as the detection strategy. With the help of symplectic matrix theory, we discuss the estimation visibility and sensitivity of output signal in lossless situation, the quantum Fisher information is also given via calculation. Finally, the impacts of two realistic factors on both visibility and sensitivity are also considered, including photon loss during the in…
Invertible maps from operators of quantum obvservables onto functions of c-number arguments and their associative products are first assessed. Different types of maps like Weyl-Wigner-Stratonovich map and s-ordered quasidistribution are discussed. The recently introduced symplectic tomography map of observables (tomograms) related to the Heisenberg-Weyl group is shown to belong to the standard framework of the maps from quantum observables onto the c-number functions. The star-product for symbols of the quantum-observable for each one of the maps (including the tomographic map) and explicit re…
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。
We consider a set of N linearly coupled harmonic oscillators and show that the diagonalization of this problem can be put in geometrical terms. The matrix techniques developed here allowed for solutions in both the classical and quantum regimes.
Classical Shadow Tomography with Mutually Unbiased Bases
本文研究将互不偏基（MUBs）电路用于经典阴影层析，发现一般可观测量方差指数增长，而AMA类可观测量的方差为多项式，且通过有偏采样可使非AMA可观测量在稀疏条件下方差降为多项式，并与Clifford电路进行了比较。
A review of the photon-number tomography and symplectic tomography as examples of star-product quantization is presented. The classical statistical mechanics is considered within the framework of the tomographic representation.
Bootstrapping Classical Shadows for Neural Quantum State Tomography
研究提出使用自回归神经量子态作为经典阴影层析的拟设，通过新的交叉熵损失估计器和重要性采样策略，克服传统方法在预测高权重和非线性可观测对象时的弱点，实现GHZ态稳定重构和混合态纯化。
Quantum-optical coherence tomography (Q-OCT) is an interferometric technique for axial imaging offering several advantages over conventional methods. Chirped-pulse interferometry (CPI) was recently demonstrated to exhibit all of the benefits of the quantum interferometer upon which Q-OCT is based. Here we use CPI to measure axial interferograms to profile a sample accruing the important benefits of Q-OCT, including automatic dispersion cancellation, but with 10 million times higher signal. Our technique solves the artifact problem in Q-OCT and highlights the power of classical correlation in o…
Quantum process tomography --- a primitive in many quantum information processing tasks --- can be cast within the framework of the theory of design of experiment (DoE), a branch of classical statistics that deals with the relationship between inputs and outputs of an experimental setup. Such a link potentially gives access to the many ideas of the rich subject of classical DoE for use in quantum problems. The classical techniques from DoE cannot, however, be directly applied to the quantum process tomography due to the basic structural differences between the classical and quantum estimation …
High-dimensional entanglement with spatial modes of light promises increased security and information capacity over quantum channels. Unfortunately, entanglement decays due to perturbations, corrupting quantum links which cannot be repaired without a tomography of the channel. Paradoxically, the channel tomography itself is not possible without a working link. Here we overcome this problem with a robust approach to characterising quantum channels by means of classical light. Using free-space communication in a turbulent atmosphere as an example, we show that the state evolution of classically …
We show that QM can be represented as a natural projection of a classical statistical model on the phase space $Ω= H\times H,$ where $H$ is the real Hilbert space. Statistical states are given by Gaussian measures on $Ω$ having zero mean value and dispersion of the Planck magnitude -- fluctuations of the ``vacuum field.'' Physical variables (e.g., energy) are given by maps $f: Ω\to {\bf R}$ (functions of classical fields). The crucial point is that statistical states and variables are symplectically invariant. The conventional quantum representation of our prequantum classical statistical mode…
The composition of the quantum potential and its role in the breakdown of classical symplectic symmetry in quantum mechanics is investigated. General expressions are derived for the quantum potential in both configuration space and momentum space representations. By comparing the configuration space and momentum space representations of the causal interpretation of quantum mechanics, the quantum potential is shown to break the symplectic symmetry that exists between these two representations in classical mechanics. In addition, it is shown that the quantum potential in configuration space may …
Quantum Machine Learning for State Tomography Using Classical Data
本文提出一种基于量子机器学习的量子态重构协议，仅使用经典测量数据在NISQ设备上实现高保真度重构，展示了在不同量子态上的成功应用。
Quantum Optical Coherence Tomography (Q-OCT) presents many advantages over its classical counterpart, Optical Coherence Tomography (OCT): it provides an increased axial resolution and is immune to even orders of dispersion. The core of Q-OCT is quantum interference of negatively correlated entangled photon pairs obtained in a Hong-Ou-Mandel configuration. This two-photon interference can be observed in the time domain in the form of dips or in the Fourier domain by means of a joint spectrum. The latter approach proved to be practical in the sense that it alleviated the requirement posed on lig…
Classical State Detection Using Quantum State Tomography
本文提出利用量子态成像技术检测混入经典态的纠缠光子，通过局部偏振测量诱导经典态，并用于提升量子网络中的经典-量子共存技术。
Quantum optical coherence tomography (Q-OCT) offers a factor-of-two improvement in axial resolution and the advantage of even-order dispersion cancellation when it is compared to conventional OCT (C-OCT). These features have been ascribed to the non-classical nature of the biphoton state employed in the former, as opposed to the classical state used in the latter. Phase-conjugate OCT (PC-OCT), introduced here, shows that non-classical light is not necessary to reap Q-OCT's advantages. PC-OCT uses classical-state signal and reference beams, which have a phase-sensitive cross-correlation, togeth…
Precision Bounds on Continuous-Variable State Tomography using Classical Shadows
该研究将连续变量量子态层析协议纳入经典阴影框架，推导了不同检测协议的测量次数严格界，通过模拟和实验验证并推广至多模场景。
Demonstrating and Benchmarking Classical Shadows for Lindblad Tomography
本研究通过随机化阴影测量方法，提高了超导跨门处理器在五量子位上的林布兰德拓扑学效率，减少了资源消耗并提升了动态学习能力。
To obtain a complete description of a quantum system, one usually employs standard quantum state tomography, which however requires exponential number of measurements to perform and hence is impractical when the system's size grows large. In this work, we introduce a self-learning tomographic scheme based on the variational hybrid quantum-classical method. The key part of the scheme is a learning procedure, in which we learn a control sequence capable of driving the unknown target state coherently to a simple fiducial state, so that the target state can be directly reconstructed by applying th…
We propose a simple scheme to reduce readout errors in experiments on quantum systems with finite number of measurement outcomes. Our method relies on performing classical post-processing which is preceded by Quantum Detector Tomography, i.e., the reconstruction of a Positive-Operator Valued Measure (POVM) describing the given quantum measurement device. If the measurement device is affected only by an invertible classical noise, it is possible to correct the outcome statistics of future experiments performed on the same device. To support the practical applicability of this scheme for near-te…
Annealing approach to quantum tomography is theoretically proposed. First, based on the maximum entropy principle, we introduce classical parameters to combine "quantum models (or quantum states)" given a prior for potentially representing the unknown target state. Then, we formulate the quantum tomography as an optimization problem on the classical parameters, by employing relative entropy of the parametrized state with the target state as the objective function to be minimized. We show that the objective function is physically implementable, in a theoretical sense at least, as an effective H…
We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of "quantum blobs" introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle in its strong Robertson--Schrödinger form. We show that these phase space units can be characterized by a simple condition of reflexivity using polar duality, thus improving previous results. We apply these geometric constructions to the characterization of pure Ga…
Direct reconstruction of the quantum density matrix elements with classical shadow tomography
本研究提出基于经典影子层析的DMP-ST框架，通过随机Clifford或有偏MUB测量实现量子密度矩阵多元素的高效全局估计，在样本复杂度等方面较传统方法实现指数级提升，推动直接层析在高维与近期量子平台的实用化。
Optical homodyne tomography is discussed in the context of classical image processing. Analogies between these two fields are traced and used to formulate an iterative numerical algorithm for reconstructing the Wigner function from homodyne statistics.
We present experimental and theoretical results for the fluctuation properties in the incomplete spectra of quantum systems with symplectic symmetry and a chaotic dynamics in the classical limit. To obtain theoretical predictions, we extend the random-matrix theory (RMT) approach introduced in [O. Bohigas and M. P. Pato, Phys. Rev. E 74, 036212 (2006)] for incomplete spectra of quantum systems with orthogonal symmetry. We validate these RMT predictions by randomly extracting a fraction of levels from complete sequences obtained numerically for quantum graphs and experimentally for microwave ne…
Shadow tomography with noisy readouts
本文提出通过测量前后随机比特翻转的 X-twirling 方法，高效缓解经典阴影层析中的读出噪声，且仅需单个 X 门，无需随机 Clifford 门。
Atoms trapped in optical lattice have long been a system of interest in the AMO community, and in recent years much study has been devoted to both short- and long-range coherence in this system, as well as to its possible applications to quantum information processing. Here we demonstrate for the first time complete determination of the quantum phase space distributions for an ensemble of $^{85}Rb$ atoms in such a lattice, including a negative Wigner function for atoms in an inverted state.
I show that in tomographic experiments measurement of a small set of observables suffices to confirm or incrementally amend prior expectations with a high degree of confidence. To this end I adapt the evidence procedure, an estimation technique used in classical image reconstruction, to use in quantum state tomography.
We show that $Ω(rd/ε)$ copies of an unknown rank-$r$, dimension-$d$ quantum mixed state are necessary in order to learn a classical description with $1 - ε$ fidelity. This improves upon the tomography lower bounds obtained by Haah, et al. and Wright (when closeness is measured with respect to the fidelity function).
The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the last is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of…
Quantum computers solve ever more complex tasks using steadily growing system sizes. Characterizing these quantum systems is vital, yet becoming increasingly challenging. The gold-standard is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge. Measurement and classical computing costs, however, increase exponentially in the system size - a bottleneck given the scale of existing and near-term quantum devices. Here, we demonstrate a scalable and practical QST approach that uses a single measurement setting, namely symmetric informationally com…
The symplectic rank of non-Gaussian quantum states
本文研究了非高斯态的辛秩，这是一种新的非高斯性单调量，具有操作性和资源理论性质。该秩定义为纯态协方差矩阵严格大于真空的辛特征值数目，并具有压缩非高斯性、限制制备复杂度等特性。
We consider quadratic tomography in star product formalism. The contraction and the behavior of the associative algebra of quadratic tomographic symbols in $\hbar\rightarrow 0$ limit are discussed. A simple $k$-deformation example is illustrated.
Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography
针对高维矩阵迹计算经典复杂度过高的问题，提出基于qudit阴影层析成像的量子方法，实现最坏二次、平均指数级加速，降低后处理复杂度，适用于任意维度，为高维数据分析开辟新路径。
Time of flight is an intuitive way to determine the velocity of particles and lies at the heart of many capabilities ranging from mass spectrometry to fluid flow measurements. Here we show time-of-flight imaging can realize tomography of a quantum state of motion of a single trapped atom. Tomography of motion requires studying the phase space spanned by both position and momentum. By combining time-of-flight imaging with coherent evolution of the atom in an optical tweezer trap, we are able to access arbitrary quadratures in phase space without relying on coupling to a spin degree of freedom. …
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。
Quantum-optical coherence tomography (Q-OCT) provides a dispersion-canceled axial-imaging method, but its practical use is limited by the weakness of the light source and by artifacts in the images. A recent study using chirped-pulse interferometry (CPI) has demonstrated dispersion-canceled and artifact-free OCT with a classical system; however, unwanted background signals still remain after removing the artifacts. Here, we propose a classical optical method that realizes dispersion-canceled, artifact-free, and background-free OCT. We employ a time-reversed system for Q-OCT with transform-limi…
We introduce the concepts of Poisson brackets for classical noise, and of canonically conjugate Wiener processes (symplectic noise). Phase space diffusions driven by these processes are considered and the general form of a stochastic process preserving the full (system and noise) Poisson structure is obtained. We show that, once the classical stochastic model is required to preserve the joint system and noise Poisson bracket, it has much in common with quantum markovian models.
We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an extension of Gromov's "principle of the symplectic camel" obtained in collaboration with N. Dias and J. Prata. This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to s…
We study the minimum volume ellipsoid estimator associates to a cloud of points in phase space. Using as a natural measure of uncertainty the symplectic capacity of the covariance ellipsoid we find that classical uncertainties obey relations similar to those found in non-standard quantum mechanics.
Scalable and Flexible Classical Shadow Tomography with Tensor Networks
本文结合局部置乱经典阴影框架与张量网络，提出适用于有限深度局部Clifford随机电路的可扩展经典阴影层析方法，在NISQ设备上以低门开销获得更高样本效率，并在准局部算子预测和保真度估计上优于Pauli测量。
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。
Online Shadow Tomography Matching the Classical Bounds
本研究针对在线影子断层成像问题，提出匹配经典最优率的算法，改进了相关参数界限，关键在于基于量子Efron–Stein分解的测量后损伤量化框架。
Classical Shadow Tomography with Locally Scrambled Quantum Dynamics
该研究将经典阴影层析成像推广至局部随机量子动力学，利用纠缠特征构建无偏估计器并界定样本复杂度，数值证明浅层电路和单次局部哈密顿演化可实现高效近似层析，拓展了近期量子设备的应用。
Tomography of a Macroscopic Quantum State influenced by Classical Self-Gravity
研究经典自引力对宏观机械振子量子态层析成像的影响，基于薛定谔 - 牛顿理论，分析测量记录差异及重建协方差变化，通过赫林格距离量化可区分性，还在非线性量子力学背景下阐述相关问题。
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