G. M. D'Ariano, L. Maccone, M. Paini
We propose a tomographic reconstruction scheme for spin states. The experimental setup, which is a modification of the Stern-Gerlach scheme, can be easily performed with currently available technology. The method is generalized to multi-particle states, analyzing the spin 1/2 case for indistinguishable particles. Some Monte Carlo numerical simulations are given to illustrate the technique.
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Tomographic probability representation of multimode electromagnetic field states in the scheme of center-of-mass tomography is reviewed. Both connection of the field state Wigner function and observable Weyl symbols with the center-of-mass tomograms as well as connection of Grönewold kernel with the center-of-mass tomographic kernel determining the noncommutative product of the tomograms are obtained. The dual center-of-mass tomogram of the photon states are constructed and the dual tomographic kernel is obtained. The models of other generalised center-of-mass tomographies are discussed. Examp…
We present a tomographic scheme, based on spacetime symmetries, for the reconstruction of the internal degrees of freedom of a Dirac spinor. We discuss the circumstances under which the tomographic group can be taken as SU(2), and how this crucially depends on the choice of the gamma matrix representation. A tomographic reconstruction process based on discrete rotations is considered, as well as a continuous alternative.
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.
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.
We investigate an error mitigated tomographic approach to the quantum circuit cutting problem in the presence of gate and measurement noise. We explore two tomography specific error mitigation techniques; readout error mitigated conditional fragment tomography, which uses knowledge of readout errors on all cut and conditional qubit measurements in the tomography reconstruction procedure; and dominant eigenvalue truncation (DEVT), which aims to improve the performance of circuit cutting by performing truncation of the individual conditional tomography fragments used in the reconstruction. We fi…
Tomographic Limits of the Petz Recovery Map
该研究探讨Petz恢复映射用于量子态断层扫描的可能性，分析直接迭代的局限性，提出通过经典分布扩展的Petz构造可恢复贝叶斯和最大似然断层扫描结构，为相关应用提供视角。
New type of tomographic probability distribution, which contains complete information on the density matrix (wave function) related to the Fresnel transform of the complex wave function, is introduced. Relation to symplectic tomographic probability distribution is elucidated. Multimode generalization of the Fresnel tomography is presented. Examples of applications of the present approach are given.
We present a protocol for quantum cryptography in which the data obtained for mismatched bases are used in full for the purpose of quantum state tomography. Eavesdropping on the quantum channel is seriously impeded by requiring that the outcome of the tomography is consistent with unbiased noise in the channel. We study the incoherent eavesdropping attacks that are still permissible and establish under which conditions a secure cryptographic key can be generated. The whole analysis is carried out for channels that transmit quantum systems of any finite dimension.
We propose an approach to reconstruct any superconducting charge qubit state by using quantum state tomography. This procedure requires a series of measurements on a large enough number of identically prepared copies of the quantum system. The experimental feasibility of this procedure is explained and the time scales for different quantum operations are estimated according to experimentally accessible parameters. Based on the state tomography, we also investigate the possibility of the process tomography.
Quantum Supremacy in Tomographic Imaging: Advances in Quantum Tomography Algorithms
本研究通过减少投影角度并增强抗伪影鲁棒性，验证了量子优化算法在断层重建中超越经典方法的量子霸权，实现了在50%投影角度和50%正弦图误差下仍能精确重建无伪影图像。
We present a tomographic approach to the study of quantum nonlocality in multipartite systems. Bell inequalities for tomograms belonging to a generic tomographic scheme are derived by exploiting tools from convex geometry. Then, possible violations of these inequalities are discussed in specific tomographic realizations providing some explicit examples.
We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction, but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a st…
Recently quantum tomography has been proposed as a fundamental tool for prototyping a few qubit quantum device. It allows the complete reconstruction of the state produced from a given input into the device. From this reconstructed density matrix, relevant quantum information quantities such as the degree of entanglement and entropy can be calculated. Generally orthogonal measurements have been discussed for this tomographic reconstruction. In this paper, we extend the tomographic reconstruction technique to two new regimes. First we show how non-orthogonal measurement allow the reconstruction…
We study the possibility of performing quantum state tomography via equidistant states. This class of states allows us to propose a non-symmetric informationally complete POVM based tomographic scheme. The scheme is defined for odd dimensions and involves an inversion which can be analytically carried out by Fourier transform.
A tomographic technique is introduced in order to determine the quantum state of the center of mass motion of neutrons. An experiment is proposed and numerically analyzed.
The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance, and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography er…
We have developed a general method for constructing a set of non-orthogonal bases with equal separations between all different basis' states in prime dimensions.It results that the corresponding bi-orthogonal counterparts are pairwise unbiased with the components of the original bases. Using these bases we derive an explicit expression for the optimal tomography in non-orthogonal bases. Special two dimensional case is analyzed separately.
Displaced Fock states of the electromagnetic field have been synthesized by overlapping the pulsed optical single-photon Fock state |1> with coherent states on a high-reflection beamsplitter and completely characterized by means of quantum homodyne tomography. The reconstruction reveals highly non-classical properties of displaced Fock states, such as negativity of the Wigner function and photon number oscillations. This is the first time complete tomographic reconstruction has been performed on a highly non-classical optical 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 propose and analyze quantum state estimation (tomography) using continuous quantum measurements with resource limitations, allowing the global state of many qubits to be constructed from only measuring a few. We give a proof-of-principle investigation demonstrating successful tomographic reconstruction of an arbitrary initial quantum state for three different situations: single qubit, remote qubit, and two interacting qubits. The tomographic reconstruction utilizes only a continuous weak probe of a single qubit observable, a fixed coupling Hamiltonian, together with single-qubit controls. I…
We propose to experimentally test the nonclassicality of quantum states through homodyne tomography. For single-mode states we check violations of inequalities involving the photon-number probability. For two-mode states we test the nonclassicality by reconstructing some suitable number-operator functions. The test can be performed with available quantum efficiency of homodyne detection, by measuring the pertaining quantities on the corresponding noisy states.
Random-phase homodyne tomography of the field intensity is a concrete example of the Quantum Roulette of Helstrom. In this paper we give the explicit POM of such measurement and compare it with direct photodetection and heterodyne detection. Effects of nonunit quantum efficiency are also considered. Naimark extensions for the roulette POM are analyzed and its experimental realization is discussed.
Conventional tomographic techniques are becoming increasingly infeasible for reconstructing the operators of quantum devices of growing sophistication. We describe a novel tomographic procedure using coherent states which begins by reconstructing the diagonals of the operator, and then each successive off-diagonal in a recursive manner. Each recursion is considerably more efficient than reconstructing the operator in its entirety, and each successive recursion involves fewer parameters. We apply our technique to reconstruct the positive-operator-valued measure (POVM) corresponding to a recentl…
Quantum tomography of Rydberg atom graphs by configurable ancillas
针对传统量子态层析需精确单量子比特控制的实验难题，提出用相互作用可调的可配置辅助粒子生成层析基测量的方法，并在里德堡原子阵列实验中验证了其高保真全量子态重构能力。
We investigate how decoherence affects the entanglement established between two quantum dots in micro cavities, and propose a tomographic scheme able to measure the entangled state. The scheme we consider establishes the entanglement via the exchange and measurement of a photon. Making the realistic assumption of noise dominated by pure-dephasing processes, we find that the photon must be exchanged and measured on timescales shorter than the quantum dots' characteristic dephasing time for appreciable levels of entanglement to be achieved. The tomographic scheme is able to reconstruct the full …
We formulate the notion of quantum channels in the framework of quantum tomography and address there the issue of whether such maps can be regarded as classical stochastic maps. In particular kernels of maps acting on probability representation of quantum states are derived for qubit and bosonic systems. In the latter case it results that a single mode Gaussian quantum channel corresponds to non-Gaussian classical channels.
We present an error analysis of various tomographic protocols based on the linear inversion for the reconstruction of an unknown two-qubit state. We solve the problem of finding a tomographic protocol which is the most robust against errors in terms of the lowest value (i.e., equal to 1) of a condition number, as required by the Gastinel-Kahan theorem. In contrast, standard tomographic protocols, including those based on mutually unbiased bases, are nonoptimal for determining all 16 elements of an unknown two-qubit density matrix. Our method is based on the measurements of the 16 generalized P…
Quantum information protocols often rely on tomographic techniques to determine the state of the system. A popular method of encoding information is on the different paths a photon may take, for example, parallel waveguides in integrated optics. However, reconstruction of states encoded onto a large number of paths is often prohibitively resource intensive and requires complicated experimental setups. Addressing this, we present a simple method for determining the state of a photon in a superposition of d paths using a rotating one-dimensional optical Fourier Transform. We establish the theory…
We introduce the Singapore protocol, a qubit protocol for quantum key distribution that is fully tomographic, more efficient than other tomographic protocols, and very robust. Under ideal circumstances the efficiency is log_2(4/3)=0.415 key bits per qubit sent. This is 25% more than the efficiency of 1/3=0.333 for the standard six-state protocol, which sets the benchmark. We describe a simple two-way communication scheme that extracts 0.4 key bits per qubit and thus gets close to the information-theoretical limit. The noise thresholds that we report for a hierarchy of eavesdropping attacks dem…
Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried in a number of different settings on a fixed experimental setup. The collected data is often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how should one optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standa…
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 introduce a self-learning tomographic technique in which the experiment guides itself to an estimate of its own state. Self-guided quantum tomography (SGQT) uses measurements to directly test hypotheses in an iterative algorithm which converges to the true state. We demonstrate through simulation on many qubits that SGQT is a more efficient and robust alternative to the usual paradigm of taking a large amount of informationally complete data and solving the inverse problem of post-processed state estimation.
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…
We show the feasibility of a tomographic reconstruction of Schrödinger cat states generated according to the scheme proposed by S. Song, C.M. Caves and B. Yurke [Phys. Rev. A 41, 5261 (1990)]. We present a technique that tolerates realistic values for quantum efficiency at photodetectors. The measurement can be achieved by a standard experimental setup.
An adaptive optimization technique to improve precision of quantum homodyne tomography is presented. The method is based on the existence of so-called null functions, which have zero average for arbitrary state of radiation. Addition of null functions to the tomographic kernels does not affect their mean values, but changes statistical errors, which can then be reduced by an optimization method that "adapts" kernels to homodyne data. Applications to tomography of the density matrix and other relevant field-observables are studied in detail.
We have performed the first experimental tomographic reconstruction of a three-photon polarization state. Quantum state tomography is a powerful tool for fully describing the density matrix of a quantum system. We measured 64 three-photon polarization correlations and used a "maximum-likelihood" reconstruction method to reconstruct the GHZ state. The entanglement class has been characterized using an entanglement witness operator and the maximum predicted values for the Mermin inequality was extracted.
We show that data from homodyne-like detection based on photon-number-resolving (PNR) detectors may be effectively exploited to reconstruct quantum states of light using the tomographic reconstruction techniques originally developed for homodyne detection based on photodiodes. Our results open new perspectives to quantum state reconstruction, and pave the way to the use of PNR-based homodyne-like detectors in quantum information science.
Pulsed homodyne quantum tomography usually requires a high detection efficiency limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency ($<50\%$) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian type. This result is obtained by applying the so-called "minimax" adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our…
The tomographic description of a quantum state is formulated in an abstract infinite dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.
We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schrödinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position a…
Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme …
It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrodinger-Dirac picture of quantum mechanics on Hilbert spaces. In this picture states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture,which has evolved in the $C^\star-$algebraic approach to quantum mechanics, starts with the algebra of observables and introduce states as a derived concept. The equivalence between these two pictures amounts essentially, to the Gelfand-N…
Using tomographic reconstruction we determine the complete internuclear quantum state, represented by the Wigner function, of a dissociating I2 molecule based on femtosecond time resolved position and momentum distributions of the atomic fragments. The experimental data are recorded by timed ionization of the photofragments with an intense 20 fs laser pulse. Our reconstruction method, which relies on Jaynes' maximum entropy principle, will also be applicable to time resolved position or momentum data obtained with other experimental techniques.
Demonstration of Fourier-domain Quantum Optical Coherence Tomography for a fast tomographic quantum imaging
本研究实验演示了傅里叶域量子光学相干层析成像（Q-OCT）结构，提出两种联合光谱预处理算法补偿分辨率下降效应，验证其在多指标上的优势，推动Q-OCT向实用化方向发展。
Quantum tomography of the superfluid-insulator transition for a mesoscopic atomtronic ring
本文提出用相空间层析半经典方法分析有限尺寸Bose-Hubbard电路的超级流体-绝缘体转变，揭示量子相变处的混沌指纹，并通过Higgs测度表征多体本征态涨落及Goldstone与Higgs带的形成。
Muon spin rotation/relaxation/resonance (MuSR) technique for studying matter structures is considered by means of a recently introduced probability representation of quantum spin states. A relation between experimental MuSR histograms and muon spin tomograms is established. Time evolution of muonium, anomalous muonium, and a muonium-like system is studied in the tomographic representation. Entanglement phenomenon of a bipartite muon-electron system is investigated via tomographic analogues of Bell number and positive partial transpose (PPT) criterion. Reconstruction of the muon-electron spin s…
We propose a method for the tomographic reconstruction of qubit states for a general class of solid state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, $XXZ$-, or XY- type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data.
I investigate the extent to which the description of quantum systems by Gibbs states can be justified purely on the basis of tomographic data, without recourse to theoretical concepts such as infinite ensembles, environments, information, or to the systems' dynamics. I show that the use of Gibbs states amounts to a relevance hypothesis, which I spell out in detail. This hypothesis can be subjected to statistical hypothesis testing and hence assessed on the basis of the experimental data.
Refined sample complexities from the tomographic rate function
本文建立协变量子态层析协议分析框架，推导多种协议的速率函数并证明其严格排序，进而得出Wasserstein型样本复杂度的严格不等式，表明速率函数是比样本复杂度更精细的性能指标。
All the $n(2n+3)$ mean and covariance parameters of an $n$-mode Gaussian states are expressed in terms of the expectation values of the same number of conjugates of the total number observable. This permits a complete tomography of the state. The same is applied to outputs of a Gaussian channel corresponding to selected coherent states to perform the complete tomography of the channel. This leads to some interesting problems concerning the distribution of the number operator and also tomographic complexity.
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