Process reconstruction: From unphysical to physical maps via maximum likelihood

Superfast maximum likelihood reconstruction for quantum tomography

Conventional methods for computing maximum-likelihood estimators (MLE) often converge slowly in practical situations, leading to a search for simplifying methods that rely on additional assumptions for their validity. In this work, we provide a fast and reliable algorithm for maximum likelihood reconstruction that avoids this slow convergence. Our method utilizes the state-of-the-art convex optimization scheme---an accelerated projected-gradient method---that allows one to accommodate the quantum nature of the problem in a different way than in the standard methods. We demonstrate the power of…

Process reconstruction from incomplete and/or inconsistent data

We analyze how an action of a qubit channel (map) can be estimated from the measured data that are incomplete or even inconsistent. That is, we consider situations when measurement statistics is insufficient to determine consistent probability distributions. As a consequence either the estimation (reconstruction) of the channel completely fails or it results in an unphysical channel (i.e., the corresponding map is not completely positive). We present a regularization procedure that allows us to derive physically reasonable estimates (approximations) of quantum channels. We illustrate our proce…

Constructing valid density matrices on an NMR quantum information processor via maximum likelihood estimation

Estimation of quantum states is one of the most important steps in any quantum information processing experiment. A naive reconstruction of the density matrix from experimental measurements can often give density matrices which are not positive, and hence not physically acceptable. How do we ensure that at all stages of reconstruction, we keep the density matrix positive and normalized? Recently a method has been suggested based on maximum likelihood estimation, wherein the density matrix is guaranteed to be positive definite. We experimentally implement this protocol and demonstrate its utili…

Maximum-likelihood method in quantum estimation

The maximum-likelihood method for quantum estimation is reviewed and applied to the reconstruction of density matrix of spin and radiation as well as to the determination of several parameters of interest in quantum optics.

Benchmarking maximum-likelihood state estimation with an entangled two-cavity state

The efficient quantum state reconstruction algorithm described in [P. Six et al., Phys. Rev. A 93, 012109 (2016)] is experimentally implemented on the non-local state of two microwave cavities entangled by a circular Rydberg atom. We use information provided by long sequences of measurements performed by resonant and dispersive probe atoms over time scales involving the system decoherence. Moreover, we benefit from the consolidation, in the same reconstruction, of different measurement protocols providing complementary information. Finally, we obtain realistic error bars for the matrix element…

Reconstruction of the Real Quantum Channel via Convex Optimization

Quantum process tomography is often used to completely characterize an unknown quantum process. However, it may lead to an unphysical process matrix, which will cause the loss of information respect to the tomography result. Convex optimization, widely used in machine learning, is able to generate a global optimal model that best fits the raw data while keeping the process tomography in a legitimate region. Only by correctly revealing the original action of the process can we seek deeper into its properties like its phase transition and its Hamiltonian. Thus, we reconstruct the real quantum ch…

Reconstruction of the spin state

System of 1/2 spin particles is observed repeatedly using Stern-Gerlach apparatuses with rotated orientations. Synthesis of such non-commuting observables is analyzed using maximum likelihood estimation as an example of quantum state reconstruction. Repeated incompatible observations represent a new generalized measurement. This idealized scheme will serve for analysis of future experiments in neutron and quantum optics.

基于凸优化的量子态与量子过程的真实实验重构

True experimental reconstruction of quantum states and processes via convex optimization

针对传统量子态与过程层析的线性反演易产生非物理矩阵的问题，提出基于半定规划求解的约束凸优化方法，在两比特NMR系统中实现了更高保真度的物理有效量子态、过程及噪声表征。

Crystallizing highly-likely subspaces that contain an unknown quantum state of light

In continuous-variable tomography, with finite data and limited computation resources, reconstruction of a quantum state of light is performed on a finite-dimensional subspace. No systematic method was ever developed to assign such a reconstruction subspace---only ad hoc methods that rely on hard-to-certify assumptions about the source and strategies. We provide a straightforward and numerically feasible procedure to uniquely determine the appropriate reconstruction subspace for any given unknown quantum state of light and measurement scheme. This procedure makes use of the celebrated statisti…

自由电子量子态的层析重建

Tomographic reconstruction of free-electron quantum states

该研究提出最大似然估计、贝叶斯反演和深度学习三种算法重建快速电子量子态，应用于阿秒电子脉冲序列数据提取密度矩阵，得到约245阿秒的脉冲持续时间并预测36%的相干度。

Image processing as state reconstruction in optics

The image reconstruction of partially coherent light is interpreted as the quantum state reconstruction. The efficient method based on maximum-likelihood estimation is proposed to acquire information from registered intensity measurements affected by noise. The connection with totally incoherent image restoration is pointed out. The feasibility of the method is demonstrated numerically. Spatial and correlation details significantly smaller than the diffraction limit are revealed in the reconstructed pattern.

Quantum phase transitions in nonhermitian harmonic oscillator

The Stone theorem requires that in a physical Hilbert space ${\cal H}$ the time-evolution of a stable quantum system is unitary if and only if the corresponding Hamiltonian $H$ is self-adjoint. Sometimes, a simpler picture of the evolution may be constructed in a manifestly unphysical Hilbert space ${\cal K}$ in which $H$ is nonhermitian but ${\cal PT}-$symmetric. In applications, unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary reconstruction of the physical Hilbert space ${\cal H}$. For a ${\cal PT}-$symmetric version of the spik…

量子纠错中探测器误差模型的重建

Reconstruction of detector error model for quantum error correction

提出基于相关性分析的超图重建算法，通过精确代数相关方程和自顶向下并发剪枝策略，从实验综合征统计中重建离散物理超图，无假阳性，并揭示密集码中连续参数提取的方差级联现象。

Maximum-Likelihood Quantum State Tomography by Cover's Method with Non-Asymptotic Analysis

We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an $O ( ( 1 / k ) \log D )$ rate, where $k$ denotes the number of iterations and $D$ denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is $O ( D ^ 3 + N D ^2 )$, where $N$ denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the $R ρR$ method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homod…