The research presented in this article concerns the stroboscopic approach to quantum tomography, which is an area of science where quantum Physics and linear algebra overlap. In this article we introduce the algebraic structure of the parametric-dependent quantum channels for 2-level and 3-level systems such that the generator of evolution corresponding with the Kraus operators has no degenerate eigenvalues. In such cases the index of cyclicity of the generator is equal $1$, which physically means that there exists one observable the measurement of which performed sufficient number of times at distinct instants provides enough data to reconstruct the initial density matrix and, consequently, the trajectory of the state. Necessary conditions for the parameters and relations between them are introduced. The results presented in this paper seem to have considerable potential applications in experiments due to the fact that one can perform quantum tomography by conducting only one kind of measurement. Therefore, the analyzed evolution models can be considered optimal in the context of quantum tomography. Finally, we also introduce some remarks concerning optimal evolution models in case of $n-$dimensional Hilbert space.
In this paper we analyze selected evolution models of $N-$level open quantum systems in order to find the minimal number of observables (Hermitian operators) such that their expectation values at some time instants determine the accurate representation of the quantum system. The assumption that lies at the foundation of this approach to quantum tomography claims that time evolution of an open quantum system can be expressed by the Kossakowski - Lindblad equation of the form $\dotρ = \mathbb{L} ρ$, which is the most general type of Markovian and time-homogeneous master equation which preserves …
In this study the determinant of the average quadratic error matrix is used as the measure of state estimation efficiency. This quantity is easily computable in some cases, so it gives us a reasonable tool to find optimal measurement setup for different quantum tomography problems. We present numerous applications for a single qubit when von Neumann measurements or a single POVM is used and a part of the parameters of the state is given. Under some restriction the optimality is found for $n$-level system as well. The optimal measurements have some complementary relation to each other or to the…
We present the first complete optimization of quantum tomography, for states, POVMs, and various classes of transformations, for arbitrary prior ensemble and arbitrary representation, giving corresponding feasible experimental schemes.
We develop a quantum process tomography method, which variationally reconstruct the map of a process, using noisy and incomplete information about the dynamics. The new method encompasses the most common quantum process tomography schemes. It is based on the variational quantum tomography method (VQT) proposed by Maciel et al. in arXiv:1001.1793[quant-ph].
Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design optimal probe states for detector estimation based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the minimum UMSE and the minimum condition number for quantum detectors and provide concrete examples that can achieve optimal detector tomography. In order to enhance the estimation precision, we also propose a two-step adaptive detector tomography algorithm to optimize the probe states adapt…
Practical quantum state tomography is usually performed by carrying out repeated measurements on many copies of a given state. The accuracy of the reconstruction depends strongly on the dimensionality of the system and the number of copies used for the measurements. We investigate the accuracy of an experimental implementation of a minimal and optimal tomography scheme for one- and two-qubit states encoded in the polarization of photons. A suitable statistical model for the attainable accuracy is introduced.
In this brief article we indicate a connection between Jordan normal form of a square matrix and the stroboscopic approach to quantum tomography. We show that the index of cyclicy of a generator of evolution, which receives much attention in the stroboscopic tomography, can be defined and computed by referring to the Jordan decomposition of a square matrix. The result presented in this work shows a relation between terminology from quantum tomography and linear algebra.
We develop a practical quantum tomography protocol and implement measurements of pure states of ququarts realized with polarization states of photon pairs (biphotons). The method is based on an optimal choice of the measuring scheme's parameters that provides better quality of reconstruction for the fixed set of statistical data. A high accuracy of the state reconstruction (above 0.99) indicates that developed methodology is adequate.
The present short review article illustrates the latest theoretical developments on quantum tomography, regarding general optimization methods for both data-processing and setup. The basic theoretical tool is the informationally complete measurement. The optimization theory for the setup is based on the new theoretical approach of quantum combs.
In this article we propose a dynamic quantum tomography model for open quantum systems with evolution given by phase-damping channels. Mathematically, these channels correspond to completely positive trace-preserving maps defined by the Hadamard product of the initial density matrix with a time-dependent matrix which carries the knowledge about the evolution. Physically, there is a strong motivation for considering this kind of evolution because such channels appear naturally in the theory of open quantum systems. The main idea behind a dynamic approach to quantum tomography claims that by per…
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
Given an experimental set-up and a fixed number of measurements, how should one take data in order to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [arXiv:quant-ph/0411093v1]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of 'minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias.…
Sample Optimal and Memory Efficient Quantum State Tomography
National University of Singapore（新加坡国立大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；University of Copenhagen（哥本哈根大学）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）；Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）
In this article we propose a dynamic quantum state tomography model for qutrits subject to laser cooling. We prove that one can reduce the number of distinct measurement setups required for state reconstruction by employing the stroboscopic approach. The results are in line with current advances in quantum tomography where there is a strong tendency to investigate the optimal criteria for state reconstruction. We believe that the stroboscopic approach can be considered an efficient tool for density matrix identification since it allows to determine the minimal number of distinct observables ne…
Quantum tomography is an essential method of the photonic technology toolbox and is routinely used for evaluation of experimentally prepared states of light and characterization of devices transforming such states. The tomography procedure consists of many different sequentially performed measurements. We present considerable tomography speedup by optimally arranging the individual constituent measurements, which is equivalent to solving an instance of the traveling salesman problem. As an example, we obtain solutions for photonic systems of up to five qubits, and conclude that already for sys…
Mutually unbiased bases determine an optimal set of measurements to extract complete information about the quantum state of a system. However, quite often a priori information about the state exist, making some of the measurement bases superfluous. This is, for example, the case when multiqubit states belong to the permutationally invariant subspace. In this paper we derive the minimal sets of mutually unbiased bases needed to tomographically reconstruct such states.
Balanced homodyning, heterodyning and unbalanced homodyning are the three well-known sampling techniques used in quantum optics to characterize all possible photonic sources in continuous-variable quantum information theory. We show that for all quantum states and all observable-parameter tomography schemes, which includes the reconstructions of arbitrary operator moments and phase-space quasi-distributions, localized sampling with unbalanced homodyning is always tomographically more powerful (gives more accurate estimators) than delocalized sampling with heterodyning. The latter is recently k…
A minimal set of measurement operators for quantum state tomography has in the non-degenerate case ideally eigenbases which are mutually unbiased. This is different for the degenerate case. Here, we consider the situation where the measurement operators are projections on individual pure quantum states. This corresponds to maximal degeneracy. We present numerically optimized sets of projectors and find that they significantly outperform those which are taken from a set of mutually unbiased bases.
It is a well-known fact that the optimal POVM for quantum state tomography is the symmetric, informationally complete, positive operator valued measure (SIC-POVM). We investigate the same problem only in the case when there are some a priori information about the state, specifically when some parameters are known. In this paper we mainly focus on solving a 3-dimensional optimization problem, which gives us a non-trivial example for the so-called conditional SIC- POVMs, a straightforward generalization of the concept of SIC-POVMs. We also present other special cases to show further applications…
We show that weighted unitary 2-designs define optimal measurements on the system-ancilla output state for ancilla-assisted process tomography of unital quantum channels. Examples include complete sets of mutually unbiased unitary-operator bases. Each of these specifies a minimal series of optimal orthogonal measurements. General quantum channels are also considered.
Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable tradeoffs inherent to quantum tomography.
A process tomography based optimization scheme for open quantum systems is used to determine the performance limits of Josephson charge qubits within current experimental means. The qubit is modeled microscopically as an open quantum system taking into account state leakage, as well as environment-induced dephasing based on experimental noise spectra. Within time-optimal control theory, we show that the competing requirements for suppression of state leakage and dephasing can be met by an external control of the effective qubit-environment interaction, yielding minimal gate fidelity losses of …
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 …
Quantum process tomography provides a means of measuring the evolution operator for a system at a fixed measurement time $t$. The problem of using that tomographic snapshot to predict the evolution operator at other times is generally ill-posed since there are, in general, infinitely many distinct and compatible solutions. We describe the prediction, in some ``maximal ignorance'' sense, of the evolution of a quantum system based on knowledge only of the evolution operator for finitely many times $0<τ_{1}<\dots<τ_{M}$ with $M\geq 1$. To resolve the ill-posedness problem, we construct this predi…
We report on the experimental implementation of a polarimeter based on a scheme known to be optimal for obtaining the polarization vector of ensembles of spin-1/2 quantum systems, and the alignment procedure for this polarimeter is discussed. We also show how to use this polarimeter to estimate the polarization state for identically prepared ensembles of single photons and photon pairs and extend the method to obtain the density matrix for generic multi-photon states. State reconstruction and performance of the polarimeter is illustrated by actual measurements on identically prepared ensembles…
The article establishes a framework for dynamic generation of informationally complete POVMs in quantum state tomography. Assuming that the evolution of a quantum system is given by a dynamical map in the Kraus representation, one can switch to the Heisenberg picture and define the measurements in the time domain. Consequently, starting with an incomplete set of positive operators, one can obtain sufficient information for quantum state reconstruction by multiple measurements. The framework has been demonstrated on qubits and qutrits. For some types of dynamical maps, it suffices to initially …
We argue that the claimed optimality of a new process tomography method suggested in [quant-ph/0601033] and [quant-ph/0601034] is based on not completely fair comparison that does not take into account the available information in an equal way. We also argue that the method is not a new process tomography scheme, but rather represents an interesting modification of ancilla assisted process tomography method. In our opinion these modifications require deeper understanding and further investigation.
In this paper, we derive analytic expressions for the starting (initial) values of the parameters of the T-matrix that is frequently employed in the construction of a theoretical density matrix in a Maximum Likelihood Estimate (MLE) procedure; this optimization procedure is used to fit the experimental data pertaining to single qubit state tomography. Appropriate starting values are critical in achieving a global minimum in the fitting process. We also indicate an analytic way of making the experimentally determined density matrix physical without resorting to the MLE process, if optimality in…
We give bounds on the average fidelity achievable by any quantum state estimator, which is arguably the most prominently used figure of merit in quantum state tomography. Moreover, these bounds can be computed online---that is, while the experiment is running. We show numerically that these bounds are quite tight for relevant distributions of density matrices. We also show that the Bayesian mean estimator is ideal in the sense of performing close to the bound without requiring optimization. Our results hold for all finite dimensional quantum systems.
Unified formalism and adaptive algorithms for optimal quantum state, detector and process tomography
School of Engineering, Australian National University（澳大利亚国立大学工程学院）；Laboratory of Quantum Information, University of Science and Technology of China（中国科学技术大学量子信息实验室）；CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China（中国科学技术大学中科院量子信息与量子物理卓越创新中心）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；University of Michigan–Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University（上海交通大学密西根学院）
A number of problems in quantum state and system identification are addressed. Specifically, it is shown that the maximum likelihood estimation (MLE) approach, already known to apply to quantum state tomography, is also applicable to quantum process tomography (estimating the Kraus operator sum representation (OSR)), Hamiltonian parameter estimation, and the related problems of state and process (OSR) distribution estimation. Except for Hamiltonian parameter estimation, the other MLE problems are formally of the same type of convex optimization problem and therefore can be solved very efficien…
The efficiency of quantum state tomography is discussed from the point of view of quantum parameter estimation theory, in which the trace of the weighted covariance is to be minimized. It is shown that tomography is optimal only when a special weight is adopted.
We present theoretical and experimental results on high-fidelity transfer of a trapped Bose-Einstein condensate into its first vibrationally excited eigenstate. The excitation is driven by mechanical motion of the trap, along a trajectory obtained from optimal control theory. Excellent agreement between theory and experiment is found over a large range of parameters. We develop an approximate model to map the dynamics of the many-body condensate wave function to a driven two-level system.
Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition
University of Pennsylvania（宾夕法尼亚大学）；Scuola Normale Superiore（意大利高等师范学院）；University Mohammed VI Polytechnic（穆罕默德六世_polytechnic_大学）；Stony Brook University（石溪大学）；Griffith University（格里菲斯大学）
In the paper we discuss possible applications of the so-called stroboscopic tomography (stroboscopic observability) to selected decoherence models of 2-level quantum systems. The main assumption behind our reasoning claims that the time evolution of the analyzed system is given by a master equation of the form $\dotρ = \mathbb{L} ρ$ and the macroscopic information about the system is provided by the mean values $m_i (t_j) = Tr(Q_i ρ(t_j))$ of certain observables $\{Q_i\}_{i=1} ^r $ measured at different time instants $\{t_j\}_{j=1}^p$. The goal of the stroboscopic tomography is to establish th…
A general law is presented for (composite) quantum systems which directly describes the time evolution of quantum states (with one or both components) through an arbitrary noisy quantum channel. It is shown that the time evolution of all quantum states through a quantum channel can be completely captured by the evolution of a single 'probe state'. Thus in order to grasp the information of the final output states subject to a quantum channel, especially an unknown one, it only requires quantum state tomography of a single probe state, which dramatically simplifies the practical operations in ex…
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…
Optimal tomography of bosonic and fermionic Gaussian states
Inria, Télécom Paris – LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎高等电信学院-通信与图像实验室，巴黎理工学院）；Scuola Normale Superiore（比萨高等师范学院）；LMU Munich & Munich Center for Quantum Science and Technology (MCQST), University of Amsterdam & QuSoft（慕尼黑大学及慕尼黑量子科学与技术中心，阿姆斯特丹大学及QuSoft）；CWI & QuSoft（荷兰数学与计算机科学研究中心及QuSoft）
Quantum state tomography (QST) represents an essential tool for the characterization, verification, and validation (QCVV) of quantum processors. Only for a few idealized scenarios, there are analytic results for the optimal measurement set for QST. E.g., in a setting of non-degenerate measurements, an optimal minimal set of measurement operators for QST has eigenbases which are mutually unbiased. However, in other set-ups, dependent on the rank of the projection operators and the size of the quantum system, the optimal choice of measurements for efficient QST needs to be numerically approximat…
