In this paper, we focus on alternate forms of the T-matrix used in the Maximum Likelihood Estimate (MLE) procedure for fitting the experimental data collected in quantum state tomography experiments. In particular, we analyze the single quantum state tomography case, deriving in the process three new valid alternate forms for achieving optimality. These alternative forms then serve as a consistency check, thus enhancing the robustness of the MLE fitting process. One form, in particular, serves as a useful compliment to the standard form normally employed. We subsequently provide a generalization of these forms to the case of multiqubit state tomography.
The deployment of intermediate- and large-scale quantum devices necessitates the development of efficient full state tomographical techniques for quantum benchmarks. Here, we introduce a matrix filling-based method for tomography of pure quantum states, called the matrix-completion quantum state tomography. This method requires only 2n + 1 local Pauli operators and minimal post-processing for n-qubit states. Numerical results show that our method is highly efficient on superconducting real quantum devices and achieves better fidelity estimates of multiqubit quantum states as compared to contem…
We present a method for quantum state tomography that enables the efficient estimation, with fixed precision, of any of the matrix elements of the density matrix of a state, provided that the states from the basis in which the matrix is written can be efficiently prepared in a controlled manner. Furthermore, we show how this algorithm is well suited for quantum process tomography, enabling to perform selective and efficient quantum process tomography.
For one-mode and multimode light, the photon-number tomograms of Gaussian quantum states are explicitly calculated in terms of multivariable Hermite polynomials. Positivity of the tomograms is shown to be necessary condition for positivity of the density matrix.
We provide a simple analytic relation which connects the density operator of the radiation field with the number probabilities. The problem of experimentally "sampling" a general matrix elements is studied, and the deleterious effects of nonunit quantum efficiency in the detection process are analyzed showing how they can be reduced by using the squeezing technique. The obtained result is particulary useful for intracavity field reconstruction states.
We show that the Fano representation leads to a particularly simple and appealing form of the quantum process tomography matrix $χ_{_F}$, in that the matrix $χ_{_F}$ is real, the number of matrix elements is exactly equal to the number of free parameters required for the complete characterization of a quantum operation, and these matrix elements are directly related to evolution of the expectation values of the system's polarization measurements. These facts are illustrated in the examples of one- and two-qubit quantum noise channels.
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 discuss details of the preparation and detection of entangled electron-nuclear spin states in 15N:C60 together with a quantitative evaluation of the complete density matrix. All four Bell states of a two qubit subsystem were analyzed. In addition we find a quantum critical temperature of Tq = 7.73 K for this system at an electron spin resonance frequency of 95 GHz.
In this note, we describe a method for reconstructing matrix product states from a small number of efficiently-implementable measurements. Our method is exponentially faster than standard tomography, and it can also be used to certify that the unknown state is an MPS. The basic idea is to use local unitary operations to measure in the Schmidt basis, giving direct access to the MPS representation. This compares favorably with recently and independently proposed methods that recover the MPS tensors by performing a variational minimization, which is computationally intractable in certain cases. O…
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a s…
We discuss characterization of experimental quantum gates by the error matrix, which is similar to the standard process matrix $χ$ in the Pauli basis, except the desired unitary operation is factored out, by formally placing it either before or after the error process. The error matrix has only one large element, which is equal to the process fidelity, while other elements are small and indicate imperfections. The imaginary parts of the elements along the left column and/or top row directly indicate the unitary imperfection and can be used to find the needed correction. We discuss a relatively…
We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in …
We present a new method for quantum process tomography. The method enables us to efficiently estimate, with fixed precision, any of the parameters characterizing a quantum channel. It is selective since one can choose to estimate the value of any specific set of matrix elements of the super-operator describing the channel. Also, we show how to efficiently estimate all the average survival probabilities associated with the channel (i.e., all the diagonal elements of its $χ$--matrix.
Multi-photon system has been studied by many groups, however the biggest challenge faced is the number of copies of an unknown state are limited and far from detecting quantum entanglement. The difficulty to prepare copies of the state is even more serious for the quantum state tomography. One possible way to solve this problem is to use adaptive quantum state tomography, which means to get a preliminary density matrix in the first step and revise it in the second step. In order to improve the performance of adaptive quantum state tomography, we develop a new distribution scheme of samples and…
With the general theorem of SQPT, we shall develop a scheme to detemine an arbitrary matrix element of $chi$, which is expanded with the Choi operators, in a scalbe way.
Quantum state tomography, which aims to find the best description of a quantum state -- the density matrix, is an essential building block in quantum computation and communication. Standard techniques for state tomography are incapable of tracking changing states and often perform poorly in the presence of environmental noise. Although there are different approaches to solve these problems theoretically, experimental demonstrations have so far been sparse. Our approach, matrix-exponentiated gradient tomography, is an online tomography method that allows for state tracking, updates the estimate…
We demonstrate complete characterization of a two-qubit entangling process - a linear optics controlled-NOT gate operating with coincident detection - by quantum process tomography. We use maximum-likelihood estimation to convert the experimental data into a physical process matrix. The process matrix allows accurate prediction of the operation of the gate for arbitrary input states, and calculation of gate performance measures such as the average gate fidelity, average purity and entangling capability of our gate, which are 0.90, 0.83 and 0.73, respectively.
We propose a procedure for tomographic characterization of continuous variable quantum operations which employs homodyne detection and single-mode squeezed probe states with a fixed degree of squeezing and anti-squeezing and a variable displacement and orientation of squeezing ellipse. Density matrix elements of a quantum process matrix in Fock basis can be estimated by averaging well behaved pattern functions over the homodyne data. We show that this approach can be straightforwardly extended to characterization of quantum measurement devices. The probe states can be mixed, which makes the pr…
We present the results of the first photonic implementation of a new method for quantum process tomography. The method (originally presented by A. Bendersky et al, Phys. Rev. Lett 100, 190403 (2008)) enables the estimation of any element of the chi-matrix that characterizes a quantum process using resources that scale polynomially with the number of qubits. It is based on the idea of mapping the estimation of any chi-matrix element onto the average fidelity of a quantum channel and estimating the latter by sampling randomly over a special set of states called a 2-design. With a heralded single…
Tomography of the two qubit density matrix shared by Alice and Bob is an essential ingredient for guaranteeing an acceptable margin of confidentiality during the establishment of a secure fresh key through the Quantum Key Distribution (QKD) scheme. We show how the Singapore protocol for key distribution is optimal from this point of view, due to the fact that it is based on so called SIC POVM qubit tomography which allows the most accurate full tomographic reconstruction of an unknown density matrix on the basis of a restricted set of experimental data. We illustrate with the help of experimen…
Linear optical circuits of growing complexity are playing an increasing role in emerging photonic quantum technologies. Individual photonic devices are typically described by a unitary matrix containing amplitude and phase information, the characterisation of which is a key task. We present a constructive scheme to retrieve the unitary matrix describing an arbitrary linear optical device using data obtained from one-photon and two-photon ensembles. The scheme is stable on the arbitrarily increasable length scale of the photon packet and independent of photon loss at input and output ports of t…
Quantum state tomography is a key technique for quantum information processing, but is challenging due to the exponential growth of its complexity with the system size. In this work, we propose an algorithm which iteratively finds the best non-negative matrix product state approximation based on a set of measurement outcomes whose size does not necessarily grow exponentially. Compared to the tomography method based on neural network states, our scheme utilizes a so-called tensor train representation that allows straightforward recovery of the unknown density matrix in the matrix product state …
State and measurement tomography make assumptions about the experimental states or measurements. These assumptions are often not justified because state preparation and measurement errors are unavoidable in practice. Here we describe how the Gram matrix associated with the states and measurement operators can be estimated via semidefinite programming if the states and the measurements are so called globally completable. This is for instance the case if the unknown measurements are known to be projective and non-degenerate. The computed Gram matrix determines the states, and the measurement ope…
We show that the density matrix of a spin-l system can be described entirely in terms of the measurement statistics of projective spin measurements along a minimum of 4l+1 different spin directions. It is thus possible to represent the complete quantum statistics of any N-level system within the spherically symmetric three dimensional space defined by the spin vector. An explicit method for reconstructing the density matrix of a spin-1 system from the measurement statistics of five non-orthogonal spin directions is presented and the generalization to spin-l systems is discussed.
We present the first NMR implementation of a scheme for selective and efficient quantum process tomography without ancilla. We generalize this scheme such that it can be implemented efficiently using only a set of measurements involving product operators. The method allows us to estimate any element of the quantum process matrix to a desired precision, provided a set of quantum states can be prepared efficiently. Our modified technique requires fewer experimental resources as compared to the standard implementation of selective and efficient quantum process tomography, as it exploits the speci…
We review single-qubit quantum process tomography for trace-preserving and nontrace-preserving processes, and derive explicit forms of the general constraints for fitting experimental data. These forms provide additional insight into the structure of the process matrix as well as reveal a tighter bound on the trace of a nontrace-preserving process than has been previously stated. We also describe, for completeness, how to incorporate measured imperfect input states.
We consider the problem of quantum-state tomography under the assumption that the state is pure, and more generally that its rank is bounded by a given value. In this scenario, new notions of informationally complete POVMs emerge, which allow for high-fidelity state estimation with fewer measurement outcomes than are required for an arbitrary rank state. We study this in the context of matrix completion, where the POVM outcomes determine only a few of the density matrix elements. We give an analytic solution that fully characterizes informational completeness and elucidates the important role …
We introduce a technique to estimate error-mitigated expectation values on noisy quantum computers. Our technique performs shadow tomography on a logical state to produce a memory-efficient classical reconstruction of the noisy density matrix. Using efficient classical post-processing, one can mitigate errors by projecting a general nonlinear function of the noisy density matrix into the codespace. The subspace expansion and virtual distillation can be viewed as special cases of the new framekwork. We show our method is favorable in the quantum and classical resources overhead. Relative to sub…
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.
Quantum process tomography is used to fully characterize the evolution of the quantum vibrational state of atoms. Rubidium atoms are trapped in a shallow optical lattice supporting only two vibrational states, which we charcterize by reconstructing the 2x2 density matrix. Repeating this process for a complete set of inputs allows us to completely characterize both the system's intrinsic decoherence and resonant coupling.
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.
Recently, Bendersky \emph{et al.} developed a method to complete the task of characterizing an arbitrary $χ$ matrix element in a scalable way, Phys. Rev. Lett. Vol. \textbf{100}, 190403(2008), where an auxiliary system was needed. In present work, we shall show that the same task can also be completed within the scheme of standard quantum process tomography (SQPT) where there is no requirement for ancilla. Our method depends on two observations: With the elaborately chosen operators basis, the SQPT may have an economical form where a single run of experiment, in which we measure the expectatio…
We present an example of quantum process tomography performed on a single solid state qubit. The qubit used is two energy levels of the triplet state in the Nitrogen-Vacancy defect in Diamond. Quantum process tomography is applied to a qubit which has been allowed to decohere for three different time periods. In each case the process is found in terms of the $χ$ matrix representation and the affine map representation. The discrepancy between experimentally estimated process and the closest physically valid process is noted.
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…
Quantum operations describe any state change allowed in quantum mechanics, including the evolution of an open system or the state change due to a measurement. In this letter we present a general method based on quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation. As input the method needs only a single entangled state. The feasibility of the technique for the electromagnetic field is shown, and the experimental setup is illustrated based on homodyne tomography of a twin-beam.
Using a relation between a bi-orthogonal set of equiseparable bases and the weak values of the density matrix we derive an explicit formula for its tomographic reconstruction completely analogous to the standard mutually unbiased bases expansion. With the simple example of a qubit is evidenced the relationship between weak values, measured probabilities and the separation between non-orthogonal bases.
We construct minimax optimal non-asymptotic confidence sets for low rank matrix recovery algorithms such as the Matrix Lasso or Dantzig selector. These are employed to devise adaptive sequential sampling procedures that guarantee recovery of the true matrix in Frobenius norm after a data-driven stopping time $\hat n$ for the number of measurements that have to be taken. With high probability, this stopping time is minimax optimal. We detail applications to quantum tomography problems where measurements arise from Pauli observables. We also give a theoretical construction of a confidence set fo…
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.
A single photon, delocalized over two optical modes, is characterized by means of quantum homodyne tomography. The reconstructed four-dimensional density matrix extends over the entire Hilbert space and thus reveals, for the first time, complete information about the dual-rail optical quantum bit as a state of the electromagnetic field. The experimental data violate the Bell inequality albeit with a loophole similar to the detection loophole in photon counting experiments.
Quantum computation has been growing rapidly in both theory and experiments. In particular, quantum computing devices with a large number of qubits have been developed by IBM, Google, IonQ, and others. The current quantum computing devices are noisy intermediate-scale quantum $($NISQ$)$ devices, and so approaches to validate quantum processing on these quantum devices are needed. One of the most common ways of validation for an n-qubit quantum system is quantum tomography, which tries to reconstruct a quantum system's density matrix by a complete set of observables. However, the inherent noise…
In this paper, we examine a variety of strategies for numerical quantum-state estimation from data of the sort commonly measured in experiments involving quantum state tomography. We find that, in some important circumstances, an elaborate and time-consuming numerical optimization to obtain 'the best' density matrix corresponding to a given data set is not necessary, and that cruder, faster numerical techniques may well be 'good enough'.
In this paper we describe in detail and generalize a method for quantum process tomography that was presented in [A. Bendersky, F. Pastawski, J. P. Paz, Physical Review Letters 100, 190403 (2008)]. The method enables the efficient estimation of any element of the $χ$--matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required to implement it scale polynomically with the number of qubits of the system. The estimation of all diagonal elements of the $χ$--matrix c…
For an initially well designed but imperfect quantum information system, the process matrix is almost sparse in an appropriate basis. Existing theory and associated computational methods (L1-norm minimization) for reconstructing sparse signals establish conditions under which the sparse signal can be perfectly reconstructed from a very limited number of measurements (resources). Although a direct extension to quantum process tomography of the L1-norm minimization theory has not yet emerged, the numerical examples presented here, which apply L1-norm minimization to quantum process tomography, s…
We present a continuous variable tomography scheme that reconstructs the Husimi Q-function (Wigner function) by Lagrange interpolation, using measurements of the Q-function (Wigner function) at the Padua points, the optimal sampling points for two dimensional reconstruction. Our approach drastically reduces the number of measurements required compared to using equidistant points on a regular grid, although reanalysis of such experiments is possible. The reconstruction algorithm produces a reconstructed function with exponentially decreasing error and quasi-linear runtime in the number of Padua…
Characterizing Photonic Partial Distinguishability Through Gram Matrix Tomography
Center for Quantum Enabled-Computing, Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心量子使能计算中心）；International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术国际实验室）；Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles（布鲁塞尔自由大学布鲁塞尔理工学院量子信息与通信中心）；Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）
The techniques of low-rank matrix recovery were adapted for Quantum State Tomography (QST) previously by D. Gross et al. [Phys. Rev. Lett. 105, 150401 (2010)], where they consider the tomography of $n$ spin-$1/2$ systems. For the density matrix of dimension $d = 2^n$ and rank $r$ with $r \ll 2^n$, it was shown that randomly chosen Pauli measurements of the order $O(dr \log(d)^2)$ are enough to fully reconstruct the density matrix by running a specific convex optimization algorithm. The result utilized the low operator-norm of the Pauli operator basis, which makes it `incoherent' to low-rank ma…
The experimental implementation of selective quantum process tomography (SQPT) involves computing individual elements of the process matrix with the help of a special set of states called quantum 2-design states. However, the number of experimental settings required to prepare input states from quantum 2-design states to selectively and precisely compute a desired element of the process matrix is still high, and hence constructing the corresponding unitary operations in the lab is a daunting task. In order to reduce the experimental complexity, we mathematically reformulated the standard SQPT …
Identifiability in Quantum State, Process, and Network Tomography
CONNECT Research Centre, School of Engineering, Trinity College Dublin（都柏林三一学院工程学院连接研究中心）；Manning College of Information and Computer Science, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校曼宁信息与计算机科学学院）；Department of Computing, School of Science, South East Technological University（东南理工大学理学院计算系）
