Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state can not be uniquely determined. In this case, among the density matrices compatible with the available data, it is commonly preferred that one which is the most uncommitted with the missing information. This is the purpose of the Maximum Entropy estimation (MaxEnt) and the Variational Quantum Tomography (VQT). Here, we propose a variant of Variational Quantum Tomography and show its relationship with Maximum Entropy methods in quantum tomographies with incomplete set of measurements. We prove their equivalence in case of eigenbasis measurements, and through numerical simulations we stress their similar behavior. Hence, in the modified VQT formulation we have an estimate of a quantum state as unbiased as in MaxEnt and with the benefit that VQT can be more efficiently solved by means of linear semidefinite programs.
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].
We propose an iterative algorithm for incomplete quantum process tomography, with the help of quantum state estimation, based on the combined principles of maximum-likelihood and maximum-entropy. The algorithm yields a unique estimator for an unknown quantum process when one has less than a complete set of linearly independent measurement data to specify the quantum process uniquely. We apply this iterative algorithm adaptively in various situations and so optimize the amount of resources required to estimate the quantum process with incomplete data.
This is a Bachelor's thesis on point estimators for incomplete tomography of qutrits as of 2014, submitted to the National University of Singapore. The main content of the thesis focuses on various methods of estimation such as maximum entropy and average estimator and show that they are quite different. Numerical simulations of these methods show however that these estimators perform very close to one another. Therefore on this basis, there is no reason to favor one method over another.
In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantum-state and qua…
We demonstrate that incomplete quantum tomography can give conclusive information in experimental realizations. We divide the state space into a union of multiple disjoint subsets and determine conclusively which of the subsets a system, prepared in completely unknown state, belongs to. In other words we construct and solve membership problems. Our membership problems are partitions of the state space into a union of four disjoint sets formed by fixing two maximally entangled reference states and boundary values of fidelity function "radius" between the reference states and the unknown prepara…
The main goal of this paper is to extend and apply the principle of maximum entropy (MaxEnt) to incomplete quantum process estimation tasks. We will define a so-called process entropy function being the von Neumann entropy of the state associated with the quantum process via Choi-Jamiolkowski isomorphism. It will be shown that an arbitrary process estimation experiment can be reformulated in a unified framework and MaxEnt principle can be consistently exploited. We will argue that the suggested choice for the process entropy satisfies natural list of properties and it reduces to the state MaxE…
State of a $d$-dimensional quantum system can only be inferred by performing an informationally complete measurement with $m\geqslant d^2$ outcomes. However, an experimentally accessible measurement can be informationally incomplete. Here we show that a single informationally incomplete measuring apparatus is still able to provide all the information about the quantum system if applied several times in a row. We derive a necessary and sufficient condition for such a measuring apparatus and give illustrative examples for qubits, qutrits, general $d$-level systems, and composite systems of $n$ q…
There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numeric…
Quantum state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements does, as a rule, not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting st…
We introduce a new method to reconstruct unknown quantum states out of incomplete and noisy information. The method is a linear convex optimization problem, therefore with a unique minimum, which can be efficiently solved with Semidefinite Programs. Numerical simulations indicate that the estimated state does not overestimate purity, and neither the expectation value of optimal entanglement witnesses. The convergence properties of the method are similar to compressed sensing approaches, in the sense that, in order to reconstruct low rank states, it needs just a fraction of the effort correspon…
Learning quantum tomography from incomplete measurements
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；Institute of Theoretical Physics, Wrocław University of Science and Technology（弗罗茨瓦夫科技大学理论物理研究所）
We reconstruct the time dependent quantum map corresponding to the relaxation process of a two-spin system in liquid-state NMR at room temperature. By means of quantum tomography techniques that handle informational incomplete data, we show how to properly post-process and normalize the measurements data for the simulation of quantum information processing, overcoming the unknown number of molecules prepared in a non-equilibrium magnetisation state ($N_j$) by an initial sequence of radiofrequency (RF) pulses. From the reconstructed quantum map, we infer both longitudinal ($T_1$) and transversa…
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…
Quantum process tomography is an experimental technique to fully characterize an unknown quantum process. Standard quantum process tomography suffers from exponentially scaling of the number of measurements with the increasing system size. In this work, we put forward a quantum machine learning algorithm which approximately encodes the unknown unitary quantum process into a relatively shallow depth parametric quantum circuit. We demonstrate our method by reconstructing the unitary quantum processes resulting from the quantum Hamiltonian evolution and random quantum circuits up to $8$ qubits. R…
We improve upon a recently introduced efficient quantum state reconstruction procedure targeted to states well-approximated by the multi-scale entanglement renormalization ansatz (MERA), e.g., ground states of critical models. We show how to numerically select a subset of experimentally accessible measurements which maximizes information extraction about renormalized particles, thus dramatically reducing the required number of physical measurements. We numerically estimate the number of measurements required to characterize the ground state of the critical 1D Ising (resp. XX) model and find th…
The need to perform quantum state tomography on ever larger systems has spurred a search for methods that yield good estimates from incomplete data. We study the performance of compressed sensing (CS) and least squares (LS) estimators in a fast protocol based on continuous measurement on an ensemble of cesium atomic spins. Both efficiently reconstruct nearly pure states in the 16-dimensional ground manifold, reaching average fidelities FCS = 0.92 and FLS = 0.88 using similar amounts of incomplete data. Surprisingly, the main advantage of CS in our protocol is an increased robustness to experim…
Centro Interdisciplinario de Telecomunicaciones Electrónica, Computación y Ciencia Aplicada (CONICET), Universidad Nacional de Rio Negro（里奥内格罗国立大学跨学科电信、计算与应用科学中心（阿根廷国家科学研究委员会））；Instituto de Ciencias de la Computación, CONICET - Universidad de Buenos Aires（布宜诺斯艾利斯大学计算机科学学院（阿根廷国家科学研究委员会））；Instituto de Física La Plata (CONICET-UNLP)（拉普拉塔物理研究所（阿根廷国家科学研究委员会-拉普拉塔国立大学））
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…
Quantum states are successfully reconstructed using the maximum likelihood estimation on the subspace where the measured projectors reproduce the identity operator. Reconstruction corresponds to normalization of incompatible observations. The proposed approach handles the noisy data corresponding to realistic incomplete observation with finite resolution.
This is a PhD dissertation on the latest numerical quantum estimation schemes as of 2012, submitted to the National University of Singapore. The main content of the thesis focuses on accessing quantum information with informationally incomplete measurements to reconstruct quantum states of large quantum systems, as well as to reduce the amount of resources to reconstruct quantum channels.
Quantum state tomography is an integral part of quantum computation and offers the starting point for the validation of various quantum devices. One of the central tasks in the field of state tomography is to reconstruct with high fidelity, the quantum states of a quantum system. From an experiment on a real quantum device, one can obtain the mean measurement values of different operators. With such a data as input, in this report we employ the maximal entropy formalism to construct the least biased mixed quantum state that is consistent with the given set of expectation values. Even though in…
We scrutinize the effects of non-ideal data acquisition on the homodyne tomograms of photon quantum states. The presence of a weight function, schematizing the effects of the finite thickness of the probing beam or equivalently noise, only affects the state reconstruction procedure by a normalization constant. The results are extended to a discrete mesh and show that quantum tomography is robust under incomplete and approximate knowledge of tomograms.
National University of Defense Technology（国防科技大学）；National Innovation Institute of Defense Technology（国防科技创新研究院）
Quantum state tomography is a key process in most quantum experiments. In this work, we employ quantum machine learning for state tomography. Given an unknown quantum state, it can be learned by maximizing the fidelity between the output of a variational quantum circuit and this state. The number of parameters of the variational quantum circuit grows linearly with the number of qubits and the circuit depth, so that only polynomial measurements are required, even for highly-entangled states. After that, a subsequent classical circuit simulator is used to transform the information of the target …
The certificate of success for a number of important quantum information processing protocols, such as entanglement distillation, is based on the difference in the entanglement content of the quantum states before and after the protocol. In such cases, effective bounds need to be placed on the entanglement of non-local states consistent with statistics obtained from local measurements. In this work, we study numerically the ability of a novel type of homodyne detector which combines phase sensitivity and photon-number resolution to set accurate bounds on the entanglement content of two-mode qu…
We consider a variant of the BB84 protocol for quantum cryptography, the prototype of tomographically incomplete protocols, where the key is generated by one-way communication rather than the usual two-way communication. Our analysis, backed by numerical evidence, establishes thresholds for eavesdropping attacks on the raw data and on the generated key at quantum bit error rates of 10% and 6.15%, respectively. Both thresholds are lower than the threshold for unconditional security in the standard BB84 protocol.
We present a formalism for self-calibrating tomography of arbitrary dimensional systems. Self-calibrating quantum state tomography was first introduced in the context of qubits, and allows the reconstruction of the density matrix of an unknown quantum state despite incomplete knowledge of the unitary operations used to change the measurement basis. We show how this can be generalized to qudits, i.e. d-level systems, and provide a specific example for a V-type three-level atomic system whose transition dipole moments are not known. We show that it is always possible to retrieve the unknown stat…
We study the performance of efficient quantum state tomography methods based on neural network quantum states using measured data from a two-photon experiment. Machine learning inspired variational methods provide a promising route towards scalable state characterization for quantum simulators. While the power of these methods has been demonstrated on synthetic data, applications to real experimental data remain scarce. We benchmark and compare several such approaches by applying them to measured data from an experiment producing two-qubit entangled states. We find that in the presence of expe…
The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomography, and investigate how the estimation error behaves with a reduction in the number of measurement settings, compared with the standard ion tomography setup. We present extensive simulation results showing that the error is robust with respect to the choice of states of a given rank, the random sel…
Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. First, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e., the sample complexity of tomography decreases with the rank. Second, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple …
A powerful tool for studying geometrical problems in Hilbert space is developed. In particular, we study the quantum pure state tomography problem in finite dimensions from the point of view of dynamical systems and bifurcations theory. First, we introduce a generalization of the Hellinger metric for probability distributions which allows us to find a geometrical interpretation of the quantum state tomography problem. Thereafter, we prove that every solution to the state tomography problem is an attractive fixed point of the so-called physical imposition operator. Additionally, we demonstrate …
We introduce and experimentally demonstrate a technique for performing quantum state tomography on multiple-qubit states despite incomplete knowledge about the unitary operations used to change the measurement basis. Given unitary operations with unknown rotation angles, our method can be used to reconstruct the density matrix of the state up to local sigma-z rotations as well as recover the magnitude of the unknown rotation angle. We demonstrate high-fidelity self-calibrating tomography on polarization-encoded one- and two-photon states. The unknown unitary operations are realized in two ways…
Many experiments in quantum information aim at creating multi-partite entangled states. Quantifying the amount of entanglement that was actually generated can, in principle, be accomplished using full-state tomography. This method requires the determination of a parameter set that is growing exponentially with the number of qubits and becomes infeasible even for moderate numbers of particles. Non-trivial bounds on experimentally prepared entanglement can however be obtained from partial information on the density matrix. The fundamental question that needs to be addressed in this context is th…
Universal compilation is a training process that compiles a trainable unitary into a target unitary and it serves vast potential applications from quantum dynamic simulations to optimal circuits with deep-compressing, device benchmarking, quantum error mitigation, and so on. Here, we propose a universal compilation-based variational algorithm for the preparation and tomography of quantum states in low-depth quantum circuits. We apply the Fubini-Study distance to be a trainable cost function under various gradient-based optimizers, including the quantum natural gradient approach. We evaluate th…
Measuring incomplete sets of mutually unbiased bases constitutes a sensible approach to the tomography of high-dimensional quantum systems. The unbiased nature of these bases optimizes the uncertainty hypervolume. However, imposing unbiasedness on the probabilities for the unmeasured bases does not generally yield the estimator with the largest von Neumann entropy, a popular figure of merit in this context. Furthermore, this imposition typically leads to mock density matrices that are not even positive definite. This provides a strong argument against perfunctory applications of linear estimat…
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.
We propose the simulation of quantum-optical systems in the ultrastrong-coupling regime using a variational quantum algorithm. More precisely, we introduce a short-depth variational form to prepare the groundstate of the multimode Dicke model on a quantum processor and present proof-of-principle results obtained via cloud access to an IBM device. We moreover provide an algorithm for characterizing the groundstate by Wigner state tomography. Our work is a first step towards digital quantum simulation of quantum-optical systems with potential applications to the spin-boson, Kondo and Jahn-Teller…
A system-ancilla bipartite state capable of containing the complete information of an unknown quantum channel acting on the system is called faithful. The equivalence between faithfulness of state and invertibility of the corresponding Jamiolkowski map proved by D'Ariano and Presti has been a useful characterization for ancilla-assisted process tomography albeit the proof was incomplete as they assumed trace nonincreasing quantum operations, not quantum channels. We complete the proof of the equivalence and introduce the generalization of faithfulness to various classes of quantum channels. We…
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 …
Modern day quantum simulators can prepare a wide variety of quantum states but the accurate estimation of observables from tomographic measurement data often poses a challenge. We tackle this problem by developing a quantum state tomography scheme which relies on approximating the probability distribution over the outcomes of an informationally complete measurement in a variational manifold represented by a convolutional neural network. We show an excellent representability of prototypical ground- and steady states with this ansatz using a number of variational parameters that scales polynomia…
Learning Entanglement Quasiprobability from Noisy and Incomplete Data
MIIT Key Laboratory of Complex-field Intelligent Sensing, School of Optics and Photonics, Beijing Institute of Technology（北京理工大学光电与光子学学院复杂场智能感知工信部重点实验室）；Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing)（北京理工大学长三角研究院（嘉兴））；Center for Photonic Quantum Precision Measurement, Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology（北京理工大学前沿交叉科学研究院光子量子精密测量中心）
Quantum state tomography is the experimental procedure of determining an unknown state. It is not only essential for the verification of resources and processors of quantum information but is also important in its own right with regard to the foundation of quantum mechanics. Standard methods have been elusive for large systems because of the enormous number of observables to be measured and the exponential complexity of data post-processing. Here, we propose a new scheme of quantum state tomography that requires the measurement of only three observables (acting jointly on the system and pointe…
We present strategies how to reconstruct (estimate) properties of a quantum channel described by the map E based on incomplete measurements. In a particular case of a qubit channel a complete reconstruction of the map E can be performed via complete tomography of four output states E[rho_j ] that originate from a set of four linearly independent test states j (j = 1, 2, 3, 4) at the input of the channel. We study the situation when less than four linearly independent states are transmitted via the channel and measured at the output. We present strategies how to reconstruct the channel when jus…
Deep Neural Network-assisted improvement of quantum compressed sensing tomography
ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology（ICFO - 光子科学研究所，巴塞罗那科学技术学院）；ICREA（加泰罗尼亚高级研究学院）；Physics Department E. Pancini - Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学E. Pancini物理系）
State preparation is of fundamental importance in quantum physics, which can be realized by constructing the quantum circuit as a unitary that transforms the initial state to the target, or implementing a quantum control protocol to evolve to the target state with a designed Hamiltonian. In this work, we study the latter on quantum many-body systems by the time evolution with fixed couplings and variational magnetic fields. In specific, we consider to prepare the ground states of the Hamiltonians containing certain interactions that are missing in the Hamiltonians for the time evolution. An op…
