Quantum inference of states and processes

2004

The theory of quantum state reconstruction is illustrated here on several examples taken from modern experimental praxis. Maximum-likelihood estimation is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system-a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for manipulating states of a neutron and internal as well as external states of a photon.

Annals of Physics, 1991

A new approach to quantum state determination is developed using data in the form of observed eigenvectors. An exceedingly natural inversion of such data results when the quantum probability rule is recognised as a conditional. The reversal of this conditional via Bayesian methods results in an inferred probability density over states which readily reduces to a density matrix estimator. The inclusion of concepts drawn from communication theory then defines an Optimal State Determination Problem which is explored on Hilbert spaces of arbitrary finite dimensionality.

Physical Review A, 2012

We present a detailed account of quantum state estimation by joint maximization of the likelihood and the entropy. After establishing the algorithms for both perfect and imperfect measurements, we apply the procedure to data from simulated and actual experiments. We demonstrate that the realistic situation of incomplete data from imperfect measurements can be handled successfully.

Maximum-Likelihood Methods in Quantum Mechanics, Lect. Notes Phys. 649, 59–112 (2004), 2004

Maximum Likelihood estimation is a versatile tool covering wide range of applications, but its benefits are apparent particularly in the quantum domain. For a given set of measurements, the most likely state is estimated. Though this problem is nonlinear, it can be effectively solved by an iterative algorithm exploiting the convexity of the likelihood functional and the manifold of density matrices. This formulation fully replaces the inverse Radon transformation routinely used for tomographic reconstructions. Moreover, it provides the most efficient estimation strategy saturating the Cramer-Rao lower bound asymptotically. In this sense it exploits the acquired data set in the optimal way and minimizes the artifacts associated with the reconstruction procedure. The idea of maximum likelihood reconstruction is further extended to the estimation of quantum processes, measurements, and discrimination between quantum states. This technique is well suited for future applications in quantum information science due to its ability to quantify very subtle and fragile quantum effects.

2004

Maximum Likelihood estimation is a versatile tool covering wide range of applications, but its benefits are apparent particularly in the quantum domain. For a given set of measurements, the most likely state is estimated. Though this problem is nonlinear, it can be effectively solved by an iterative algorithm exploiting the convexity of the likelihood functional and the manifold of density matrices. This formulation fully replaces the inverse Radon transformation routinely used for tomographic reconstructions. Moreover, it provides the most efficient estimation strategy saturating the Cramer-Rao lower bound asymptotically. In this sense it exploits the acquired data set in the optimal way and minimizes the artifacts associated with the reconstruction procedure. The idea of maximum likelihood reconstruction is further extended to the estimation of quantum processes, measurements, and discrimination between quantum states. This technique is well suited for future applications in quantum information science due to its ability to quantify very subtle and fragile quantum effects.

2020

In this paper, inspired by the "Minimum Description Length Principle" in classical Statistics, we introduce a new method for predicting the outcomes of a quantum measurement and for estimating the state of a quantum system with minimum quantum complexity, while, at the same time, avoiding overfitting.

AIP Conference Proceedings

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.

Fortschritte der Physik, 2001

We investigate the relative merits of techniques for recovery of density matrices of two qubits from experimental data. Our results are applied to measure the states of photons produced in down-conversion experiments.

Physical Review A, 2010

Performance of quantum process estimation is naturally limited by fundamental, random, and systematic imperfections of preparations and measurements. These imperfections may lead to considerable errors in the process reconstruction because standard data-analysis techniques usually presume ideal devices. Here, by utilizing generic auxiliary quantum or classical correlations, we provide a framework for the estimation of quantum dynamics via a single measurement apparatus. By construction, this approach can be applied to quantum tomography schemes with calibrated faulty-state generators and analyzers. Specifically, we present a generalization of the work begun by M. Mohseni and D. A. Lidar [Phys. Rev. Lett. 97, 170501 (2006)] with an imperfect Bell-state analyzer. We demonstrate that for several physically relevant noisy preparations and measurements, classical correlations and a small data-processing overhead suffice to accomplish the full system identification. Furthermore, we provide the optimal input states whereby the error amplification due to inversion of the measurement data is minimal.

2011

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 structure of such estimators will also be explored.