Optimal, reliable estimation of quantum states

Accurately inferring the state of a quantum device from the results of measurements is a crucial task in building quantum information processing hardware. The predominant state estimation procedure, maximum likelihood estimation (MLE), generally reports an estimate with zero eigenvalues. These cannot be justified. Furthermore, the MLE estimate is incompatible with error bars, so conclusions drawn from it are suspect. I propose an alternative procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues, its eigenvalues provide a bound on their own uncertainties, and it is the most accurate procedure possible. I show how to implement BME numerically, and how to obtain natural error bars that are compatible with the estimate. Finally, I briefly discuss the differences between Bayesian and frequentist estimation techniques.Comment: RevTeX; 14 pages, 2 embedded figures. Comments enthusiastically welcomed

Effect of nonnegativity on estimation errors in one-qubit state tomography with finite data

We analyze the behavior of estimation errors evaluated by two loss functions, the Hilbert-Schmidt distance and infidelity, in one-qubit state tomography with finite data. We show numerically that there can be a large gap between the estimation errors and those predicted by an asymptotic analysis. The origin of this discrepancy is the existence of the boundary in the state space imposed by the requirement that density matrices be nonnegative (positive semidefinite). We derive an explicit form of a function reproducing the behavior of the estimation errors with high accuracy by introducing two approximations: a Gaussian approximation of the multinomial distributions of outcomes, and linearizing the boundary. This function gives us an intuition for the behavior of the expected losses for finite data sets. We show that this function can be used to determine the amount of data necessary for the estimation to be treated reliably with the asymptotic theory. We give an explicit expression for this amount, which exhibits strong sensitivity to the true quantum state as well as the choice of measurement.Comment: 9 pages, 4 figures, One figure (FIG. 1) is added to the previous version, and some typos are correcte

Anisotropy and internal field distribution of MgB2 in the mixed state at low temperatures

Magnetization and muon spin relaxation on MgB2 were measured as a function of field at 2 K. Both indicate an inverse-squared penetration depth strongly decreasing with increasing field H below about 1 T. Magnetization also suggests the anisotropy of the penetration depth to increase with increasing H, interpolating between a low Hc1 and a high Hc2 anisotropy. Torque vs angle measurements are in agreement with this finding, while also ruling out drastic differences between the mixed state anisotropies of the two basic length scales penetration depth and coherence length.Comment: 4 pages, 4 figure

Exponential speed-up with a single bit of quantum information: Testing the quantum butterfly effect

We present an efficient quantum algorithm to measure the average fidelity decay of a quantum map under perturbation using a single bit of quantum information. Our algorithm scales only as the complexity of the map under investigation, so for those maps admitting an efficient gate decomposition, it provides an exponential speed up over known classical procedures. Fidelity decay is important in the study of complex dynamical systems, where it is conjectured to be a signature of quantum chaos. Our result also illustrates the role of chaos in the process of decoherence.Comment: 4 pages, 2 eps figure

Bell's Theorem from Moore's Theorem

It is shown that the restrictions of what can be inferred from classically-recorded observational outcomes that are imposed by the no-cloning theorem, the Kochen-Specker theorem and Bell's theorem also follow from restrictions on inferences from observations formulated within classical automata theory. Similarities between the assumptions underlying classical automata theory and those underlying universally-unitary quantum theory are discussed.Comment: 12 pages; to appear in Int. J. General System

A review of the radiological treatment

The Draft Waste Management Programmatic Environmental Impact Statement (WM PEIS) was released by the U.S. Department of Energy (DOE) for public comment on September 22, 1995. Prepared in accordance with the National Environmental Policy Act (NEPA), the Final WM PEIS is currently scheduled for release in late summer 1996. The Draft WM PEIS was published after about 3 years of effort to select and evaluated the best alternatives for treating, storing, and disposing of the 50-year legacy of radioactive and chemically hazardous wastes existing within the DOE complex. The evaluation examined the potential health and environmental impacts of integrated waste management alternatives for five categories of waste types at 54 DOE sites. A primary consideration as a potential source of human health impacts at all sites is that of radiological releases resulting from postulated accidents involving facilities used to treat radioactive wastes. This paper first provides a brief, updated summary of the approach used to define and perform treatment facility accident analyses in the Draft WM PEIS. It reviews the selection of dominant sequences for the major sites most affected by the preferred waste management alternatives and highlights the salient accident analysis results. Finally, it summarizes and addresses key public and state and federal agency comments relating to accident analysis that were received in the public comment process

Waste management facility accident analysis (WASTE ACC) system: software for analysis of waste management alternatives

This paper describes the Waste Management Facility Accident Analysis (WASTE{underscore}ACC) software, which was developed at Argonne National Laboratory (ANL) to support the US Department of Energy`s (DOE`s) Waste Management (WM) Programmatic Environmental Impact Statement (PEIS). WASTE{underscore}ACC is a decision support and database system that is compatible with Microsoft{reg_sign} Windows{trademark}. It assesses potential atmospheric releases from accidents at waste management facilities. The software provides the user with an easy-to-use tool to determine the risk-dominant accident sequences for the many possible combinations of process technologies, waste and facility types, and alternative cases described in the WM PEIS. In addition, its structure will allow additional alternative cases and assumptions to be tested as part of the future DOE programmatic decision-making process. The WASTE{underscore}ACC system demonstrates one approach to performing a generic, systemwide evaluation of accident risks at waste management facilities. The advantages of WASTE{underscore}ACC are threefold. First, the software gets waste volume and radiological profile data that were used to perform other WM PEIS-related analyses directly from the WASTE{underscore}MGMT system. Second, the system allows for a consistent analysis across all sites and waste streams, which enables decision makers to understand more fully the trade-offs among various policy options and scenarios. Third, the system is easy to operate; even complex scenario runs are completed within minutes

Hamiltonian Determination with Restricted Access in Transverse Field Ising Chain

We propose a method to evaluate parameters in the Hamiltonian of the Ising chain under site-dependent transverse fields, with a proviso that we can control and measure one of the edge spins only. We evaluate the eigenvalues of the Hamiltonian and the time-evoultion operator exactly for a 3-spin chain, from which we obtain the expectation values of $\sigma_x$ of the first spin. The parameters are found from the peak positions of the Fourier transform of the expectation value. There are four assumptions in our method, which are mild enough to be satisfied in many physical systems.Comment: 15pages, 4 figure

Robust Online Hamiltonian Learning

In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.Comment: 24 pages, 12 figures; to appear in New Journal of Physic