A discrete invitation to quantum filtering and feedback control

The engineering and control of devices at the quantum-mechanical level--such as those consisting of small numbers of atoms and photons--is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory, while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.Comment: 76 pages, 12 figures. A PDF file with high resolution figures can be found at http://minty.caltech.edu/papers.ph

On the exchange of intersection and supremum of sigma-fields in filtering theory

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page

Sliding mode control of quantum systems

This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). Sliding mode control is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e., eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum sliding mode control systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of main features of the proposed method is that the designed control laws can guarantee desired control performance in the presence of uncertainties in the system Hamiltonian. This sliding mode control approach provides a useful control theoretic tool for robust quantum information processing with uncertainties.Comment: 18 pages, 4 figure

How much time does a measurement take?

We consider the problem of measurement using the Lindblad equation, which allows the introduction of time in the interaction between the measured system and the measurement apparatus. We use analytic results, valid for weak system-environment coupling, obtained for a two-level system in contact with a measurer (Markovian interaction) and a thermal bath (non-Markovian interaction), where the measured observable may or may not commute with the system-environment interaction. Analysing the behavior of the coherence, which tends to a value asymptotically close to zero, we obtain an expression for the time of measurement which depends only on the system-measurer coupling, and which does not depend on whether the observable commutes with the system-bath interaction. The behavior of the coherences in the case of strong system-environment coupling, found numerically, indicates that an increase in this coupling decreases the measurement time, thus allowing our expression to be considered the upper limit for the duration of the process.Comment: REVISED VERSION: 17 pages, 2 figure

Drosophila Insulin Pathway Mutants Affect Visual Physiology and Brain Function Besides Growth, Lipid, and Carbohydrate Metabolism

OBJECTIVE—Type 2 diabetes is the most common form of diabetes worldwide. Some of its complications, such as retinop-athy and neuropathy, are long-term and protracted, with an un-clear etiology. Given this problem, genetic model systems, such as in flies where type 2 diabetes can be modeled and studied, offer distinct advantages. RESEARCH DESIGN AND METHODS—We used individual flies, control, and mutant individuals with partial loss-of-function insulin pathway genes. We measured wing size and tested body weight for growth phenotypes, the latter by means of a microbal-ance. We studied total lipid and carbohydrate content, lipids by a reaction in single fly homogenates with vanillin-phosphoric acid, and carbohydrates with an anthrone-sulfuric acid reaction. Cholinesterase activity was measured using the Ellman method in head homogenates from pooled fly heads, and electroretinogram

A Quantum Langevin Formulation of Risk-Sensitive Optimal Control

In this paper we formulate a risk-sensitive optimal control problem for continuously monitored open quantum systems modelled by quantum Langevin equations. The optimal controller is expressed in terms of a modified conditional state, which we call a risk-sensitive state, that represents measurement knowledge tempered by the control purpose. One of the two components of the optimal controller is dynamic, a filter that computes the risk-sensitive state. The second component is an optimal control feedback function that is found by solving the dynamic programming equation. The optimal controller can be implemented using classical electronics. The ideas are illustrated using an example of feedback control of a two-level atom

Quantum projection filter for a highly nonlinear model in cavity QED

Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite-dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case.) One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a stongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.Comment: 19 pages, 6 figures. A version with high quality images can be found at http://minty.caltech.edu/papers.ph

Heat release by controlled continuous-time Markov jump processes

We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Bellman-Jacobi-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.Comment: final version, section 2.1 revised, 26 pages, 3 figure

Reliability of Bioelectrical Impedance Analysis for Estimating Whole‐Fish Energy Density and Percent Lipids

We evaluated bioelectrical impedance analysis (BIA) as a nonlethal means of predicting energy density and percent lipids for three fish species: Yellow perch Perca flavescens, walleye Sander vitreus, and lake whitefish Coregonus clupeaformis. Although models that combined BIA measures with fish wet mass provided strong predictions of total energy, total lipids, and total dry mass for whole fish, including BIA provided only slightly better predictions than using fish mass alone. Regression models that used BIA measures to directly predict the energy density or percent lipids of whole fish were generally better than those using body mass alone (based on Akaike’s information criterion). However, the goodness of fit of models that used BIA measures varied widely across species and at best explained only slightly more than one‐half the variation observed in fish energy density or percent lipids. Models that combined BIA measures with body mass for prediction had the strongest correlations between predicted and observed energy density or percent lipids for a validation group of fish, but there were significant biases in these predictions. For example, the models underestimated energy density and percent lipids for lipid‐rich fish and overestimated energy density and percent lipids for lipid‐poor fish. A comparison of observed versus predicted whole‐fish energy densities and percent lipids demonstrated that models that incorporated BIA measures had lower maximum percent error than models without BIA measures in them, although the errors for the BIA models were still generally high (energy density: 15‐18%; percent lipids: 82‐89%). Considerable work is still required before BIA can provide reliable predictions of whole‐fish energy density and percent lipids, including understanding how temperature, electrode placement, and the variation in lipid distribution within a fish affect BIA measures.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/141722/1/tafs1519.pd

Single hadron response measurement and calorimeter jet energy scale uncertainty with the ATLAS detector at the LHC

The uncertainty on the calorimeter energy response to jets of particles is derived for the ATLAS experiment at the Large Hadron Collider (LHC). First, the calorimeter response to single isolated charged hadrons is measured and compared to the Monte Carlo simulation using proton-proton collisions at centre-of-mass energies of sqrt(s) = 900 GeV and 7 TeV collected during 2009 and 2010. Then, using the decay of K_s and Lambda particles, the calorimeter response to specific types of particles (positively and negatively charged pions, protons, and anti-protons) is measured and compared to the Monte Carlo predictions. Finally, the jet energy scale uncertainty is determined by propagating the response uncertainty for single charged and neutral particles to jets. The response uncertainty is 2-5% for central isolated hadrons and 1-3% for the final calorimeter jet energy scale.Comment: 24 pages plus author list (36 pages total), 23 figures, 1 table, submitted to European Physical Journal