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 Quantum Phase Operator for Coherent States

In papers by Lynch [Phys. Rev. A41, 2841 (1990)] and Gerry and Urbanski [Phys. Rev. A42, 662 (1990)] it has been argued that the phase-fluctuation laser experiments of Gerhardt, B\"uchler and Lifkin [Phys. Lett. 49A, 119 (1974)] are in good agreement with the variance of the Pegg-Barnett phase operator for a coherent state, even for a small number of photons. We argue that this is not conclusive. In fact, we show that the variance of the phase in fact depends on the relative phase between the phase of the coherent state and the off-set phase $\phi_0$ of the Pegg-Barnett phase operator. This off-set phase is replaced with the phase of a reference beam in an actual experiment and we show that several choices of such a relative phase can be fitted to the experimental data. We also discuss the Noh, Foug\`{e}res and Mandel [Phys.Rev. A46, 2840 (1992)] relative phase experiment in terms of the Pegg-Barnett phase taking post-selection conditions into account.Comment: 8 pages, 8 figures. Typographical errors and misprints have been corrected. The outline of the paper has also been changed. Physica Scripta (in press

Training a perceptron in a discrete weight space

On-line and batch learning of a perceptron in a discrete weight space, where each weight can take $2 L+1$ different values, are examined analytically and numerically. The learning algorithm is based on the training of the continuous perceptron and prediction following the clipped weights. The learning is described by a new set of order parameters, composed of the overlaps between the teacher and the continuous/clipped students. Different scenarios are examined among them on-line learning with discrete/continuous transfer functions and off-line Hebb learning. The generalization error of the clipped weights decays asymptotically as $exp(-K \alpha^2)$/$exp(-e^{|\lambda| \alpha})$ in the case of on-line learning with binary/continuous activation functions, respectively, where $\alpha$ is the number of examples divided by N, the size of the input vector and $K$ is a positive constant that decays linearly with 1/L. For finite $N$ and $L$, a perfect agreement between the discrete student and the teacher is obtained for $\alpha \propto \sqrt{L \ln(NL)}$. A crossover to the generalization error $\propto 1/\alpha$, characterized continuous weights with binary output, is obtained for synaptic depth $L > O(\sqrt{N})$.Comment: 10 pages, 5 figs., submitted to PR

Parisi Phase in a Neuron

Pattern storage by a single neuron is revisited. Generalizing Parisi's framework for spin glasses we obtain a variational free energy functional for the neuron. The solution is demonstrated at high temperature and large relative number of examples, where several phases are identified by thermodynamical stability analysis, two of them exhibiting spontaneous full replica symmetry breaking. We give analytically the curved segments of the order parameter function and in representative cases compute the free energy, the storage error, and the entropy.Comment: 4 pages in prl twocolumn format + 3 Postscript figures. Submitted to Physical Review Letter

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

Storage capacity of a constructive learning algorithm

Upper and lower bounds for the typical storage capacity of a constructive algorithm, the Tilinglike Learning Algorithm for the Parity Machine [M. Biehl and M. Opper, Phys. Rev. A {\bf 44} 6888 (1991)], are determined in the asymptotic limit of large training set sizes. The properties of a perceptron with threshold, learning a training set of patterns having a biased distribution of targets, needed as an intermediate step in the capacity calculation, are determined analytically. The lower bound for the capacity, determined with a cavity method, is proportional to the number of hidden units. The upper bound, obtained with the hypothesis of replica symmetry, is close to the one predicted by Mitchinson and Durbin [Biol. Cyber. {\bf 60} 345 (1989)].Comment: 13 pages, 1 figur

Minimum-Uncertainty Angular Wave Packets and Quantized Mean Values

Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates therefore must produce packets with integer angular momentum.Comment: accepted for publication in Physical Review

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

A canonical ensemble approach to graded-response perceptrons

Perceptrons with graded input-output relations and a limited output precision are studied within the Gardner-Derrida canonical ensemble approach. Soft non- negative error measures are introduced allowing for extended retrieval properties. In particular, the performance of these systems for a linear and quadratic error measure, corresponding to the perceptron respectively the adaline learning algorithm, is compared with the performance for a rigid error measure, simply counting the number of errors. Replica-symmetry-breaking effects are evaluated.Comment: 26 pages, 10 ps figure

A Pathwise Ergodic Theorem for Quantum Trajectories

If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings the time averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states.Comment: 8 page