Abstracts QRANDOM Seminar

(started Jan. 2002 at EURANDOM) OVERVIEW TALKS 2002, 2003

2004

W. de Rock,  March 31, 2004

Steady state fluctuations of the dissipated heat in a quantum stochastic model

We introduce a quantum stochastic dynamics for heat conduction. A multi-level subsystem is coupled to reservoirs at different temperatures. Energy quanta are detected in the reservoirs allowing the study of steady state fluctuations of the entropy production. Our main result states a symmetry in its large deviation rate function.

B. Janssens, March 31, 2004

Quantum Measurement

At least in case of a qubit, we'll show collapse of the wave packet on the original system to be a mathematical consequence of information gain and unitary evolution, rather than a condition to be imposed on that evolution. Perhaps also some related topics, if time permits.

Luc Bouten, (University of Nijmegen), February 25, 2004

Squeezing enhanced control

We study an open quantum system in contact with its environment, the electromagnetic field which will be taken to be in a squeezed state.We derive a stochastic differential equation for the state of the open system conditioned on the result of a measurement that is being performed in the field. Then we will apply some action on the open system depending on the measurement result in order to control its state. It will turn out that the squeezing of the field will enhance our control of the system.

Michael Keyl, (TU Braunschweig), February 25, 2004

Quantum state estimation and large deviations

In this talk we investigate the large deviation behaviour of estimation schemes for arbitrary mixed states of finite dimensional quantum systems. In this context we present two main results: Using relations to quantum hypothesis testing, we show that the rate function is always bounded from above by the quantum relative entropy. In addition we consider an explicit scheme which is based on covariant observables and representation theory of unitary groups, and which extends previous results concerning the estimation of the spectrum of the density operator. The corresponding rate function, however, is smaller than the relative entropy, and this leads to the question whether the proposed scheme can be further improved.

Luigi Accardi, (University of Rome II), January 22, 2004

Dissipation, transport and coherence in quantum physics

The stochastic limit is the mathematical theory of dissipation and transport in quantum phenomena. The main results of the stochastic limit are condensed in the stochastic golden rules, which are natural generalizations of the Fermi golden rule. They allow to deduce the relevant physical information for wide classes of systems from relatively simple calculations. Practically all the phenomenological equations used in quantum physics, in particular all the master and Langevin equations, can be deduced in a simple and rigorous way using this technique. The series of lectures is aimed at providing an introduction to the simplest golden rules and at illustrating them with some applications to different physical phenomena such as: SQUID, quantum optics (electromagnetic induced transparency, coherent population trapping, Kerr media, ...), quantum conductivity, quantum interacting particles, ... A detailed programme of the lectures, especially the balance between the new mathematical developments motivated by the stochastic limit and the physical applications and examples, will be tailored according to the interests of the participants.

2003

Bas Janssens, (University of Nijmegen), November 14, 2003

Quantum Theory of Measurement and Macroscopic Observables

The generation of probabilities from probability amplitudes in a quantum mechanical measurement process is discussed in the framework of infinite quantum systems ( i.e. quasi-local c*-algebras ). A reduction of the wave-packet will be viewed as the gradual destruction of coherence in a superposition of two vectorstates. The core of the argument will essentially be the content of an article by Klaus Hepp, "Quantum Theory of Measurement and Macroscopic Observables", Helv. Phys. Acta Vol. 45, 237 (1972).

Dominique Spehner, (University of Essen), October 30, 2003

Quantum trajectories in QED cavities

We study the time evolution of the quantum field inside a cavity coupled to a beam of two-level atoms of temperature T, given that each atom, after having crossed the cavity, interacts with a classical field E and finally with a detector measuring its state. It is found that, if the coupling between the atoms and the quantum field is weak and E is not too small, for any given realization of the measurements, an arbitrary initial state of the field localizes after some time into squeezed states. The centre \alpha of the squeezed state moves randomly in time in the complex plane, but the squeezing amplitude r and the phase \phi show very small fluctuations. Their mean values \bar{r} and \bar{phi} are independent of the random results of the measurements, of the initial state of the atom-field coupling constant \lambda. The time evolution of r and \phi is determined analytically by deriving and solving the quantum state diffusion equation describing the field dynamics in the limit of small \lambda, keeping E finite. It is shown that \bar{r} increases with T, i.e., the squeezing is enhanced by increasing the temperature of the atomic beam.

Manuel Ballester, (University of Utrecht), July 16, 2003

Estimation of unitary quantum operations

In this talk the problem of optimally estimating an unknown unitary quantum operation is treated. A comparison is made between separable and non-separable measurements and it is shown that non-separable measurements are always better at least by a factor of two. It is also shown that if separable measurements are used, classical communication does not improve the estimation. In two dimensions it is also shown that measuring one of the bell states against the other three is as good as any separable measurement and has the advantage of having only two outcomes.

Hans Maassen, (Katholieke Universteit Nijmegen), July 16, 2003

Continuous time limit of repeated measurement

We present a recent result of St\'ephane Attal and Yan Pautrat (Grenoble), who showed how the process of repeated Kraus measurements on a quantum system can be embedded into a tensor product of Fock spaces and the quantum system itself, and how this process converges to the solution of a quantum stochastic differential equation

Hans Maassen, (Katholieke Universteit Nijmegen), April 16, 2003

Madalin Guta (EURANDOM),April 16, 2003

Covariance: CP-maps, POVM's, Instruments... Optimal covariant measurements

Covariance with respect to group actions imposes restrictions on the transformations encountered in quantum physics, channels (CP-maps), CP-semigroups of dissipative evolution, or measurements with their associated positive operator valued measures (POVM) and instruments. I will give a comprehensive review of the properties of such objects, relations among each other and to Mackey's transitive imprimitivity theorem. In the second part I will describe Holevo's results on state estimation for covariant classes of states. It turns out that the "optimal" measurement must be a covariant one and in general it does not correspond to an observable. This clarifies for example the "problem"of non-existence of time observable, and the time-energy uncertainty relation.

Jan-Ake Larsson, (Linköping University), February 6, 2003

Qubits from Number States and Bell Inequalities for Number Measurements.

Bell inequalities for number measurements are derived via the observation that the bits of the number indexing a number state are proper qubits. Violations of these inequalities are obtained from the output state of the Non-degenerate Optical Parametric Amplifier.

2002

Madalin Guta (EURANDOM), December 19, 2002

Sieve maximum likelihood estimation of the density matrix through quantum tomography (II) 

The state of the one-mode light field or quantum harmonic oscillator is derscribed by a density matrix $\rho$, that is a positive operator of trace one on an infinite dimensional separable Hilbert space $H$. A general feature of quantum systems is that they cannot be observed without perturbing their state. If the state is unknown, one can obtain information about it by performing measurements on the system, or rather on a big number of identically prepared systems. The results of the measurement are i.i.d. with probability distribution $p_\rho(x)$. If the map $T: \rho \to p_\rho$ is injective it means that the state is in principle identifiable given the distribution $p_\rho$ by inverting $T$. This is the case with the "quantum tomography" map in the homodyne detection experimental setup. In practice we have to successively estimate the state from the results of the first $n$ measurements . The estimation method which we use is sieve maximum likelihood. The estimator at the n-th step, $\hat{\rho}_n$ is chosen from the density matrices over a finite dimensional subspace of $H$ whose dimension grows with a certain rate $N(n)$. We are interested in the convergence of $\hat{\rho}_n$ to $\rho$ in the appropriate norm on density matrices. For this we use results by van de Geer and Wong & Shen from the theory of non-parametric maximum likelihood estimation, in which a central role is played by the bracketing entropy of the sieves. Moreover one needs to estimate the norm of the inverse of the restriction of the map $T$ to the finite matrices of dimension $N(n)$. The talk should be accessible both to quantum physicists and statisticians.

Richard Gill (EURANDOM, Universiteit Utrecht), December 19, 2002

Optimal design of the experiment of the century

Consider four binary random variables X_1, X_2, Y_1, Y_2 such that one can only collect data from the joint distribution of each of the pairs (X_i,Y_j), i=1,2, j=1,2, but never from the complete four-tuple. There is an experimental set-up in physics where this picture is applicable, where moreover two competing theories say something very different about the possible probability distributions of the data: according to classical physics (C), the four bivariate distributions from which data can be collected, are bivariate marginals of a single four-variate distribution, while according to quantum mechanics (Q), this need not necessarily be the case. Now there are a lot of variations possible in an experimental test of Q's claims. The literature contains competing claims that one or another experimental design is the best one, usually based on rather vague criteria. I will argue that an objective way to compare different possible experiments is in terms of the weight of statistical evidence provided, per observation, for Q versus the competing classical theories C. I will explain how this weight of evidence can be quantified using the Kullback-Leibler divergence between the two competing probability models for the data [the mean value under Q of the log-likelihood ratio of Q versus C]. Next I will show how finding the optimal experimental design corresponds to finding a saddle-point in a certain two-person zero-sum game, played between the quantum experimentalist Q and the classical theorist C, in which Q chooses various features of experimental design while C tries to come up with a theory which explains the resulting data. Some of the games we will consider turn out to have solutions (saddlepoints) and hence values, which lead to an objective comparison between various proposals. This is joint-work-in-progress with Wim van Dam (Berkeley) and Peter Grunwald (CWI). Some of our results are numerical and are obtained using programs developed by Piet Groeneboom for solving nonparametric missing data problems - the connection will be explained.

Madalin Guta (EURANDOM), September 13, 2002

Non-commutative convergence theorems

I will discuss quantum analogues of classical convergence theorems such as the dominated convergence, using the theory of trace ideals of Calkin. Subsequently, I will  present a number of distances on the space of quantum states together with their relations with classical distances on probability distibutions. Such results are an important ingredient in the proof of the consistency of maximum likelihood estimator for quantum tomography which will be presented in a future talk.

Luc Bouten (University Nijmegen), September 13, 2002

The Belavkin equation I

This is the first talk in a series of two or three talks on the work of V.P. Belavkin in quantum filtering theory. First a short introduction to quantum stochastic calculus for Bose noise will be given. Then we will investigate the problem of quantum stochastic nonlinear filtering with respect to nondemolition, continuous measurement. Goal of the talks is to discuss the stochastic calculus of a posteriori conditional expectations in quantum observed systems and more specific to derive a general quantum filtering stochastic equation. In the second or third talk applications to quantum optics will be discussed.

Wim van Dam (University of California), August 23, 2002

Statistical strenght of non-locality proofs

Professor Viacheslav P Belavkin (Nottingham University, UK), June 21, 2002

Cramer-Rao type inequalities for quantum channels

In this talk I will give a brief overview of the quantum Cramer-Rao and van Trees inequalities introduced by Helstrom and Gill for estimation of the efficiency of quantum measurements, and will outline their generalizations for truly quantum channels.

Luc Bouten (Nijmegen University), March 8, 2002

A Stochastic Schrödinger Equation

Last update 23-02-09

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