Hans Maassen

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• Prof.
• Professor Emeritus at Radboud University

The no-broadcasting theorem in quantum information says that a set of states on a quantum system admits a common broadcasting (copying) operation if and only if their density matrices belong to a commuting family. We discuss and prove this theorem, as well as the closely related “no-cloning theorem” in the context of quantum probability theory, i.e...

In a series of papers in the 1980s Alexander Holevo proved a classification theorem for continuous quantum measurement processes, or, as they would today be called, stationary quantum trajectories in continuous time. His main tools were functional analytic in character: starting from a Bochner-type inequality he employed dilation techniques for pos...

A stochastic calculus based on integral kernels is developed for the Wiener process. The application of integral kernels to other types of noise is lndicated.

We translate inequalities and conjectures for immanants and generalized matrix functions into inequalities in the L\"owner order. These have the form of trace polynomials and generalize the inequalities from [FH, J. Math. Phys. 62 (2021), 2, 022203].

Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal...

Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal...

The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be non-optimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work we establish optimal uncertainty relations by characterising the...

We consider a finite but otherwise general measurement on a finite quantum system, repeated infinitely often. We prove that observation of the asymptotic or `macroscopic' behavior of the measurement record amounts to a von Neumann measurement on the system. In the course of time the type of asymptotic behavior can be viewed as establishing itself,...

On this bicentenary of George Boole of Cork, we consider the role of Boolean logic in physics, and contrast it with the quantum logic proposed by Birkhoff and von Neumann in 1936. Although the latter work is not widely known, it could arguably be considered as a major breakthrough in our thinking about the material world. Boolean thinking needs rep...

From its very birth in the 1920s, quantum theory has been characterized by a certain strangeness: It seems to run counter to the intuitions that we humans have about the world we live in. According to these “realistic” intuitions all things have their definite place and sharply determined qualities, such as speed, color, and weight. Quantum theory...

In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by “coding” the state space of the system before transmission into a “protected” part of a larger state space, and by applying a proper “decoding” map afterwards. By a version of the Heisenberg Principle, wh...

We consider the question of convergence of particular series of integrals, which are labeled by rooted trees. Necessary and sufficient criteria for convergence are obtained, together with an explicit expression for the sum. The technique used is strongly reminiscent of the generating function approach of Galton and Watson to branching processes. Th...

In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by 'coding' the state space of the system before transmission into a 'protected' part of a larger state space, and by applying a proper 'decoding' map afterwards. We investigate two kinds of protected subspa...

A stochastic calculus based on integral kernels is developed for the Wiener process. The application of integral kernels to other types of noise is indicated.

A brief description is given of dilation theory in connection with systems theory.

A description is introduced of operators on Fock space by way of integral kernels. In terms of these kernels, the quantum stochastic differential equation for a Markov process over the n×n matrices can be explicitly solved. As an example the Wigner-Weisskopf atom is treated.

Without Abstract

We attempt to clarify certain puzzles concerning state collapse and decoherence. In open quantum systems decoherence is shown to be a necessary consequence of the transfer of information to the outside; we prove an upper bound for the amount of coherence which can survive such a transfer. We claim that in large closed systems decoherence has never...

We prove that the quantum trajectory of repeated perfect measurement on a finite quantum system either asymptotically purifies, or hits upon a family of 'dark' subspaces, where the time evolution is unitary.

We prove that the quantum trajectory of repeated perfect measurement on a finite quantum system either asymptotically purifies, or hits upon a family of `dark' subspaces, where the time evolution is unitary.

In this introductory course we sketch the framework of quantum probability in order to discuss open quantum systems, in particular the damped harmonic oscillator.

Laser-based optical diagnostics, such as planar laser-induced fluorescence and, especially, Raman imaging, often require selective spectral filtering. We advocate the use of an imaging spectrograph with a broad entrance slit as a spectral filter for two-dimensional imaging. A spectrograph in this mode of operation produces output that is a convolut...

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 state...

A derivation of Belavkin's stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system's quantum state gi...

Nuclear Overhauser effect (NOE) data are an indispensable source of structural information in biomolecular structure determination by NMR spectroscopy. The number and type of experimental restraints used in the structure calculation and the RMS deviation of the restraints are usually reported. We present a new method for quantifying the information...

We consider a quantum version of a well-known statistical decision problem, whose solution is, at first sight, counter-intuitive to many. In the quantum version a continuum of possible choices (rather than a finite set) has to be considered. It can be phrased as a two person game between a player P and a quiz master Q. Then P always has a strategy...

Starting point is a given semigroup of completely positive maps on the 2 times 2 matrices. This semigroup describes the irreversible evolution of a decaying 2-level atom. Using the integral-sum kernel approach to quantum stochastic calculus we couple the 2-level atom to an environment, which in our case will be interpreted as the electromagnetic fi...

A new approach to the generalised Brownian motion introduced by M.

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species F gives rise to an endofunctor F of...

For a quantum-mechanical counting process we show ergodicity, under the condition that the underlying open quantum system approaches equilibrium in the time mean. This implies equality of time average and ensemble average for correlation functions of the detection current to all orders and with probability 1.

A new approach to the generalised Brownian motion introduced by M. Bożejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor V of the category of Hilbert spaces with contractions....

A new approach to the generalised Brownian motion introduced by M. Bozejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor F_V of the category of Hilbert spaces with contraction...

The preparation of an arbitrary target state of the quantized single model radiation field under a simple, time-independent atom-field interaction, without any final state projection, and independently of the initial state of the field is demonstrated. It is shown that, given finite number N of atoms, the state preparation is possible with very hig...

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' $\K$ are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species $F$ gives rise to an endofunctor...

Contents 1. The Framework of Quantum Probability 1.1. Making probability non-commutative 1.2. Events and random variables 1.3. Interpretation of quantum probability 1.4. The quantum coin toss: `spin' 1.5. Positive denite kernels 2. Some Quantum Mechanics 2.1. Position and momentum 2.2. Energy and time evolution 2.3. The harmonic oscillator 2.4. The...

We present an algorithm for generating a random weak order of m objects in which all possible weak orders are equally likely. The form of the algorithm suggests analytic expressions for the probability of a Condorcet winner both for linear and for weak preference orders. 1.

B: !(P i ) (or !(Q i )) is the probability to nd the value i (or i ) when measuring the observable A (or B). One now denes the uncertainty H(A;!) of A in the state ! as the Shannon entropy of this probability distribution: H(A;!) = n X i=1 !(P i ) log !(P i ): The question was raised ([BBM], [Deu], [Kra]), what can be said about H(A;!) +H(B;!), mor...

The q-deformed commutation relation aa # - qa # a = 11 for the harmonic oscillator is considered with q # [-1, 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a # in the vacuum state is explicitly calculated. This distri...

We give an explicit proof of the pair partitions formula for the moments of the q-harmonic oscillator, and of the claim made by G. Parisi that the q-deformed lattice Laplacian on the d-dimensional lattice tends to the q-harmonic oscillator in distribution for d !1. 1

: Stability under small nonlinear perturbations is proved for a class of linear quantum dynamical systems, including the harmonic oscillator coupled to a free Bose eld and the innite harmonic chrystal. The main tool is an estimate of Dyson's time-dependent perturbation series, based on a labeling of its terms by rooted trees. 1. Introduction. In re...

This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is devel...

The q-deformed commutation relation aa Gamma qa a = 11 for the harmonic oscillator is considered with q 2 [Gamma1; 1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a + a in the vacuum state is explicitly calculated. This dist...

We consider two independent q-Gaussian random variables X 0 and X 1 and a function fl chosen in such a way that fl(X 0 ) and X 0 have the same distribution. For q 2 (0; 1) we find that at least the fourth moments of X 0 +X 1 and fl(X 0 )+X 1 are different. We conclude that no q-deformed convolution product can exist for functions of independent q-G...

This paper is organised as follows. Sections 1 and 2 give the necessary background. Sections 3 and 4 develop criteria for the existence of scattering operators. They are applied to various situations in Sections 5, 6 and 7. x1. Probability spaces and stochastic processes. By a non-commutative probability space we shall mean a pair (A; ') consisting...

this article during a stay at the "Mathematisches Forschungsinstitut Oberwolfach", financed also from the European Human Capital and Mobility Program on Quantum Probability. finite coin tosses in the third section. Suprisingly, they lead quite quickly to recent developments. In an appendix we indicate the steps which lead to the full mathematical m...

A relativistic oscillator whose period is independent of its energy is of great fundamental importance in both relativistic classical mechanics and relativistic quantum mechanics. In this work theoretical and computational investigations of such a constant period oscillator are reported, with emphasis on basic mathematical and physical properties o...

A Fock space representation is given for the quantum Lorentz gas, i.e., for random Schro¨dinger operators of the form H(ω)=p2+Vω=p2+∑ &Jgr;(x−xj(ω)), acting in H=L2(Rd), with Poisson distributed xjs. An operator H is defined in K=H⊗P=H⊗L2(Ω,P(dω))=L2(Ω,P(dω);H) by the action of H(ω) on its fibers in a direct integral decomposition. The stationarity...

A direct proof is given of Voiculescu's addition theorem for freely independent real-valued random variables, using resolvents of self-adjoint operators. In contrast to the original proof, no assumption is made on the existence of moments above the second.

Using (non-adapted) quantum stochastic calculus as our main tool we calculate the dynamical Stark effect, i.e., the splitting of the flourescent spectrum of a two-level atom in a strong laser beam. We introduce the scattering operator for this system, and establish a well-known relation between the scattered spectrum and the autocorrelation functio...

The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebraM n with stationary faithful state π. If (ℳ, μ) is the quantum measure space in question (ℳ a von Neumann algebra and μ a faithful normal weight), then the semigroupe tL of transitio...

A new class of uncertainty relations is derived for pairs of observables in a finite-dimensional Hilbert space which do not have any common eigenvector. This class contains an ``entropic'' uncertainty relation which improves a previous result of Deutsch and confirms a recent conjecture by Kraus. Some comments are made on the extension of these rela...

An introduction is given to some fundamental concepts in quantum probability, such as (quantum) probability spaces and (quantum) stochastic processes. Recent results are described relating to the question, what transition probabilities for an n-level quantum system are theoretically possible in a quantum Markov process.

For identity and trace preserving one-parameter semigroups {T t} t Tt = \smallint Aut( Mn ) adrt ( a)T_t = \smallint _{Aut\left( {M_n } \right)} \alpha d\rho _t \left( \alpha \right) , and this condition is then characterised in terms of the generator ofT t. There is a one-to-one correspondence between essentially commutative Markov dilations, wea...

Without Abstract

A brief introduction is given to Markov dilations, i.e. quantum Markov processes as constructed from their transition probability semigroups. The dilation is constructed of the two-level atom, decaying to its ground state, under the assumption of cononical commutation relations for the outside world.

A quantum-mechanical treatment of the evolution of an anharmonic oscillator coupled to a heat bath is given. It is shown that for a certain class of anharmonic potentials the heat bath drives the oscillator to an equilibrium state, close to the quantum Gibbs state associated to the potential. Thus a partial proof is provided for a conjecture of R....

Local perturbations of the dynamics of infinite quantum systems are considered. It is known that, if the Møller morphisms associated to the dynamics and its perturbation are invertible, the perturbed evolution is isomorphic to the unperturbed one, and thereby shares its ergodic properties. It was claimed by V. Ya. Golodets [Theor. Math. Phys. 23, 5...

It is shown that a quantum harmonic oscillator, coupled to a massless scalar field in three dimensions, satisfies a quantum Langevin equation in the point charge limit.