Quantum Probability and Infinite Dimensional Analysis

FOREWORD

CONTENTS

Assuming that a probability measure on ℝ^d has finite moments of any order, its moments are completely determined by two family of operators. The first family is composed of the neutral (preservation) operators. The second family consists of the commutators between the annihilation and creation operators. As a confirmation of this fact, a characterization of the Gaussian probability measures in terms of these two families of operators is given. The proof of this characterization relies on a simple combinatorial identity.

A Feynman formula is a representation of a solution of Cauchy problem for an evolution partial differential (or pseudodifferential) equation by a limit of some Gaussian or complex Gaussian finite dimensional integrals when their multiplicity tends to infinity. Some Feynman formulas for solutions of heat and Schrödinger type equations with Levy Laplacians on the infinite dimensional manifold which is the set of mappings of a real segment into a Riemannian manifold are obtained.

We describe the “no-go” theorems recently obtained by Accardi-Boukas-Franz in [1] for the Boson case, and by Accardi-Boukas in [2] for the q-deformed case, on the issue of the existence of a common Fock space representation of the renormalized powers of quantum white noise (RPWN).

Our attempts to establish a Fock representation for the renormalized higher powers of white noise, involve the assignement of a meaning to the powers of the Dirac delta function. In this paper we give meaning to the expression where n ≥ 2, δ^(k) is the k-th derivative of the Dirac delta function and are arbitrary.

We apply new techniques based on the distributional theory of Fourier transforms to study, in the stochastic limit of quantum theory, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schrödinger equation obtained in such limit. This approach permits us to dispense with the “analytical condition” and other restrictions usually considered and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field.

In this paper, we give a new criterion for positivity of generalized functions and positive operators on test functions space of entire function on the dual of a nuclear space N′ and with θ order growth condition, denoted by . This definitions are used to prove that every positive operator has integral representation given by positive Radon measure and these measures are characterized by integrability conditions. This new criterion of positivity can be easily applied to several examples.

In the paper the Bayesian and the least squares methods of quantum state tomography are compared for a single qubit. The quality of the estimates are compared by computer simulation when the true state is either mixed or pure. The fidelity and the Hilbert-Schmidt distance are used to quantify the error. It was found that in the regime of low measurement number the Bayesian method outperforms the least squares estimation. Both methods are quite sensitive to the degree of mixedness of the state to be estimated, that is, their performance can be quite bad near pure states.

Some bounds on the entropic informational quantities related to a quantum continual measurement are obtained and the time dependencies of these quantities are studied.

In this paper a new concept of q-symmetric tensor product is defined, where q is a parameter in the interval (−1,1]. Duality theorems are established for new spaces called generalized q–Fock spaces and some of their features are indicated.

A characterisation is made of not necessarily bounded quantum stochastic flows with bounded generator matrices in the general algebraic setting of von Neumann and Itô algebras. This characterisation is then used to construct dilations of filtering and contraction flows which are homomorphic and reduce to the original flow, giving stochastic generalisations of the Stinespring and Evans–Hudson dilations. Flows covariant with respect to a group action are also studied, along with their generators and covariant dilations.

The isomorphism between quantum semimartingale algebras is used to explain the relationship that exists between the evolution equation of Alicki and Fannes and that of Hudson and Parthasarathy.

The aim of this work is to prove that the second quantization of a solution of the Boolean quantum stochastic differential equation (B-QSDE) is solution of Hudson-Parthasarathy quantum stochastic differential equation (HP-QSDE).

In this paper several results obtained for some evolutionary equations on a compact Riemannian manifold are presented. In particular, representations of the solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a manifold are obtained in the form of limits of finite-dimensional integrals. These limits coincide with integrals over Smolyanov surface measures on the set of trajectories in a manifold and over the Wiener measure, generated by Brownian motion in a domain with absorption on the boundary. Integrands are combinations of elementary functions of coefficients of the equation and geometric characteristics of the manifold. Also representations of the solution of the Cauchy problem for the Schroedinger equation on a compact Riemannian manifold are obtained in the form of functional integrals over Smolyanov surface measures. In the proof a substantial role is played by Smolyanov-Weizsaecker-Wittich asymptotic estimates for Gaussian integrals over a manifold, by the Chernoff theorem and by the method of transition from the Schroedinger to the heat equation going back to Doss.

We use the formalism of quantum mechanics in the framework of the Bohmian (pilot wave) model to describe stochastisity of the financial market. We interpret nonclassical contribution to stochasticity as a psycho-financial (information) field ψ(q) describing expectations of agents of the financial market.

We present a new proof of the central limit theorem performed in [2] for symmetric measures based on a different approach.

We address the problem of finding the optimal joint unitary transformation on system + ancilla which is the most efficient in programming any desired channel on the system by changing the state of the ancilla. We present a solution to the problem for dim(H) = 2 for both system and ancilla.

In this paper we derive the state equation of the optical cavity in interacting Fock space as well as in boson Fock space. We design closed-loop feedback control system of a composite cavity QED in boson Fock space using beam splitter device and prove with the help of Nyquist stability criterion that the system is stable. The other physical characteristics, such as, phase margin and gain margin of the closed-loop feedback control system are also discussed.

We review the definition of Markov states on quasi–local algebras, their structure and the main properties on known models.

In this paper we recall some open problems in Information Geometry.

Determinantal point processes on a measure space (χ, Σ, μ) whose kernels represent trace class Hermitian operators on L²(χ) are associated to “quasifree” density operators on the Fock space over L²(χ).

There are many ways of understanding the time operator. In this report the time operator will be discussed from the view point of stochastic analysiss. First, a stationary stochastic process is taken to be a representation of the ordinary time t, which is an abelian group and is linearly ordered set. Together with the one parameter group of the time shift we can find the well known commutation relations in terms of generators. Second, we observe a semi-group that defines the propagation of a diffusion process. Its generator satisfies another interesting commutation relations with the differential operators acting on the space of functions on the configuration space. There one can see the transversal relations among the generators, so thatthe theory of dynamical systems can be applied.

The study of ageing phenomena leads to the investigation of a maximal parabolic subalgebra of 𝔠𝔬𝔫𝔣₃ which we call 𝔞𝔩𝔱. We investigate its Lie structure, prove some results concerning its representations and characterize the related Appell systems.

The double product integral is constructed as a family of unitary operators in double Fock space satisfying quantum stochastic differential equations with unbounded operator coefficients.

Ohya and Volovich have proposed a new quantum computation model with chaotic amplification to solve the SAT problem, which went beyond usual quantum algorithm. In this paper, we generalize quantum Turing machine by rewriting usual quantum Turing machine in terms of channel transformation. Moreover, we define some computational classes of generalized quantum Turing machine and show that we can treat the Ohya-Volovich (OV) SAT algorithm.

We study dynamics in field models appearing in a level-truncation scheme for non-commutatively interacting string field theory. Distinguishing property of such models is that the corresponding equations of motion contain infinite number of derivatives. We study existence of physically interesting solutions of these equations in special approximation for interacting open-closed string model. We also present a general relation for stress tensor as well as energy conservation law for the case of arbitrary (finite) number of levels. Recent applications of such models include cosmological inflation and dark energy problems.

Logarithmic Sobolev inequalities are an essential tool in the study of interacting particle systems, cf. e.g. [4, 5]. In this note we show that the logarithmic Sobolev inequality proved on the configuration space ℕ^ℤd under Poisson reference measures in [1] can be extended to geometric reference measures using the results of [2]. As a corollary we obtain a deviation estimate for an interacting particle system.

We propose a solution of the problem of hidden variables. In our approach QM can be reconstructed as an asymptotic projection of statistical mechanics of classical fields. Determinism can be reestablished in QM, but the price is the infinite dimension of the phase space. Our classical→ quantum projection is based on Gaussian integration on the Hilbert space and the Taylor expansion (up to the second order term) of functionals of classical fields. Our solution of the problem of hidden variables is given in the framework which differs essentially from the conventional one (cf. Einstein-Polosky-Rosen, von Neumann, Kochen-Specker, …, Bell). The crucial point is that quantum mechanics is just an asymptotic image of prequantum classical statistical field theory (PCSFT).

We study quantization in the hyperbolic Hilbert space: harmonic oscillator and electromagnetic field.

We present the classical-type dynamical model which poses some distinguishing properties of quantum mechanics. The model is based on a conjecture of existence of fundamental indivisible time quantum. The resulting formalism – discrete time dynamics – is constructed and used to study the energy spectrum for motion in central potential. We show that the spectrum is discrete and present a method of reconstruction of the corresponding potential for which the discrete time dynamics leads to the experimentally observed spectra.

In this paper we consider the deformation of conditionally free convolution, connected with the free cosh–law. Because that measure is freely infinitely divisible, we can define a new, associative convolution using the theory from [5]. We calculate the central and Poisson measure for that convolution and show that the coefficients of the continued fraction form of the Cauchy transform for the central and Poisson limit measures of that convolution are equal to the respective coefficients of the underlying measure starting from the third level.

In this note we investigate the problem of calculating the state of a system if the state of a subsystem is given.

A map φ : M_m(ℂ) → M_n(ℂ) is decomposable if it is of the form φ = φ₁ + φ₂ where φ₁ is a CP map while φ₂ is a co-CP map. A partial characterization of decomposability for maps φ : M₂(ℂ) → M₃(ℂ) is given.

This paper is a summary of my talk on 26th conference of QP and IDA in Levico (Trento), February, 2005 with some supplemental explanation. Thermodynamical formulations such as characterizations of equilibrium states and spontaneous symmetry breaking are given for general quasi-local systems. We show that the unbroken symmetry of fermion grading (the univalence super-selection rule) follows essentially only from the local structure that satisfies the graded commutation relations.

No abstract received.

In this paper we shall discuss the Lévy Laplacian as an operator acting on some class of the Lévy functionals. The Laplacian acts on Gaussian and Poisson functionals in the class as a scalar operator. Based on this result, we introduce some domain of the Laplacian on which is a self-adjoint operator. We also discuss associated semigroups and associated stochastic processes.

An important tool for integral transformation in symmetric Fock space is the so-called integral-sigma or -lemma. It exists in several versions depending on the description of the random point configurations [4, 17, 15, 20]. Usually, one has to assume that the point configurations have no multiple points.

The goal of this paper is to present a detailed proof of a very general form of it including configurations with multiple points.

No abstract received.

We shall compare Gaussian (white) and Poisson noises in order to characterize the latter. First, their characteristics are discussed, then we show that approximations of the two noises indicate significant difference between them from the viewpoints of invariance under transformation group and optimality of information. There one can recongnize dissimilarity in addition to similarity. The dissimilarity is more significant. In fact, there are many different properties between the two noises in parallel, so that we can see the contrast which is most interesting for us. Then, we shall come to a characterization of the so-called fractional power distribution with the help of the significant properties of Poisson noise. It is our hope that special methods of approximation to the two noises would further reflect a duality between them. One is related to the rotation group which is continuous, while the other is to the symmetry group which is discrete.

I present a review of two conjectures, each one about an inequality in the Segal-Bargmann space in the case when the dimension of the underlying phase space is finite. One concerns a reverse log-Sobolev inequality, and the other concerns a Hirschman inequality. If either conjecture is true, it allows us to prove the corresponding inequality in the case of infinite dimensional phase space.

The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983. In this paper, we compare with mutual entropy-type measures and show some resuls for quantum capacity.